Probabilistic Algorithms for Geometric Elimination

AAECC 9, 463–520 (1999)

Probabilistic Algorithms for Geometric Elimination ∗

Guillermo Matera 1,2

1 Laboratorio de Computacion, Universidad Favaloro, Solıs 453 (1078) Buenos Aires, Argentina2 Instituto de Desarrollo Humano, Universidad Nacional de General Sarmiento, Roca 850 (1663)San Miguel, Argentina (e-mail: [email protected])

Abstract. We develop probabilistic algorithms that solve problems of geomet-ric elimination theory using small memory resources. These algorithms are ob-tained by means of the adaptation of a general transformation due to A. Borodinwhich convertsuniformboolean circuit depth into sequential (Turing machine)space. The boolean circuits themselves are developed using techniques basedon the computation of a primitive element of a suitable zero-dimensional alge-bra and diophantine considerations.

Our algorithms improve considerably the space requirements of the elim-ination algorithms based on rewriting techniques (Grobner solving), havingsimultaneously a time performance of the same kind of them.

Keywords: Probabilistic algorithms, Elimination theory, Boolean circuits,Arithmetic circuits

Table of Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464

2 On Arithmetic and Boolean Circuits. . . . . . . . . . . . . . . . . . . . . . . . . . 4672.1 Uniform linear algebra algorithms. . . . . . . . . . . . . . . . . . . . . . . . 4702.2 Manipulation of univariate polynomials. . . . . . . . . . . . . . . . . . . . . 4772.3 Manipulation of arithmetic circuits. . . . . . . . . . . . . . . . . . . . . . . . 479

3 The Resolution of Polynomial Equation Systems:Primitive Element Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4823.1 The construction of a primitive element. . . . . . . . . . . . . . . . . . . . . 4843.2 The computation of the isolated points of a variety. . . . . . . . . . . . . . . . 489

∗ Work partially supported by the following Argentinian grants: UBACYT TW80, PIP CONICET4571, ANPCYT PICT 03-00000-01593.

464 G. Matera

4 The Division Modulo a Reduced Complete Intersection Ideal. . . . . . . . . . . . . 4944.1 General trace theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4954.2 The computation of the matrix that performs the division. . . . . . . . . . . . 497

5 Applications to Geometric Elimination Problems. . . . . . . . . . . . . . . . . . . 5015.1 The consistency of polynomial equation systems

and the effective Nullstellensatz. . . . . . . . . . . . . . . . . . . . . . . . . 5015.2 The membership and representation problems in the case

of complete intersection ideals. . . . . . . . . . . . . . . . . . . . . . . . . . 5095.3 An effective version of Quillen–Suslin Theorem. . . . . . . . . . . . . . . . . 511

1 Introduction

We use standard notions and notations for boolean complexity models andboolean complexity classes as can be found in [4], [3] or [63]. We recall thatthe classes NCi are defined as the set ofO(logi n)-uniform families of booleancircuits of polynomial size and depthO(logi n) with bounded fan-in. A familyof boolean circuit is calledS(n)-space uniformif its standard encoding can bebuilt using deterministic spaceS(n).

We will also use the model ofdivision-free arithmetic circuits(see e.g.[22], [23] or [1]). LetR be a ring. Anarithmetic circuitβ overR is a directedacyclic graph (dag for short)C(β), where all nodes have bounded indegree ofeither 0 or 2. The nodes of indegree 0 representinginput nodesare labeledby indeterminatesX1, . . . , Xn, the nodes of indegree 0 representingparameternodesare labeled by elements ofR and the nodes of indegree 2 (calledinternalnodes) are labeled by arithmetic operations. The elements ofR occurring inparameter node are calledparametersof the arithmetic circuitβ. Finally, nodeswith out-degree 0 are calledoutput nodes.

An arithmetic circuitβ is calleddivision-freeif it contains no divisions. Wedefine thesizeof a division-free arithmetic circuitβ as the number of internalnodes occurring inβ. The depth ofβ is defined as the length of the longest pathwhich joins an input node with an output node. A nodeρ of β is nonscalarif ithas the following property: the nodeρ has indegree 2, the arithmetic operationof ρ is a multiplication and predecessor nodes ofρ are not parameter nodes.Thenonscalar depthof β is defined as the length of the longest path joining aninput node with an output node when only nonscalar nodes are considered.

In the sequel we are going to use a general argument due to A. Borodin[7] (see also [21]), which allows to turn uniformbooleancircuit depth (paralleltime) intosequentialspace. This argument allows us to re-interpret the propertyof well parallelizability of a boolean function as the use of a very small amountof working space in the scope of sequential computing. Moreover, Borodin’sargument is constructive, so that given a family of boolean circuits in NCi andan algorithm which witnesses its uniformity, we are going to use this argumentfor the transformation of the given data into an equivalent sequential algorithmwhich works in spaceO(logi n).

Probabilistic Algorithms for Geometric Elimination 465

Our intention is to adapt this idea to the context ofgeometriceliminationtheory. The first systematic attempts in this direction were made in [53] and[52] (see also [54] and [47] in the context ofalgebraicelimination theory),wheredeterministicalgorithms were designed for the resolution of selectedproblems of geometric elimination. These algorithms, although very efficientwith respect to the management of memory space, have the drawback of a poortime performance. This unsuitable behaviour with respect to running time makesprohibitive the practical applicability of these algorithms. In order to speed upthe computations we are going to introduce randomness in our procedures.

The central idea of the method of [53] consists in the reduction of elimina-tion problems to linear algebra computations. These reductions rely on suitableversions of effective Nullstellensatze (cf. [35] or [58]). Unfortunately, the ma-trices occurring in these reductions have exponential size, mainly due to thesyntactical aspect of the kind of problems under consideration. By this syn-tactical aspect we refer to the codification of multivariate polynomials whichrepresent the basic objects of the language.

Taking into account that ann-variate polynomial of degreed has

(d + n

n

)distinct monomials (this quantity can be estimated asO(dn)), we see that theusual dense or sparse codification of polynomials (i.e. the representation ofpolynomials by means of the vector of all or all nonzero coefficients) implies anexponential behaviour of such a data structure. In order to avoid this exponentialbehaviour we are going to introduce an alternative codification of intermediateand final results by means ofarithmetic circuits. The fact that the polynomialswhich occur in our procedures are “special” in some sense will allow us to takeadvantage of this codification.

The codification of polynomials by means of arithmetic circuits has thedisadvantage of its nonuniqueness. We useprobabilistictests for the verificationof identities of polynomials given by arithmetic circuits. Such probabilistic testsconsist in the evaluation of the two polynomials under consideration in somepoints chosen at random in a suitable set. Throughout this work we are goingto consider two alternative test methods: the Schwartz–Zippel test (see e.g.[37], Corollary 2.1; see also [60], [64], [41], [38]) and the Heintz–Schnorr test(see e.g.[36], Theorem 4.4; [45], Corollary 19 or [58]).

An important point of our algorithmic method is the representation of alge-braic varieties by means of primitive element techniques. The idea – which isoriginally due to Kronecker [46] and has been applied in several papers suchas [2], [11], [13], [14], [24], [26], [28], [29], [30], [43], [45], [49], [51], [56]– consists in finding a “pseudo-parameterization” of the variety under consid-eration which, at the same time, provides valuable algebraic information. Thispseudo-parameterization is given by a curve parameterized by a primitive ele-mentu (which is a linear form thatseparatespoints) together with a polynomialcondition onu in order to determine which of the points of this curve belongto the variety under consideration.

466 G. Matera

The information we obtain from such a representation is applied system-atically in order to solve the following geometric elimination problems: thedecision of the consistency of a given system of polynomial equations, themembership problem for complete intersection ideals, the computation of aBezout identity in the case of inconsistent polynomial equation systems, the re-duction of a given polynomial modulo a given complete intersection ideal, andthe determination of the dimension of a given algebraic variety. As a furtherapplication we obtain an algorithmic version of Quillen–Suslin theorem thatrequires – for the first time – only polynomial space.

Let us also mention that we apply a particular method of size reduction tothe matrices occurring during our procedures. This method of reduction relieson arguments on the regularity of the Hilbert function of a suitable graduatering as in [51], [9] or [28] (see also [61]) or the division modulo a completeintersection ideal by means of a trace formula as in [20] or [45] (see also [2],[5], [12] and [59]).

Applying all these tools we build arithmetic circuits which have certainnodes that are generated randomly. Then we translate these arithmetic circuitsinto boolean circuits following a general method which allows an efficient trans-lation and show the uniformity of the families of boolean circuits obtained inthis way. For this purpose we prove a “Macro expansion lemma” which allowsus to simplify considerably the discussion on the uniformity of the familiesof boolean circuits we obtain. Finally we apply Borodin’s argument in orderto get sequential algorithms with small space requirements and improved timeperformance with respect to those of [53]. The random integers we use in ourprocedures are generated at the beginning of the execution of the algorithm andfully stored in memory space.

We describe our algorithms in the model of bounded error probabilisticTuring machines (see e.g. [4]). They work in spaceO

(n4 log2(shd)

)and in

time (shd)O(n2 log(shd)), wheres in the number of input polynomials,n is thenumber of variables they contain andd andh are upper bounds for their degreeand logarithmic height respectively (with logarithmic height we refer to themaximal binary size of the coefficients of the polynomials involved). Thesebounds improve considerably the (simultaneous)O

(n4 log2(shd)

)space and

(shd)O(n4 log(shd)) time bounds of [53], and the(shd)O(n2) space and(shd)O(n2)

time bounds of the algorithms based on Grobner basis computations (see [25],[44], [17] and [55]).

Let us mention in this context that the probabilistic(hsdn)O(1) spaceand (hsdn)O(1) time bounds of the algorithms of [45] and the probabilistic(nhδη)O(logn) space and(nhδη)O(logn) time bounds of the algorithms of [29],[26], [57] and [31] (hereδ andη are geometric parameters that can be estimated,in worst case, bydn andhdn respectively) have a better time performance thanour procedures but require much more space than ours.


Let us also mention theO(n2 log2(hδηd)

)space and(hηδd)

O

(n2 log(hδηd)

)time bounds of [52] that, although worse than ours in worst case, may be ofpractical interest for special well suited input polynomial equation systems.

2 On Arithmetic and Boolean Circuits

The elimination problems we are considering here can be naturally describedby division-free arithmetic circuits. Borodin’s argument demands a translationprocess of these arithmetic circuits into their boolean counterparts, once a boundfor the size of the integers occurring as inputs in the arithmetic circuits has beenfixed.

The control of the logarithmic height of the integers occurring in our arith-metic circuits is unavoidable for the sake of an efficient translation. In orderto perform this control we are going to pay special attention to thenonscalardepthof our arithmetic circuits, that is, the depth of essential multiplications.As shown in [45], there is a close relation between the nonscalar depth andthe logarithmic height of the output, while restricting the arithmetic circuits tointeger inputs and constants.

The translation process from arithmetic circuits to boolean circuits can bedescribed roughly as a “macro expansion” procedure, in which every arithmeticoperation is replaced by a boolean circuit that performs the arithmetic operationon integer numbers of prescribed logarithmic height. In order to achieve effi-ciency in this process we implement the addition of several integers by meansof carry save addercircuits (see e.g. [63]).

Another important point is the uniformity of the families of boolean circuitsunder consideration. This condition is unavoidable to keep the requirements ofmemory space within reasonable limits. In the sequel we are going to discuss theuniformity of the arithmetic circuits we develop rather than the uniformity of theboolean circuits we obtain after the translation. As we will show in Lemma 2 ourtranslation procedure maps uniform families of arithmetic circuits into uniformfamilies of boolean circuits.

Furthermore, the discussion of the uniformity of our families of arithmeticcircuits can be simplified further by allowing the introduction of “arithmeticmacros”, whose uniformity is known. For this purpose we have to deal withcircuits overZ with unbounded fan-in and unbounded fan-out, which we aregoing to callgeneralized arithmetic circuit.

Definition 1 A generalized arithmetic circuitβ over Z is a directed acyclicgraph (dag for short) C(β) whose nodes are labeled. The nodes ofC(β) ofindegree 0 are labeled either by indeterminatesX1, . . . , Xn (the input nodes)or by elements ofZ (theparameter nodes). The remaining nodes are labeled by

468 G. Matera

a fixed set of polynomial maps with integer coefficientsΦ1, . . . , Φr . The nodeswith out-degree0 are calledoutput nodes.

For 1 ≤ i ≤ r we define thesizeof β in Φi as the number of nodes labeledby Φi occurring inβ. The depth ofβ in Φi is defined as the highest numberof occurrences of nodes labeled byΦi in a path joining an input node with anoutput node.

We assume that the nodes of a generalized arithmetic circuit withn inputnodes are numbered in such a way that the input nodes are numbered from 1 ton.The standard encoding of a generalized arithmetic circuitC is a list having oneentry for every node appearing in the dag associated toC in the following way:

• every node labeled by an integerm is encoded by(ρ, m), whereρ denotesthe node number.

• every node labeled by a polynomial map is encoded by(ρ, op, ρ1, . . . , ρr),whereρ denotes the node number,op identifies the polynomial map andρj is a pair indicating the node number and coordinate whose output is thej -th input ofρ.

A family of generalized arithmetic circuitsis a sequence of generalizedarithmetic circuits{Cn}n∈N together with a sequence of finite sets of polyno-mials maps{Φ(n)

1 , . . . , Φ(n)rn

}n∈N, such that the circuitCn hasnO(1) input nodesand its nodes are labeled by the polynomial mapsΦ

(n)1 , . . . , Φ(n)

rn. A family of

generalized arithmetic circuits{Cn}n∈N is calleduniform in spaceS(n) if thereexists a deterministic Turing machineM such that, on inputn, computes thestandard encoding of the circuitCn using spaceS(n).

The main result of this section, Lemma 2, is the key tool which allows us togive a systematic treatment of the uniformity question of the algorithms whichwe are going to develop throughout this contribution. Lemma 2 shows that afamily of generalized arithmetic circuits which is uniform in small space, whosenodes are labeled by a sequence of polynomial maps which is uniform in smallspace has an efficient boolean translation. In order to prove this result we needto introduce some terminology.

Let C be a generalized arithmetic circuit overZ with n inputs and leth bea fixed positive integer. We say that a boolean circuitD is associated toC atlogarithmic heighth if the circuitD computes, given the binary representationof n integersm1, . . . , mn of h bits as input, the binary representation of theoutput ofC on input(m1, . . . , mn). Given a family of generalized arithmeticcircuits{Cn}n∈N and a sequence{hn}n∈N, we say that a family of boolean circuits{Dn}n∈N is associated to{Cn}n∈N at logarithmic height{hn}n∈N if for anyn ∈ N

the circuitDn is associated toCn at logarithmic heighthn. Now we can statethe main result of this section:

Lemma 2 (Macro expansion lemma)Let {C(1)n }n∈N, . . . , {C(r)

n }n∈N be fami-lies of division-free arithmetic circuits of sizeK1(n), . . . , Kr (n) and nonscalar


depthλ1(n), . . . , λr(n) respectively. Assume that the families{C(1)n }n∈N, . . . ,

{C(r)n }n∈N use parameters of logarithmic height bounded by{η(n)

1 }n∈N, . . . ,

{η(n)r }n∈N, respectively and that they are uniform in spaceS1(n), . . . , Sr (n)

respectively. For1 ≤ k ≤ r we denote byΦ(k)n the polynomial map com-

puted by the arithmetic circuitC(k)n . Assume further that for any sequence

{(h(n)1 , . . . , h(n)

r )}n∈N ⊆ (Z+)r there exist families of boolean circuits{C(1)

n (h(n)1 )}n∈N, . . . , {C(r)

n (h(n)r )}n∈N associated to{C(1)

n }n∈N, . . . , {C(r)n }n∈N at

logarithmic height{(h(n)1 , . . . , h(n)

r )}n∈N with sizeL1(n, h(n)1 ), . . . , Lr(n, h(n)

r )

and depth 1(n,h(n)1 ), . . . , `r(n, h(n)

r ), which are uniform in spaceS1(n,h(n)1 ),. . .,

Sr(n, h(n)r ) respectively. Let{Cn}n∈N be a family of generalized arithmetic cir-

cuits using parameters of logarithmic height bounded by{ηn}n∈N which is uni-form in spaceS(n). Assume that the nodes of the circuitCn are labeled byΦ

(n)1 , . . . , Φ(n)

r and thatCn has sizeCi(n) and depthγi(n) in Φ(n)i for 1 ≤ i ≤ r.

Then, for any sequence{hn}n∈N ⊆ Z+ there exists a family of boolean circuits{Cn(hn)}n∈N associated to{Cn}n∈N at logarithmic height{hn}n∈N with size∑r

k=1

∑γi(n)Li(n, Hn) and depth

∑ri=1 γi(n)ì(n, Hn) which is uniform in

space∑r

i=1

(Si(n)+Si(n, Hn)

)+S(n)+ log( ∑r

i=1 γi(n)Li(n, Hn))+ logHn,

whereHn denotes the number

Hn = 21+

r∑i=1

λi(n)γi (n)(

log

( r∑i=1

0i(n)Ki(n)

)+ maxi{η(n)

i , ηn} + hn

).

Proof. The construction of the family of boolean circuits{Cn(hn)}n∈N can beseen as a macro expansion procedure. This macro expansion follows three basicrules: first, every input nodeρ of Cn is replaced byhn input nodes ofCn(hn)

representing thehn binary digits of the integer corresponding to the nodeρ.Secondly, every node ofCn labeled by an integer parameterα is replaced byηn nodes ofCn(hn) labeled by the binary digits ofα. Finally, every occurrenceof a nodeρ of Cn labeled byC(i)

n is replaced inCn(hn) by a boolean circuitC(i)

n (H(ρ)n ) that computes the binary representation of the output ofC(i)

n oninteger numbers with logarithmic heightH

(ρ)n . We chooseH(ρ)

n as the worst-case logarithmic height of the inputs of the nodeρ under consideration whenthe inputs ofCn are integers of logarithmic height bounded byhn.

In order to compute the numberH(ρ)n we observe that, if the circuitCn

appliesC(ρ)

i (n) times the morphismΦ(n)i with depthγ

(ρ)

i (n) for 1 ≤ i ≤ r tocompute the result of the nodeρ, applying [45], Proposition 15 we see that thenumberH(ρ)

n can be bounded by 21+∑ri=1 λi(n)γ

(ρ)

i (n)(

log( ∑r

i=1 Ki(n)C(ρ)

i (n))+

maxi{η(n)i , ηn} + hn

). Note that the circuitCn can be executed by performing∑r

i=1 Ki(n)Ci(n) arithmetic operations with nonscalar depth at most∑ri=1 λi(n)γi(n), which implies that for every nodeρ of Cn the numberH(ρ)

n

is bounded by

470 G. Matera

Hn := 21+∑ri=1 λi(n)γi (n)

(log

( r∑i=1

K(ρ)

i (n)C(ρ)

i (n)

)+ maxi{η(n)

i , ηn} + hn

).

Therefore, the macro expansion procedure yields a boolean circuitCn(hn)

associated toCn at logarithmic heighthn whose size is bounded by∑r

i=1 0i(n)

Li(n, Hn) and its depth is at most∑r

i=1 γi(n)ì(n, Hn).There remains to prove the uniformity of the family of boolean circuits

{Cn(hn)}n∈N. The hypotheses of Lemma 2 imply that there exist determinis-tic Turing machinesM, M1, M1, . . . , Mr, Mr working in spaceS(n), S1(n, h),

S1(n), . . . , Sr (n), Sr(n, h) such that, given a tuple(n, h) as input, they com-pute the standard encoding of the circuitsCn, C

(1)n (h), C(1)

n , . . . , C(r)n (h), C(r)

n

respectively. We are going to construct a deterministic Turing machineN suchthat, given a tuple(n, hn) as input, it computes the standard encoding ofCn(h)

working in space∑r

i=1

(Si(n)+Si(n, Hn)

)+S(n)+log( ∑r

i=1 0i(n)Li(n, Hn))

+ logHn.First of all, the machineN call the machinesM, M1, . . . , Mr to compute

the numbers maxi{η(n)i , ηn} by simple inspection of the standard encoding of

the arithmetic circuitsCn, C(1)n , . . . , C(r)

n .ThenN computes the standard encoding of the circuitCn(hn) in the fol-

lowing way: N calls the machineM which generates the standard encodingof Cn node by node.N stores in memory space a counter for the numberof the next node ofCn(hn) to be generated. Every time that the machineM

generates an input node ofCn N generateshn input nodes ofCn(hn). Sim-ilarly N generatesη nodes ofCn(hn) for any node ofCn labeled by a pa-rameter. Finally, when a nodeρ of Cn labeled byC(i)

n is found,N computesthe numbers

∑ri=1 λi(n)γ

(ρ)

i (n) and∑r

i=1 Ki(n)C(ρ)

i (n). For this purposeNcomputesγ (ρ)

i , C(ρ)

i for 1 ≤ i ≤ r by a depth-first search on the dag asso-ciated toCn andλi(n), Ki(n) by a simple inspection of the dag associated toC(i)

n for 1 ≤ i ≤ r. ThenN calls the machineM1 on input(n, H(ρ)n ) which

generates the standard encoding of the boolean subcircuit ofCn(hn) corre-sponding to the nodeρ of Cn. The working space required byN is obtainedby adding the working space of the machinesM, M1, M1, . . . , Mr, Mr , thespace required to compute and store the numberHn and the space necessaryfor the counter of the number of the next node of the boolean circuitCn(hn)

to be generated. Therefore, we have that the machineN has working space∑ri=1

(Si(n) + Si(n, Hn)

)+ S(n) + log

( ∑ri=1 Ci(n)Li(n, Hn)

)+ logHn.

2.1 Uniform linear algebra algorithms

The algorithms which we are going to develop throughout this contribution relystrongly on linear algebra calculations. Therefore, the uniformity of the familiesof boolean circuits which we are going to develop is closely related to theuniformity of the linear algebra circuits we use. Although several linear algebra


algorithms are announced to be space-uniform, no proof of such statements isexplicitly shown (see e.g. [6] or [8]).

This section is devoted to review the linear algebra algorithms we need,giving simple proofs of their space-uniformity based on our Macro expansionlemma. These linear algebra algorithms are generally well-known and havebeen chosen taking into account their space-uniformity and low nonscalar depthrather than the number of arithmetic operations they perform (a similar pointof view of ours is followed in [33], [53] and [52]). The only exceptions are theresolution of linear equation systems (Lemma 9) where some additional care istaken in order to avoid branchings in the computation and the computation ofthe characteristic polynomial (Lemma 6) because the hypotheses of the Macroexpansion lemma do not apply.

Lemma 3 There exists a family of boolean circuits{Cn(h)}n∈N which computesthe inner product

∑ni=1 xiyi of two vectors(x1, . . . , xn)and(y1, . . . , yn)ofZn of

logarithmic height bounded byh with size O(n3h2) and depthO(log(nh)), which is uniform in spaceO

(log(nh)

).

Proof. We can describe a generalized arithmetic circuitCn computing the innerproduct

∑ni=1 xiyi by using two basic operations: the product of two integers

and the addition ofn integers. As it is well known (see for example [42] or[63]) there exists a family of boolean circuits that computes the product of twointegers of logarithmic heighth with sizeO(h2) and depthO(logh) which isuniform in spaceO(logh). Using the well-knownthree-for-two trick(see e.g.[63]) we obtain a family of boolean circuit computing the addition ofn integersof logarithmic heighth with size O

(n(h + logn)2

)and depthO

(log(nh)

)which is uniform in spaceO(log(nh)). Observe thatCn performsn productswith depth 1 and one addition ofn integers. Applying the Macro expansionlemma using the product of two integers and the addition ofn integers aspolynomial maps yields a family of boolean circuits of sizeO

(n3h2) and depth

O(

log(nh))

which is uniform in spaceO(

log(nh)).

Lemma 4 There exist a family of boolean circuits computing the product ofan (m × n) integer matrix by an(n × p) integer matrix of logarithmic heighth with sizeO

(n3mph2 log2 n

)and depthO

(log(nmph)

), which is uniform in

spaceO(

log(nmph)).

Proof. We can easily describe a circuit computing a matrix product in terms ofinner product. Taking into account that we have to performmp inner productsof n-dimensional vectors, the statement of Lemma 4 follows easily from thebounds of Lemma 3 and the Macro expansion lemma.

Lemma 5 There exist a family of boolean circuits computing the powersA2, A3, . . . , Akn of a given(n × n) integer matrixA of logarithmic heighth of

472 G. Matera

sizeO(k2nn

5h2 log4(nkn))

and depthO(

log(kn) log(knnh))

which is uniform inspaceO

(log(nknh)

).

Proof. The computation of the powersA2, . . . , Akn can be organized in atree withkn nodes and depth logkn. In the l-th level of the tree we computeA2l+1, . . . , A2l+1

multiplying the matricesA, A2, . . . , A2l

by A2l

. Thereforethe computation ofA2, . . . , Akn can be easily described in spaceO(logkn) interms of matrix product. Taking into account that 2logkn(logn + h) is an upperbound for the logarithmic height of the integers appearing during the wholecomputation, the application of the bounds of the Macro expansion lemma andLemma 4 yields a family of boolean circuits which satisfies the statement ofLemma 5.

Lemma 6 There exist a family of boolean circuits computing the characteristicpolynomial of a given(n×n) integer matrixA of logarithmic heighth with sizenO(1)h2 and depthO

(log(n) log(nh)

), which is uniform in spaceO

(log(nh)

).

Proof. We compute the characteristic polynomial by using Berkowitz’salgorithm [6]. LetA := (aij ) be ann×n matrix. For 1≤ k ≤ n, we defineRk = (ak,k+1, . . . , ak,n), Sk = (ak+1,k, . . . , an,k), and for 2≤ k ≤ n

Mk = ak,k · · · ak,n

.... . .

...

an,k · · · an,n

Further, we define the lower triangular matrixCk of (n − k + 1) × (n − k) bycki,j = −1 if i = j , ck

i,j = ak,k if i = j + 1, andcki,j = −Rk · M

i−j−2k · Sk if

i − j ≥ 2. Then we have the equality(p0, p1, . . . , pn)t = ∏

1≤k≤n Ck where(p0, p1, . . . , pn) are the coefficients of the characteristic polynomial ofA.

Therefore, a family of boolean circuit computing the characteristic poly-nomial ofA can be obtained in the following way: in a first step we computesimultaneously all(Mk)

j ’s for 0 ≤ k ≤ nand 0≤ j ≤ n−k−2. From Lemma 5we deduce that this part has sizeO

( ∑nk=1 k7h2 log4(k)

) = O(n8h2 log4 n) anddepthO

(log(n) log(nh)

), and the resulting integers have logarithmic height

O(nh logn).In the second step we first compute the product(Mk)

j · Sk for 0 ≤ k ≤ n

and 0≤ j ≤ n − k − 2 in parallel and then, using this results, we computeRk · (Mk)

j · Sk for eachk, j in parallel. Applying Lemma 4 we conclude thatthis step has sizeO

(n6h2 log4 n

)and depthO

(log(nh)

).

Finally we compute the coefficients of the characteristic polynomial ofA bythe formula(p0, p1, . . . , pn)

t = ∏1≤k≤n Ck. We perform this stage by means

of a tree of matrix products. Taking into account that there aren products ofmatrices of logarithmic height bounded byO(nh logn), we conclude that thisstep can be performed with sizeO

(n9h2 log6(n)

)and depthO

(log(n) log(nh)

).


Adding the complexities obtained in each step we obtain a family of booleancircuits of sizenO(1)h2 and depthO

(log(n) log(nh)

)computing the character-

istic polynomial ofA.Now we show that this family is uniform in spaceO

(log(nh)

). From

Lemma 4 we have that there exists a deterministic Turing machineM(1) suchthat on input(m, n, p, h) M(1) computes the standard encoding of booleancircuit computing the product of an(m × n)-matrix by an(n × p)-matrix oflogarithmic heighth using spaceO

(log(nmph)

). Similarly, from Lemma 5 we

have that there exists a deterministic Turing machineM(2) such that on input(n, k, h) M(2) computes the standard encoding of boolean circuit computingthe powersA2, . . . , Ak of an(n × n)-matrix of logarithmic heighth.

A deterministic Turing machineM that computes the standard encoding ofthe family of boolean circuits described above can be constructed in the fol-lowing way: M starts callingM(2) on input (n − 1, n − 3, h), . . . , (4, 2, h)

to produce the standard encoding of the computation ofM21, . . . , Mn−3

1 , . . . ,

M2n−4, modifying the output ofM(2) in such a way that the input of these ma-

trix powering are the proper submatrices of the input matrix. ThenM calls themachineM(1) n − 2 times on input(n − 1, n − 1, 1, nh logn), n − 3 times oninput(n−2, n−2, 1, nh logn), and so on, to produce the standard encoding ofM2

1S1, . . . , Mn−31 S1, . . . , Mn−4Sn−4, M

2n−4Sn−4, Mn−3Sn−3. Similarly the stan-

dard encoding of the computation of theRkMj

k Sk ’s products is produced.Finally, a repeated calling toM(2) on input

(n+1, n, n1, O(nh logn)

), . . . ,

produces the standard encoding of the product of the matricesC1 · · ·Cn. Takinginto account that all the matrices appearing in the procedure have size at most(n + 1) × n and that the logarithmic height of the integers occurring during theprocedure is bounded byO(nh logn), we conclude that the machineM worksin spaceO

(log(nh)

).

Lemma 7 There exist a family of boolean circuits computing the rank of agiven(n×m) integer matrixA of logarithmic heighth with sizenO(1)h2 log2 m

and depthO(

log(n) log(nmh)), which is uniform in spaceO

(log(nmh)

).

Proof. Let B := A · At . As it is well-knownn − rk(A) equals the multiplicityof zero in the characteristic polynomial ofB. This consideration yields analgorithm for the computation of the rank of an integer matrix.

Therefore, we compute the rank ofA in the following way: first we computethe matrixB = A · At . Then we compute the coefficients(p0, . . . , pn) ofthe characteristic polynomial ofB. Finally, we compute the numberrk(A) =max{i; 1 ≤ i ≤ n ∧ pi 6= 0} from the boolean encoding of thepi ’s. Forthis sake, we design subcircuits “pi = 0” and “pi 6= 0”, which have the sameboolean value of truth as the quoted statement. Suppose thatpi = (pis , . . . , pi0)

is the boolean representation ofpi . Thenpi = 0 if and only if all its digits arezero, i.e. if and only ifpis ∨· · ·∨pi0 = 0. Therefore, the operator¬(pis ∨· · ·∨pi0)

474 G. Matera

has the same value of truth thatpi = 0, and the same is true forpis ∨ · · · ∨ pi0

and pi 6= 0. Then, for every 1≤ i ≤ n we compute “p0 = 0 ∧ p1 =0 ∧ · · · ∧ pi 6= 0” by means of a tree of conjunctions of the previouslyconstructed subcircuits. Finally, given a natural numberi such that 1≤ i ≤ n

with boolean representationi = (is, . . . , i0), we definei = (is, . . . , i0) byik := (

p0 = 0 ∧ p1 = 0 ∧ · · · ∧ pi 6= 0) ∧ ik, which returns thek-th digit

of i if and only if the affirmationp0 = 0 ∧ p1 = 0 ∧ · · · ∧ pi 6= 0 is trueand 0 otherwise. Therefore, the disjunction

∨1≤i≤n ik outputs thek-th digit of

max{i; 1 ≤ i ≤ n ∧ pi 6= 0}.The computation of the matrixB and its characteristic polynomial has size

nO(1)h2 log2 m and depthO(

log(n) log(nmh)). The computation of the rank of

A from the coefficients of the characteristic polynomial has sizeO(n3h) anddepthO

(log(n) log(nmh)

). Therefore, we have a family of boolean circuits of

sizenO(1)h2 log2 m and depthO(

log(n) log(nmh))

which computes the rankof an integer matrix.

There remains to prove the uniformity of this family. The computation ofthe matrixB and its characteristic polynomial consists of a matrix product andthe computation of a characteristic polynomial. Therefore, applying the Macroexpansion lemma and Lemmata 4 and 6 we obtain a deterministic Turing ma-chine working in spaceO

(log(nmh)

)which computes the standard encoding

of the computation ofB and its characteristic polynomial. The remaining partcan be easily described using spaceO

(log(nmh)

).

Lemma 8 There exists a family of boolean circuits computing the adjoint ma-trix of an (n × n) integer matrixA of logarithmic heighth with sizenO(1)h2

and depthO(

logn log(nh)), which is uniform in spaceO

(log(nh)

).

Proof. Let B := Adj (A) be the adjoint matrix of an integer(n × n)-matrixA.Letχ(T ) := T n +pn−1T

n−1+· · ·+p1T +p0 be the characteristic polynomialof A. Then we have that

B = −An−1 − pn−1An−2 − · · · − p1In×n (1)

(hereIn×n denotes then × n-identity matrix).Therefore, from equation (1) we deduce an algorithm to compute the ad-

joint matrix of A. This algorithm computes the matricesA2, . . . , An−1, thecoefficients of the characteristic polynomial ofA and then it performs the lin-ear combination of the matricesIn×n, A, A2, . . . , An−1 indicated in equation(1). We apply Lemmata 5 and 6 to compute the matricesA2, . . . , An−1 andthe coefficients of the characteristic polynomial ofA. Then we compute thematrices−An−1, −pn−1A

n−2, . . . ,−p1In×n and we add these matrices usingthe three-for-two trick. As a consequence of the bounds of Lemmata 5 and 6we conclude that there exists a family of boolean circuits computing the adjointmatrix of sizenO(1)h2 and depthO

(log(n) log(nh)

).


In order to prove the uniformity of the algorithm described above we applythe Macro expansion lemma with matrix powering, computation of the char-acteristic polynomial, product of two integers and addition ofn integers asarithmetic macros. We observe that the multiplications and additions requiredto compute the linear combination of equation (1) involve integers of loga-rithmic heightO(nh logn). Therefore, from the bounds of Lemmata 5 and 6and the fact that the product of two integers and the addition ofn integers oflogarithmic heightO(nh logn) are uniform in spaceO

(log(nh)

), we conclude

that the algorithm for the computation of the adjoint matrix described above isuniform in spaceO

(log(nh)

).

Lemma 9 There exists a family of boolean circuits of sizenO(1)h2 and depthO

(logn log(nh)

)which is uniform in spaceO

(log(nh)

)such that it checks

whether there exists a solution of a given linear equation systemA · X = b

(hereA is integer(n×m)-matrix of logarithmic heighth, X is a(m×1)-vectorof unknowns andb an integer(n × 1)-vector of logarithmic heighth), and,if this is the case, it computes numerators and denominators of a particularsolution of the system.

Proof. The idea is to reduce our problem to the resolution of a system witha nonsingular square matrix. The second system is then solved applying justCramer’s rule.

The reduction of the problem is done by an algorithm which finds a nonsin-gular square submatrix of maximal rank ofA. Let us denote this submatrix byA. Deleting fromb all the entries which do not correspond to rows ofA we ob-tain a column vectorb and deleting fromX all entries which do not correspondto the columns ofA we obtain a new column vectorX of unknowns.

Solving now the reduced nonsingular square system of linear equationsA · X = b we obtain easily a solution of the original systemA · X = b.

Therefore, we just need to find a square submatrixAofAwith maximal rank.Let us denote byAi thei-th row ofA. We compute (in parallel)rk(A1, . . . , Ai)

for i = 1, . . . , n. Every time whenrk(A1, . . . , Ai−1) < rk(A1, . . . , Ai)occurs,we keep the indexi. These indices are indicated in a vectorR = (r1, . . . , rn) ∈{0, 1}n as the nonzero entries ofR and correspond to the rows which will occurin our matrix A. Similarly we compute a vectorS = (s1, . . . , sn) ∈ {0, 1}mwhose nonzero entries correspond to the columns which we choose to buildA. The procedure to compute the vectorR requires the computation of therank of n submatrices of the matrixA and the comparison of the last digitsof the numbersrk(A1, . . . , Ai−1) andrk(A1, . . . , Ai) for 1 ≤ i ≤ n (sincerk(A1, . . . , Ai) is eitherrk(A1, . . . , Ai−1)orrk(A1, . . . , Ai−1)+1). Therefore,it can be performed with sizenO(1)h2 log2 m and depthO

(log(n) log(nmh)

).

We compute the vectorS in the same way with sizenO(1)h2 log2 m and depthO

(log(n) log(nmh)

).

476 G. Matera

Assume thatn ≥ m (the other case can be solved in a similar way). Wemultiply the input matrixA at right by an(m × n)-matrix with all its entrieszero except for its diagonal which is given by the vectorS. Then we multiplythe resulting matrix and the vectorb at left by a diagonal(n×n)-matrix whosediagonal is given by the vectorR. The resulting matrixB is an(n × n)-matrixwith the same entries asA whenever they correspond to the rows and columnswe have chosen to formA and zero otherwise. Similarly, the vectorc obtained inthis way has the same entries asb whenever they correspond to the coordinateswe have chosen to formb and zero otherwise.

Letri1, . . . , rik andsj1, . . . , sjkdenote the zero coordinates ofR andS (where

S is ann-dimensional vector whose lastn−m coordinates are zero). We definethe (n × n) matrix B which has as only nonzero entries those correspondingto (ri1, si1), . . . , (rik , sik ), which are define to be 1. Adding this matrix toBwe obtain a nonsingular(n × n)-matrix C having the property that the firstm

coordinates of the solution the linear equation systemC · Y = c is a particularsolution of the linear equation systemA · X = b. We compute the solution ofthe linear equation systemC · Y = c asY = 1

det (C)Adj (C) · c.

Applying Lemmata 7, 8 and 4 we see that the whole procedure can beexecuted by means of a family of boolean circuits of sizenO(1)h2 and depthO

(logn log(nh)

).

Now we show the uniformity of the family of boolean circuits describedabove. From Lemma 7 we have that there exists a deterministic Turing machineM1 such that, on input(n, m, h), computes the standard encoding of a family ofboolean circuits computing the rank of an(n×m) integer matrix of logarithmicheighth.

The deterministic machineM which computes the standard encoding of thefamily of boolean circuits solving a linear equation system callsM1 on input(n, 1, h), (n, 2, h), . . . , (n, m, h) to produce the standard encoding of a booleancircuit computing the rank of the matrices(A1), (A1, A2), . . . , (A1, . . . , Am) =A. Similarly,M produces the standard encoding of a boolean circuit computingthe rank of the matrices(A(1)), (A(1)|A(2)), . . . , (A(1)| . . . |A(m)) = A (hereA(i)

denotes thei-th column ofA) by calling toM1 on input(1, m, h), (2, m, h), . . . ,

(n, m, h). ThenM produces the standard encoding of the subcircuit computingthe vectorsR andS and the matrixC with spaceO

(log(nh)

). The remaining

part of the circuit can be described in terms of matrix products and computa-tion of the adjoint matrix. Therefore, applying Lemmata 4 and 8 and the Macroexpansion lemma the statement in Lemma 9 follows.

On the basis of these linear algebra algorithms we are going to build the basicalgorithms we need. These algorithms are concerned with the manipulation ofunivariate polynomials and arithmetic circuits. We start with the considerationof the manipulation of univariate polynomials.


2.2 Manipulation of univariate polynomials

LetR be the polynomial ringZ[X1, . . . , Xn], K := Q(X1, . . . , Xn) its quotientfield and letT be an indeterminate overK. During this subsection we will beconcerned with algebraic manipulations with univariate polynomialsf, g ∈R[T ]. All these manipulations will be well-parallelizable (see [4] or [63]) andwill contain no divisions by elements ofR (this is important since divisionsby elements ofR may cause serious problems while evaluating the variablesX1, . . . , Xn in some concrete point ofZn).

The first problem to be consider, assuming thatg|f holds inK[T ] is thecomputation of a polynomial ofK[T ] which represents the quotientf

g.

Lemma 10 Let {Cn,d,e,h}n,d,e,h∈N be a family of boolean circuits which evalu-ates polynomialsf0, . . . , fd, g0, . . . , ge of Z[X1, . . . , Xn] on integers of loga-rithmic heighth. LetL := L(n, d, e, h) and` := `(n, d, e, h) be the size anddepth ofCn,d,e,h respectively, and assume that the family{Cn,d,e,h}n,d,e,h∈N isuniform in spaceS := S(n, d, e, h). Let H := H(n, d, e, h) the worst-caselogarithmic height of the output ofCn,d,e,h. Let f := fdT

d + · · · + f0 andlet g := geT

e + · · · + g0. Assume thatd ≥ e holds and thatg dividesf

overQ(X1, . . . , Xn)[T ]. Then, there exists a family of boolean circuits of sizeL + dO(1)H 2 and depthO

(` + log(d) log(dh)

)computing aZ[X1, . . . , Xn]-

multiple of the polynomial ofQ(X1, . . . , Xn)[T ] representing the quotientfg,

which is uniform in spaceS + O(

log(dHL)).

Proof. Considering the polynomialh := f

gwith indeterminate coefficients,

the conditionf = gh can be translated into a linear equation system whichis obtained by means of the comparison of the coefficients off andgh cor-responding to the same monomials. This linear system hasd equations andd − e + 1 unknowns and has the following form:

ge

ge−1. . .

.... . . ge

g0 ge−1. . .

...

g0

he−d

...

h0

=

fd

fd−1...

f0

(2)

We apply the Macro expansion lemma using the polynomial maps definedby the polynomialsf0, . . . , fd, g0, . . . , ge and the particular solution of thesystem (2) obtained by the application of Lemma 9. Taking into account thatthe coefficients of the matrix of the system (2) has logarithmic heightH weconclude that we have a family of boolean circuits computing numerators anddenominators of a particular solution of the system (2) of sizeL+dO(1)H 2 and

478 G. Matera

depthO(` + log(d) log(dh)

), which is uniform in spaceS + O

(log(dH) +

log(L + dH)), as stated.

Let us now discuss the problem of the computation of the greatest commondivisor off andg.

Lemma 11 Let {Cn,d,e,h}n,d,e,h∈N be a family of boolean circuits which eval-uates polynomialsf0, . . . , fd, g0, . . . , ge of Z[X1, . . . , Xn] on integers of log-arithmic heighth. LetL :=L(n, d, e, h) and` := `(n, d, e, h) be the size anddepth ofCn,d,e,h respectively, and assume that the family{Cn,d,e,h}n,d,e,h∈N

is uniform in spaceS := S(n, d, e, h). Let H := H(n, d, e, h) the worst-case logarithmic height of the output ofCn,d,e,h. Assume thatd ≥ e. Letf = fdT

d + · · · + f0 and letg = geTe + · · · + g0. Then, there exists a fam-

ily of boolean circuits of sizeL + dO(1)H 2 and depthO(` + log(d) log(dh)

)computing aZ[X1, . . . , Xn]-multiple of the greatest common divisor off andg overQ(X1, . . . , Xn)[T ], which is uniform in spaceS + O

(log(dHL)

).

Proof. We apply techniques based on subresultants (see [15], [10]) in the ver-sion of [8]: it turns out that the minimal natural numberj such that thej -thprincipal submatrixPj(f, g) of the Sylvester matrix is nonsingular equals thedegree ofgcd(f, g) (see Theorem 2.3 in [22]). Furthermore, the last column ofthe inverse matrix ofPj(f, g) provides the coefficients of polynomialsr ands

such that the equalitygcd(f, g) = rf + sg holds.Following the ideas of Lemma 9, instead of computing the last column of

Pj(f, g)−1 we compute the last column of the matrixAdj(Pj(f, g)

)which

yields adet(Pj(f, g)

)-multiple of the greatest common divisor off andg.

Therefore, we compute in parallelc0 :=det(P0(f, g)

), . . . , ce :=

det(Pe(f, g)

). Then we compute the boolean formulas “c0 = 0”, . . . ,“ce = 0”

which have the same value of truth as the quoted statement, with the sameidea as in the proof of Lemma 7. From these formulas we obtain the leastnatural numberj such thatcj is nonzero evaluating the boolean formulasdi := (c0 = 0) ∨ · · · ∨ (ci−1 = 0) ∨ (ci 6= 0) for 1 ≤ i ≤ e. Then we computethe last column of the matrixdiAdj

(Pi(f, g)

)for 1 ≤ i ≤ e, obtaining thus

coefficients of some polynomialsri, si which are nonzero if and only ifi = j .Finally, we compute the polynomialgcd(f, g) = ∑e

i=0 rif +sig = rjf +sjg.Taking into account that the matricesPi(f, g) andAdj

(Pi(f, g)

)have logarith-

mic heightH andO(dH) respectively, we obtain a family of boolean circuitsof sizeL + dO(1)H 2 and depth +O

(log(n) log(nH)

). The productsrif, sig

for 0 ≤ i ≤ e can be thought as matrix–vector products.To prove the uniformity of the family of boolean circuits we first construct

the standard encoding of the family of boolean circuits which evaluates thepolynomialsf0, . . . , fd, g0, . . . , ge with spaceS. The mapping which producesthe matricesPi(f, g) from f0, . . . , fd, g0, . . . , ge can be described easily inspaceO(logd). Then we call the machine which witnesses the uniformity


of the computation of the determinant on input(d + e, H), . . . , (d − e, H)

to construct the standard encoding of the computation of the determinants ofthe matricesP(f, g), . . . , Pe(f, g) using spaceO

(log(dH)

). The standard

encoding of the formulas fordi requires spaceO(logd). Then we constructthe standard encoding of the computation of the last columns of the matricesAdj

(Pi(f, g)

)with additional spaceO

(log(dH)

)and finally useO

(log(dH)

)space to describe the computation of the polynomialgcd(f, g) = ∑e

i=0 rif +sig. During all these computations we use a counter to know the number of thenext node to be generated which requiresO

(log(L + dH)

)cells of working

space. Therefore, we need spaceS + O(

log(dH)) + O

(log(L + dH)

)to

compute the standard encoding of the whole circuit.

The last problem concerning manipulations with univariate polynomialsis the determination of aseparable representationof f , that is, a polynomialg ∈ K[T ] that has the same zeros asf over the algebraic closureK of K andis squarefree.

Lemma 12 Let {Cn,d,h}n,d,h∈N be a family of boolean circuits which evaluatespolynomialsf0, . . . , fd of Z[X1, . . . , Xn] on integers of logarithmic heighth. Let L := L(n, d, h) and ` := `(n, d, h) be the size and depth ofCn,d,h

respectively, and assume that the family{Cn,d,h}n,d,h∈N is uniform in spaceS := S(n, d, h). LetH := H(n, d, h) the worst-case logarithmic height of theoutput ofCn,d,h. Letf = fdT

d +· · ·+f0. Then, there exists a family of booleancircuits of sizeL + dO(1)H 2 and depthO

(` + log(d) log(dh)

)computing a

Z[X1, . . . , Xn]-multiple of the separable representationg = f

gcd(f,f ′) , which isuniform in spaceS + O

(log(dHL)

).

Proof. SinceQ(X1, . . . , Xn) has characteristic zero, we observe that the poly-nomial g = f

gcd(f,f ′) verifies the required conditions, wheref ′ denotes thederivative off with respect toT (notice that the coefficients off ′ can becomputed immediately from the coefficients off ). Applying the Macro expan-sion lemma using the polynomial maps defined by the polynomialsf0, . . . , fd,

g0, . . . , ge and the computation of the greatest common divisor and quotientgiven by the application of Lemmata 11 and 10 we obtain a family of booleancircuits of sizeL + dO(1)H 2 and depthO

(` + log(d) log(dh)

)computing a

Z[X1, . . . , Xn]-multiple of the separable representationg = f

gcd(f,f ′) , which isuniform in spaceS + O

(log(dHL)

).

2.3 Manipulation of arithmetic circuits

During this subsectionF will denote a polynomial inR := Z [X1, . . . , Xn] ofdegree at mostd given by an arithmetic circuitβ of sizeL and nonscalar depth`. Let again beK :=Q(X1, . . . , Xn) and letT be an indeterminate overK.

480 G. Matera

Different tasks concerning the manipulation of arithmetic circuits are con-sidered in this subsection. As we will see later on, all these manipulations havea uniform structure and only depend on the degree of the input polynomialF . This observation will allows us to conclude that these manipulations areuniform.

We first consider the problem of the interpolation ofF with respect to onevariable:

Lemma 13 Let {Cn,d,h}n,d,h∈N be a family of boolean circuits which evaluatesa polynomialF of Z[X1, . . . , Xn] of degreed on integers of logarithmic heighth. LetLh :=L(n, d, h) and`h := `(n, d, h) be the size and depth ofCn,d,h re-spectively, and assume that the family{Cn,d,h}n,d,h∈N is uniform in spaceSh :=S(n, d, h). LetHh :=H(n, d, h) the worst-case logarithmic height of the outputof Cn,d,e,h. Then, there exists a family of boolean circuits computing the denserepresentation ofF with respect toXn with sizeLmax{h,logd} +dO(1)Hmax{h,logd}and depth`max{h,logd} + O

(log(d) log(dHmax{h,logd})


spaceSmax{h,logd} + O(

log(dHmax{h,logd}Lmax{h,logd})).

Proof. Since deg(F ) = d, in particulardegXn(F ) ≤ d, and therefore we can

find the dense representation ofF with respect to the variableXn by interpo-lation in d + 1 points ofQ. We writeF = f0 + f1Xn + · · · + fdX

dn where

f0, . . . , fd are polynomials ofZ[X1, . . . , Xn−1]. Choosing the interpolationpoints 0, . . . , d we have:

1 0 . . . 01 1 . . . 1...

......

1 d . . . dd

f0

f1...

fd

=

F(X1, . . . , Xn−1, 0)

F (X1, . . . , Xn−1, 1)...

F (X1, . . . , Xn−1, d)

(3)

Therefore solving the system (3) yields the the coefficients ofF with respectto Xn.

The computation is performed by computing the polynomialsF(X1, . . . , Xn−1, 0), F (X1, . . . , Xn−1, 1), . . . , F (X1, . . . , Xn−1, d) in parallel,constructing the Vandermonde matrix of (3) and solving the system (3). Finallywe performd+1 (exact) divisions of the solutions of the system (3) given by theapplication of Lemma 9 by the determinant of the matrix of the system (3). Ap-plying the Macro expansion lemma using as arithmetic macros the computationofF , the resolution of a linear equation system and the exact division of integers,we obtain a family of boolean circuits of sizeLmax{h,logd}+dO(1)Hmax{h,logd} anddepth`max{h,logd} + O

(log(d) log(dHmax{h,logd})

), which is uniform in space

Smax{h,logd} +O(

log(dHmax{h,logd})+ log(Lmax{h,logd} + dO(1)Hmax{h,logd})).

We now consider the decomposition ofF in homogeneous components,that is, the computation of polynomialsF0, . . . , Fd ∈ R with the followingproperties:


• F =d∑

k=0

Fk

• Fork = 0, . . . , d, the polynomialFk is zero or homogeneous of degreek.

Lemma 14 Let {Cn,d,h}n,d,h∈N be a family of boolean circuits which evaluatesa polynomialF of Z[X1, . . . , Xn] of degreed on integers of logarithmic heighth. Let Lh := L(n, d, h) and `h := `(n, d, h) be the size and depth ofCn,d,h

respectively, and assume that the family{Cn,d,h}n,d,h∈N is uniform in spaceSh :=S(n, d, h). LetHh :=H(n, d, h) be the worst-case logarithmic height ofthe output ofCn,d,h. Then, there exists a family of boolean circuits computing thehomogeneous components ofF with sizeLmax{h,logd} + dO(1)Hmax{h,logd} anddepth`max{h,logd} + O(log(d) log(dHmax{h,logd})), which is uniform in spaceSmax{h,logd} + O(log(dHmax{h,logd}Lmax{h,logd})).

Proof. SinceF(T X1, . . . , T Xn) = ∑dk=0 T kFk(X1, . . . , Xn), the problem can

be reduced to an interpolation with respect to the variableT .Therefore, we can apply the Macro expansion lemma with the computation

of the polynomialF and interpolation as arithmetic macros. From the boundsof Lemma 13 we deduce the statement of Lemma 14.

Our last problem concerning the manipulation of arithmetic circuits is thatof avoiding divisions in arithmetic circuits. LetF0, . . . , Fm be polynomialsof R such thatF0 is nonzero and thatF0 dividesFk in Q [X1, . . . , Xn] fork = 1, . . . , m.

The problem consists in computing the polynomialsF1F0

, . . . , Fm

F0by means

of an arithmetic circuitwithout divisionsby nonconstant polynomials ofR.For this sake we are going to use the well-known procedureVermeidung vonDivisionen[62] in the version of [45] which allows to control the height of theparameters involved.

Lemma 15 Let{Cn,m,d,h}n,m,d,h∈N be a family of boolean circuits which evalu-ates polynomialsF0, . . . , Fm of Z[X1, . . . , Xn] of degreed on integers of loga-rithmic heighth. LetLh :=L(n, m, d, h) and`h :=`(n, m, d, h) be the size anddepth ofCn,m,d,h respectively, and assume that the family{Cn,m,d,h}n,m,d,h∈N isuniform in spaceSh :=S(n, m, d, h). LetHh :=H(n, m, d, h) be the worst-caselogarithmic height of the output ofCn,m,d,h. Suppose further thatF0 is nonzeroand thatF0 dividesFk in Q [X1, . . . , Xn] for k = 1, . . . , m.

Then, there exists a family of boolean circuits which, on inputCn,m,d,h andan n-tupleγ ∈ Z n of logarithmic heighth1 such thatF0(γ ) 6= 0, computesa nonzero integerθ ∈ Z and polynomialsP1, . . . , Pm ∈ Z[X1, . . . , Xn] suchthatPk = θ Fk

F0holds fork = 1, . . . , m. This family of boolean circuits has size

LN + (mndHN)O(1) and depth N + O(

log(d) log(dHN)), and is uniform in

spaceSN + O(

log(dHNLN)), whereN :=max{h, h1, logd}.

482 G. Matera

Proof. Letγ = (γ1, . . . , γn) ∈ Z n be then-tuple of the statement of Lemma 15.Let ρ := F0(γ ) and let G0, . . . , Gm be the polynomials defined byGk(X1, . . . , Xn) = Fk(X1 +γ1, . . . , Xn +γn). LetQ :=ρ −G0 andθ :=ρd+1.Then, it holds true thatθGk

G0equals the sum of all the homogeneous components

of degree bounded byd of the polynomialQk :=(∑d

i=0 ρd−iQi)Gk (see [62]).We observe that the polynomialsQ1, . . . , Qm have degree bounded by

d2+d and can be computed by a family of boolean circuits of sizeLmax{h,h1}+1+mn(dHmax{h,h1}+1)

O(1) and depthO(`max{h,h1}+1 + log(d) log(dHmax{h,h1}+1

),

which is uniform in spaceSmax{h,h1}+1 + O(

log(ndHmax{h,h1}+1)).

Applying the Macro expansion lemma and Lemma 14 we obtain a family ofboolean circuits which computes the polynomialsθGk

G0for 1 ≤ k ≤ m with size

O(LN +mndO(1)HN)) and depth N +O(

log(d) log(dHN))

which is uniformin spaceO

(SN + log(nmdHNLN)

).

In order to conclude, we only have to notice that the identity

Pk := θFk

F0= θGk(X1 − γ1, . . . , Xn − γn)

G0(X1 − γ1, . . . , Xn − γn)

holds, and thatP1, . . . , Pm are polynomials belonging toZ[X1, . . . , Xn] whichcan be computed by a family of boolean circuits which satisfies the statementof the lemma.

3 The Resolution of Polynomial Equation Systems:Primitive Element Techniques

Let F1, . . . , Fs be polynomials inQ[X1, . . . , Xn] that define an equidimen-sional algebraic variety

V :={(x1, . . . , xn) ∈ C n; F1(x1, . . . , xn) = · · · = Fs(x1, . . . , xn) = 0} .

Since we are concerned withgeometricelimination, our problems are referred tothevarietyV rather than to the polynomialsF1, . . . , Fs themselves. A commonstrategy is to reduce the computational problems to the 0-dimensionalcase (i.e.to the case whereV is nonempty and finite), because in this case there existmethods that allow us important reductions of the complexity of the algorithms.In this section we are therefore going to assume thatV is zero-dimensional.

As it is well-known, the cardinalityδ of V equals the dimension of the finitedimensionalQ-vector space

B :=Q[X1, . . . , Xn]/√

(F1, . . . , Fs)

where√

(F1, . . . , Fs) denotes the radical of the ideal(F1, . . . , Fs) generatedby F1, . . . , Fs overQ[X1, . . . , Xn].

A central point is the description of thisQ-algebraB by means of someelements which, at the same time, provide a useful “pseudo-parameterization”


of V . For this aim, we observe that any linear formU ∈ Z[X1, . . . , Xn] thatseparatesthe points of the varietyV (i.e., U satisfiesU(x) 6= U(y) for anypair of distinct pointsx, y ∈ V ) yields a basis ofB in the following way: beingu the image ofU in B, the set of all the powers{1, u, . . . , uδ−1} forms sucha basis. If this is the case, we callu (and also the linear formU ) a primitiveelementof B (see [24], [43] or [26] for details).

From a primitive elementu we obtain a representation ofB as the quotientof Q[T ] modulo certain principal ideal (the ideal generated by the minimalpolynomial ofu). More precisely we have the following situation:

Let mu ∈ Z[T ] be the minimal primitive polynomial ofu and observethatmu has degreeδ. Then there exists a nonzero integerρ and polynomialsv1, . . . , vn ∈ Z[T ] of degree strictly less thanδ such that inQ[X1, . . . , Xn] theidentity:

(F1, . . . , Fs) =(mu(U), ρX1 − v1(U), . . . , ρXn − vn(U)

)holds (by the way the polynomialsv1, . . . , vn are uniquely determined up toscaling). With this notation we have the following identities betweenQ-algebras:

B = Q [X1, . . . , Xn]/(mu(U), ρX1 − v1(U), . . . , ρXn − vn(U))

∼= Q [T ]/(mu(T )

).

The multiplication tensor ofB can be easily obtained as follows: letM bethe companion matrix of the polynomialmu and letMXi

be the matrix of thehomothesy induced byXi in B with respect to the basis{1, u, . . . , uδ−1} for1 ≤ i ≤ n. Then we haveMXi

:=ρ−1vi(M).Notice that the curvec(T ) := (

v1(T ), . . . , vn(T ))

contains all the pointsof V and that the roots ofmu(T ) are the values of the parameterT whichyield the points of this curve which belong toV . This is what we mean by a“pseudo-parameterization” ofV .

Properly translated, these notions can be applied to the more general caseof equidimensional varieties of positive dimension. First, it is possible to per-form a linear change of variables(X1, . . . , Xn) → (Y1, . . . , Yn) such that thenew variablesY1, . . . , Yn−r are free with respect toV and that the followingextension of commutative rings, namely

R :=Q[Y1, . . . , Yn−r ] −→ Q[Y1, . . . , Yn]/√

(F1, . . . , Fs) =: B

is integral. Such a linear change of variables is called aNoether normalizationand we shall say that the variablesY1, . . . , Yn are in Noether positionwithrespect to the varietyV (see e.g. [28] for details).

It is clear that theQ-algebraB is reduced. Furthermore,B is a freeR-module of finite rank (cf. [32]). In such case, an elementu ∈ B is said aprimitive elementof the ring extensionR ⊆ B if it is the image of a linear form

484 G. Matera

and if the degree of the minimal polynomialmu of u equals the rank ofB asR-module.

Let K be the quotient field ofR andB ′ = K ⊗R B. An elementu ∈ B isa primitive element ofB if and only if, for δ := rankRB = dimKB ′, the set{1, u, . . . , uδ−1} is a basis of theK-vector spaceB ′.

For the aim of this paper it will not be necessary to describe theR-algebraB explicitly. In fact a suitable description of theK-algebraB ′ suffices. TheK-algebraB ′ is characterized by the following items:

• a basis of theK-vector spaceB ′

• for n − r + 1 ≤ i ≤ n, the matrixMXiof the homothesyηXi

: B ′ → B ′

induced by the multiplication byXi in B ′ with respect to the given basis(these matrices describe the multiplication tensor of theK-algebraB ′ andhence, of theR-algebraB)

These items are what we define as ageometric solutionof the varietyV (cf.[26]). The basis ofB ′ is obtained from a suitable primitive element ofB.

In this paper, the primitive elementu ∈ B will be chosen as the image inB ofa generic linear form in the variablesXn−r+1, . . . , Xn with coefficients inZ, thatis,u will be the image of a linear formU = λn−r+1Xn−r+1 + · · · + λnXn, withλi ∈ Z for n− r + 1 ≤ i ≤ n. BeingT a new indeterminate, the minimal poly-nomialmu(T ) of u as element of theR-algebraB (or equivalently as element oftheK-algebraB ′) will always be a monic polynomial ofQ[X1, . . . , Xn−r , T ](this is a consequence of the basic assumption that the variablesX1, . . . , Xn arein Noether position with respect toV ). This minimal polynomial will alwaysbe chosen as an element of the ringZ[X1, . . . , Xn−r , T ] = R[T ].

Finally, since by assumption{1, u, . . . , uδ−1} is a basis of theK-vectorspaceB ′, there exist forn − r + 1 ≤ i ≤ n polynomialsv

(u)i ∈ R[T ] and

nonzero elementsρ(u)i ∈ R such thatρ(u)

i Xi − v(u)i (U) belongs to the extension

ideal√

F1, . . . , Fs(K)

of√

F1, . . . , Fs in K[Xn−r+1, . . . , Xn]. In particular, wehave the following identity between ideals ofK[Xn−r+1, . . . , Xn]:√

(F1, . . . , Fs)(K) = (

mu(U), ρ(u)1 Xr+1 − v

(u)r+1(U), . . . , ρ(u)

n Xn − v(u)n (U)

)Furthermore, ifM is the companion matrix of the homothesyηu with respect tothe basis{1, u, . . . , uD−1}, the matricesMXn−r+1, . . . , MXn

which characterizethe multiplication tensor of theK-algebraB ′ (or equivalently of theR-algebraB) are obtained from the identity:

ρ(u)i · MXi

= v(u)i (M)

for n − r + 1 ≤ i ≤ n.

3.1 The construction of a primitive element

We develop in this subsection the tools we need in the construction of a primi-tive elementu (and the pseudo-parameterization associated to it). This is done


following the ideas of [45], with the additional aim to prove the uniformity ofthe underlying algorithms.

First, we are going to treat the particular case of a variety defined by twopolynomials in two separated variables.

Lemma 16 Let{Cm,δ1,δ2,h}m,δ1,δ2,h∈N be a family of boolean circuits which eval-uates polynomialsf0, . . . , fδ1, g0, . . . , gδ2 of Z[Y1, . . . , Ym] on integers of log-arithmic heighth. LetLh :=L(m, δ1, δ2, h) and`h :=`(m, δ1, δ2, h) be the sizeand depth ofCm,δ1,δ2,h, and assume that the family{Cm,δ1,δ2,h}m,δ1,δ2,h∈N is uni-form in spaceSh :=S(m, δ1, δ2, h). LetHh :=H(m, δ1, δ2, h) be the worst-caselogarithmic height of the output ofCm,δ1,δ2,h.

Let R := Z[Y1, . . . , Ym], K := Q(Y1, . . . , Ym) and letK be the algebraicclosure ofK. LetF(X) :=f0 + · · · + fδ1X

δ1 andG(Y) :=g0 + · · · + gδ2Yδ2 be

polynomials which are squarefree inK[X, Y ], let I := (F(X), G(Y )

)be the

ideal generated byF(X) andG(Y) in K[X, Y ] and letW ⊂ K2

be the zero-dimensional variety defined byF(X) andG(Y). LetU :=αX + Y be a linearform that represents a primitive element of theK-algebraB := K[X, Y ]/Iand letδ :=δ1 · δ2.

Then, there exists a family of boolean circuits with sizeLh + (δh1Hh)O(1)

and depth`h + O(

log(δ) log(δh1Hh)), which is uniform in spaceSh +

O(

log(δHhLh))

such that, given an integerα of logarithmic heighth1 such thatαX+Y is a primitive element ofB, computes on integers of logarithmic heighth

the coefficients inZ[Y1, . . . , Ym] of polynomialsq, v1, v2 ∈ Z[Y1, . . . , Ym][T ]and an elementρ ∈ Z[Y1, . . . , Ym] with the following properties:

• deg(q) = δ andmax{deg(v1), deg(v2)} < δ.• (q(U), ρX − v1(U), ρY − v2(U)) ⊆ I.

Proof. We follow the ideas of [45, Lemma 26]. It is clear thatB is aK-vectorspace of dimensionδ and that the image inB of the set

B :={XiY j/0 ≤ i < δ1, 0 ≤ j < δ2}defines a basis asK-vector space ofB.

For the sake of definiteness let us fix the lexicographical order on the setB. Then the matricesMX, MY ∈ Kδ×δ of the homothesiesηX andηY of theK-algebraB with respect to the basisB can be obtained directly from thecoefficients ofF andG. Letu be the image of the linear formU in theK-algebraB and denote byMu the matrix of the homothesyηu induced by multiplicationby u in B with respect to the basisB. Observe that the identityMu = α ·MX +MY holds. Notice that, beinga, b ∈ R the leading coefficients ofF andG

respectively, the matrixabMu has all its entries inR.By the Cayley–Hamilton theorem, the characteristic polynomialχ ∈ k[T ]

of the matrixMu annihilatesu over B. Since deg(χ) = δ holds, we havethat χ is the minimal polynomial ofu over B. Defineq(T ) := χabMu

(abT )

486 G. Matera

and observe that(ab)δχMu(T ) = χabMu

(abT ) = q(T ) holds. Thus we haveq(T ) ∈ R[T ] and in particularq(U) ∈ I. Moreover the coefficients ofq can becomputed by a division-free arithmetic circuit inR of sizeδO(1) and nonscalardepthO(logδ).

In order to find the elementρ ∈ R and the polynomialsv1, v2 ∈ R[T ] ofthe statement of the lemma, we observe that

B′ := {1, u, . . . , uδ−1}is a K-vector space basis ofB, and that there exists a nonsingular matrixM ∈ Kδ×δ such that moduloI the following congruence relation holds:

1U

U2

...

Uδ−1

≡ M ·

1X

Y...

Xdeg(F )−1Y deg(G)−1

.

Multiplying both sides of this congruence relation by the adjoint matrixAdj (M)

of M, we conclude:

Adj (M) ·

1U

U2

...

Uδ−1

≡ det (M) ·

1X

Y...

Xdeg(F )−1Y deg(G)−1

.

Since thei-th column of the matrixM can be obtained as the first columnof the matrix (Mu)

i for i = 0, . . . , δ − 1, we see that(ab)δM has all itsentries inR. Thus(ab)δ

2Adj (M) = (ab)δAdj

((ab)δM

)has its entries inR and

(ab)δ2 · det (M) belongs toR. Letρ := (ab)δ

2−δ · det (M) and letv1(T ), v2(T )

be the polynomials ofR[T ] of degree less thanδ whose coefficient vectors areobtained from the second and third row of(ab)δAdj

((ab)δM

)respectively.

Then the degree ofv1, v2 is strictly less thanδ and modulo the idealI we havethe following congruences:

ρ · X ≡ v1(U)

ρ · Y ≡ v2(U).

We now analyze the underlying algorithm, in order to prove that it can be per-formed by a family of boolean circuits whose size, depth and space-uniformitysatisfy the statement of the lemma. For this purpose we apply the Macro expan-sion lemma using the polynomial maps defined by the polynomialsf0, . . . , fδ1,g0, . . . , gδ2, the integer product, the matrix product of(δ × δ)-matrices, the co-efficients of the characteristic polynomial of a(δ × δ)-matrix and the adjointmatrix of a(δ × δ)-matrix. The algorithm described above computes the poly-nomialsf0, . . . , fδ1, g0, . . . , gδ2 and computes the matricesabMX, abMY and


abMu. We observe that the matricesaMX andaMY can be obtained by a simplemapping from the polynomialsf0, . . . , fδ1, g0, . . . , gδ2 using spaceO(logd).We also observe that the entries of the matricesabMX andabMY are polyno-mials ofZ[Y1, . . . , Ym] with worst-case logarithmic heightO(Hh) on integersof logarithmic heighth.

Then we compute the matrixabMu and the coefficients of its characteristicpolynomial using Lemma 6. An inner product between the vector formed by thecoefficients of this characteristic polynomial and the vector

(1, ab, (ab)2, . . . ,

(ab)δ)yields the coefficients of the polynomialχabMu(abT ), which have worst-

case logarithmic heightO(δh1Hh) on integers of logarithmic heighth.Then we obtain the columns of the matrix(ab)δM by computing the first

column of the matrices(ab)δ−1abMu, (ab)δ−2(abMu)2, . . . , (abMu)

δ. Finallywe computeρ = det

((ab)δM

)and the second and third row of the matrix(ab)δ

Adj((ab)δM). Notice thatM has logarithmic heightO(δh1Hh) on integers of

logarithmic heighth. Therefore, combining the Macro expansion lemma withLemmata 4, 5, 6 and 8 we obtain a family of boolean circuits satisfying thestatement of the lemma.

Now we are going to reduce the general case of an equidimensional varietyto the case of two polynomials in two separated variables. We assume now thatwe have a procedure that determines, given a linear formU , a polynomialp thatannihilatesU on the given zero-dimensional varietyV under consideration (inthe next section we are going treat the problem how to find such a polynomialp).

For the rest of this subsection we will use the following notation:R will denote a polynomial ringZ[Y1, . . . , Ym] with quotient fieldK andK

will denote the algebraic closure ofK. Likewise, we will consider polynomialsF1, . . . , Fs ∈ R[X1, . . . , Xn] that define a zero-dimensional varietyV overK

n

of cardinalityδ and generate a radical idealI in K[X1, . . . , Xn].Let T1, . . . , Tn be new indeterminates. We shall denote byZj the linear

formZj := T1X1+· · ·+Tj−1Xj−1+Tj+1Xj+1+· · ·+TnXn for j = 1, . . . , n

and byIT the ideal generated byF1, . . . , Fs in K(T1, . . . , Tn)[X1, . . . , Xn].We shall use the following result:

Lemma 17 ([45]; Lemma 25) Let be given the following items:

• for everyj = 1, . . . , n, a polynomialGj ∈ R[T ], squarefree inK[T ], ofdegreedeg(Gj) ≤ δ such thatGj(Xj) belongs toI.

• for everyj = 1, . . . , n, a polynomialHj ∈ R[T1, . . . , Tn][T ], squarefreein K(T1, . . . , Tn)[T ] and monic inT except for factors inR, of degreedegT (Hj ) ≤ δ, such thatHj(Zj ) belongs toIT

Then, there exists an arithmetic circuitβ of sizeO(nδO(1)) and nonscalardepthO

(log(nδ)

)that computes a polynomialQ ∈ R[T1, . . . , Tn] which ver-

ifies the following property: for everyn-tuple (λ1, . . . , λn) ∈ R n such that

488 G. Matera

F(λ1, . . . , λn) 6= 0 holds, the linear formU := λ1X1 + · · · + λnXn is aprimitive element ofV .

From Lemma 17 and the Schwartz–Zippel test we obtain the coefficients(λ1, . . . , λn) ∈ Zn of a linear form which constitutes a primitive element forthe varietyV defined by the polynomialsF1, . . . , Fs . In the following lemmawe describe how we obtain a geometric solution of the equidimensional varietyV which is defined by the polynomialsF1, . . . , Fs .

Lemma 18 Let notations and assumptions be as in Lemma17. Let{Cm,n,δ,h}m,n,δ,h∈N be a family of boolean circuits which evaluates the coef-ficients inR[T1, . . . , Tn] of the polynomialsG1, . . . , Gn, H1, . . . , Hn on inte-gers of logarithmic heighth. LetLh := L(m, n, δ, h) and`h := `(m, n, δ, h)

be the size and depth ofCm,n,δ,h respectively, and assume that the family{Cm,n,δ,h}m,n,δ,h∈N is uniform in spaceSh := S(m, n, δ, h). LetHh := H(m, n,

δ, h) be the worst-case logarithmic height of the output ofCm,n,δ,h.Assume further that we are given ann-tuple(λ1, . . . , λn) of Zn of logarith-

mic heighth1 such that the linear formU := λ1X1+· · ·+λnXn is a primitive ele-ment of the algebraic varietyV and that the variablesX1, . . . , Xn are in Noetherposition. Then there exists a family of boolean circuits with sizeLmax{h,h1} +(nδh1Hmax{h,h1})

O(1) and depth`max{h,h1}+ O(

log(δ) log(nδh1Hmax{h,h1})),

which is uniform in spaceSmax{h,h1}+ O(

log(nδh1Hmax{h,h1}Lmax{h,h1}))

suchthat, given the integersλ1, . . . , λn computes on integers of logarithmic heighth

elementsρ1, . . . , ρn ∈ R and the coefficients inR of polynomialsq, v1, . . . , vn ∈R[T ] with the following properties:

• degT (q) = δ andq(U) ∈ I• for everyi = 1, . . . , n we havedegT (vi) < δ andρiXi − vi(U) ∈ I.

Proof. Following [45, Proposition 27], letZj(λ) := λ1X1+· · ·+ λjXj +· · ·+λnXn. By replacing the variables(T1, . . . , Tn) by (λ1, . . . , λn) in the polynomi-alsH1, . . . , Hn we obtain polynomialsh1(T ) := H1(λ1, . . . , λn, T ), . . . , hn(T )

:= Hn(λ1, . . . , λn, T ) in R[T ] such thathj

(Zj(λ)

) ∈ I holds. Observe that thecoefficients with respect toT of the polynomialsh1, . . . , hn can be computedby a family of boolean circuits of sizeLmax{h,h1} and depth max{h,h1} which isuniform in spaceSmax{h,h1}.

Since the linear formU represents a primitive element ofV , we see thatUseparates the points of the set{Gj(Xj) = 0, hj

(Zj(λ)

) = 0} ⊂ K2. In order to

apply Lemma 16 we compute a square-free representationhj (T ) of hj (T ) for1 ≤ j ≤ n. Applying Lemma 12 we see that the polynomialsh1, . . . , hn can becomputed by a family of boolean circuits with sizeLmax{h,h1}+nδO(1)Hmax{h,h1}and depthO

(`max{h,h1} + log(δ) log(δHmax{h,h1})

)which is uniform in space

Smax{h,h1} + O(

log(nδHmax{h,h1}Lmax{h,h1})).

Applying for each 1≤ j ≤ n Lemma 16 to theK-algebrak[Xj, Zj (λ)]/(Gj(Xj), hj (Zj (λ))) with U = λjXj +Zj(λ) as primitive element, we obtainelementsρj ∈ R and polynomialsvj ∈ R[T ] that verify the condition


ρjXj − vj (U) ∈(Gj(Xj), hj

(Zj(λ)

)) ⊂ I

Applying the Macro expansion lemma and Lemma 16 we conclude that allthese computations can be performed by a family of boolean circuits with sizeLmax{h,h1} + nh1(δHmax{h,h1})

O(1) and depth `max{h,h1} + O(

log(δ)

log(δh1Hmax{h,h1})), which is uniform in spaceSmax{h,h1} + O

(log(nδh1

Hmax{h,h1}Lmax{h,h1})).

3.2 The computation of the isolated points of a variety

Let R := Z[Y1, . . . , Ym] be a polynomial ring, letK := Q(Y1, . . . , Ym) bethe quotient field ofR and letK be the algebraic closure ofK. Let be givenpolynomialsF1, . . . , Fs in R[X1, . . . , Xn] and letV ⊂ K

nbe the algebraic

variety defined as the set of common zeroes of these polynomials.In this subsection we are going to compute the isolated points of the variety

V . This is a necessary preliminary step for the determination of the multiplica-tion tensor of a reduced complete intersection and allows also to compute thedimension of the varietyV .

As we have seen in the previous subsection, we need to develop a procedurewhich computes, for a given linear formU ∈ R[X1, . . . , Xn], a polynomialp ∈ R[T ] that annihilatesU on the isolated points ofV .

First we deal with the case where the isolated points ofV are locally com-plete intersection.

Lemma 19 Let{Cm,n,d,h}m,n,d,h∈N be a family of boolean circuits which evalu-ates the coefficients inR of polynomialsF1, . . . , Fn ofR[X1, . . . , Xn] of degreed and the coefficientsλ1, . . . , λn ∈ R of a linear formU := λ1X1+· · ·+λnXn

on integers of logarithmic height bounded byh. Assume further that thesecoefficients have degree bounded byd. Let Lh := L(m, n, d, h) and `h :=`(m, n, d, h) be the size and depth ofCm,n,d,h respectively, and assume thatthe family {Cm,n,d,h}m,n,d,h∈N is uniform in spaceSh := S(m, n, d, h). LetHh := H(m, n, d, h) be the worst-case logarithmic height of the output ofCm,n,d,h.

Let(F1, . . . , Fn) denote the ideal generated byF1, . . . , Fn overR[X1, . . . ,

Xn] and letV be the algebraic variety defined byF1, . . . , Fn in Kn. Then,

there exists a family of boolean circuits which, given random integersγε, γ0, . . . ,

γn, γT of logarithmic heightN := O(n log(nd)

), computes the coefficients of a

polynomialp ∈ R[T ] such thatp(U) vanishes on the isolated points of the vari-etyV . This family of boolean circuits has sizeLmax{N,h}+

((nd)nHmax{N,h}

)O(1),

depthO(`max{N,h} + n2 log(nd) log(ndHmax{N,h})

)and is uniform in space

Smax{N,h} + O(n log(ndHmax{N,h}Lmax{N,h}

).

Proof. Following e.g. [28], we produce a homotopic deformation of the givenequation system in order to reduce the problem to the zero-dimensional

490 G. Matera

projective case. For this purpose, we introduce new indeterminatesX0 andε, and consider for 1≤ i ≤ n the polynomials

Gi := X0hFi + εX

1+deg(Fi)

i

(herehFi denotes the homogenization ofFi with respect to the homogenizationvariableX0). As elements ofR [ε][X0, . . . , Xn], the polynomialsG1, . . . , Gn

are homogeneous and define a zero-dimensional subvariety ofPn(K (ε))

[28, Lemme 3.3.3]. The dense representation ofG1, . . . , Gn is obtained di-rectly from that ofF1, . . . , Fn.

We apply now arguments on the regularity of the Hilbert function of thegraduated ring

A := K (ε)[X0, . . . , Xn]/(G1, . . . , Gn) = A0 ⊕ A1 ⊕ · · · ⊕ Aj ⊕ · · ·Let us notice the following facts (see [51] or [9]):

• LetN := nd. ThenAN andAN+1 are finite dimensionalK (ε)-vector spacesof the same dimension, sayD, with D ≤ (d + 1)n.

• Since the ideal(G1, . . . , Gn) does not have any zero in the hyperplaneX0 = 0, the homothesyηX0 : AN −→ AN+1 is an isomorphism.

We consider the endomorphism:

φ := η−1X0

ηu : AN −→ AN

whereu is the image ofU in A. By the theorem of Cayley–Hamilton, the char-acteristic polynomialP(T ) of φ verifiesP(φ) ≡ 0, and hence so doesηD

X0P(φ).

SinceX0 is not a zero divisor ofA, the nonzero polynomialP(X0, T ) :=XD

0 P(

TX0

)= det (X0Mφ − T · Id) has the property thatP(X0, U) belongs to

the ideal generated byG1, . . . , Gn in K (ε)[X0, . . . , Xn] (hereMφ denotes thematrix of the linear endomorphismφ with respect to a suitableK (ε)-vectorspace basis ofAN ).

In order to compute the polynomialP(X0, T ), it is necessary to computefirst the matricesMX0 andMU of the homothesiesηX0 andηU in some suitablebasisB andB′ of AN andAN+1 respectively. For this purpose we shall follow[45].

Let Fs := {m1, . . . , mrs} be the set of all monomials of degrees in the

variablesX0, . . . , Xn (recall thatrs =(

s + n

n

)holds). Multiplying all the

monomials of degreeN − (1 + deg(Fi)

)by the polynomialGi for 1 ≤ i ≤ n,

we obtain the matrixQ1 ∈ R[ε]rn−(d+1)×rN of the relations of the monomials inFN moduloIN . Computing a square submatrixQ1 of Q1 of maximal rankwe obtain a monomial basis ofAN : we consider the subsetB := {e1, . . . , eD}of FN in which the monomialsej correspond to the columns ofQ1 not chosento form Q1.


Similarly, we compute a square submatrix of maximal rankQ2

of the matrixQ2 of relations inAN+1. Then we extract a monomial basisB′ := {e′

1, . . . , e′D} ⊆ FN+1 of AN+1, and its remainder set{v′

1, . . . , v′t} :=

FN+1 \ B′. Moreover, we obtain an elementα := det (Q2) ∈ R and a matrixB ∈ Rt×D, given by the remaining columns ofAdj (Q2)Q2, such that

α

v′1...

v′t

= B

e′1...

e′D

holds.Now we are able to compute for 1≤ i ≤ n matricesMαXi

= αMXi,

whereMαXiandMXi

represent the matrices of theK (ε)-algebra morphismsηαXi

: AN → AN+1 andηXi: AN → AN+1 defined by multiplication of the

elements ofAN by αXi andXi respectively with respect to the basisB andB′.For anyei ∈ B, the elementαXiei ∈ AN+1 is either a monomial of the formαe′

j or a monomial of the formαv′j . In the first case the corresponding column

of MαXiis a vector with all its coordinates zero except for itsj -th column where

α occurs. In the second case this column is simply thej -th row of the matrixB. In this way we compute easily the matricesMαX0 andMαu = ∑n

i=0 λiMαXi.

In order to avoid divisions in the computation of the polynomialP (X0, T ),we compute the following multiple1(ε, X0, T ) of it in R [ε][X0, T ]:

1(ε, X0, T ) =D∑

j=1

pj(ε)TjX

D−j

0 := det (MλX0)DP (X0, T )

= αDdet (MαX0)DP (X0, T )

= det (X0Adj (MαX0)Mαu − det (MαX0)T · Id) .

Dividing the polynomial1(ε, X0, T ) by the greatest possible power ofε, weobtain a new polynomial11(ε, X0, T ) such thatp(T ) := 11(0, 1, T ) is thepolynomial we are looking for [28, Proposition 3.3.4].

We observe that the whole procedure manipulates matrices whose entries arepolynomials in the variablesY1, . . . , Ym, ε of degree bounded by(nd)O(n). Thisfollows from the hypothesis on the degree of the coefficients of the polynomialsF1, . . . , Fn, U and the size of the matrices involved. Therefore, from Schwartz–Zippel test we deduce that there exist integer parametersγ, γ1, . . . , γn of log-arithmic heightO

(n log(nd)

)such that, applying the strategy of Lemma 9 to

compute submatrices of maximal rank of the matricesP(γε, γ1, . . . , γn) andQ(γε, γ1, . . . , γn) (hereP(γε, γ1, . . . , γn) andQ(γε, γ1, . . . , γn) denote the in-teger matrices obtained by the evaluation of the entries of the matricesP andQ in ε = γε, Y1 = γ1, . . . , Ym = γm) also yields submatricesP , Q of maximalrank of the matricesP, Q. We assume that our input parametersγε, γ1, . . . , γn

satisfy this property.

492 G. Matera

We now analyze the algorithm to prove that it can be performed by afamily of boolean circuits whose size, depth and space-uniformity satisfy thestatement of the lemma. Notice that the dense representation of the poly-nomialsG1, . . . , Gn can be computed by family of boolean circuits of sizeLh + O(dnHh) and depthO

(`h + n log(dHh)


Sh + n log(dHh).The mapping computing the entriespij , qij of the matricesP, Q from

the coefficients of the polynomialsG1, . . . , Gn can be easily described inspaceO

(n log(nd)

), because it only requires to manipulate the monomials in

n + 1 variables of degree at mostrN+1 = nd + 1, which form aK (ε)-vectorspace of dimension(nd)O(n). We evaluate the entries of the matricesP, Q inε = γε, Y1 = γ1, . . . , Ym = γm and determine which rows and columns ofP

andQ are chosen to form the submatricesP andQ of maximal rank. For thispurpose we apply the ideas of Lemma 9 to the matricesP(γε, γ1, . . . , γn) andQ(γε, γ1, . . . , γn). The family of boolean circuits which performs these opera-tions has sizeLN + (

(nd)nHN

)O(1), depthO

(`N +n2 log(nd) log(ndHN)

)and

is uniform in spaceSN + O(n log(ndHNLN)

).

Once we have computed the rows and columns ofP andQ chosen to formthe submatricesP andQ, we obtain the matricesP andQ. We now apply theMacro expansion lemma using the polynomial maps defined by the entries ofthe matricesP andQ and the computation of the determinant, adjoint matrixand matrix product of

((d + 1)n × (d + 1)n

)-matrices to prove that there exists

a family of boolean circuits computing the matrixB with sizeLmax{N,h} +((nd)nHmax{N,h}

)O(1), depthO

(`max{N,h} + n2 log(nd) log(ndHmax{N,h})

)and

is uniform in spaceSmax{N,h} + O(n log(ndHmax{N,h}Lmax{N,h})

). We observe

that the entries of the matrixB have worst-case logarithmic height(dnHh)O(1)

on integers of logarithmic heighth.The mapping which computes the matricesMαX0, . . . , MαXn

from the en-tries of the matrixB can be easily described in space(nd)O(n). Applyingagain the Macro expansion lemma using this mapping and the computationof adjoint matrix and determinant as polynomial maps we obtain a family ofboolean circuits computing the polynomialK(ε, X0, T ) with sizeLmax{N,h}++(

(nd)nHmax{N,h})O(1)

, depthO(`max{N,h}+n2 log(nd) log(ndHmax{N,h})

)and

is uniform in spaceSmax{N,h} + O(n log(ndHmax{N,h}Lmax{N,h})

). The coeffi-

cients of the polynomial K(ε, X0, T ) have worst-case logarithmicheight(dnHh)

O(1) on integers of logarithmic heighth.In order to determine the greatest power ofε which divides the poly-

nomial K(ε, X0, T ) we observe thatK(ε, X0, T ) is a polynomial of degreeat most(nd)O(n) and hence its coefficients with respect toε are polynomi-als of Z[X0, Y1, . . . , Ym, T ] of degree at most(nd)O(n). Therefore applyingSchwartz–Zippel test we see that there exist integers(γ0, γ1, . . . , γn, γT ) oflogarithmic heightO

(n log(nd)

)such that any coefficientp of the dense repre-

sentation ofK(ε, X0, T ) with respect toε is the polynomial zero if and only the


evaluationp(γ0, γ1, . . . , γn, γT ) of p in X0 = γ0, Y1 = γ1, . . . , Yn = γn, T =γT equals zero. We assume that the input parametersγ0, γ1, . . . , γn, γT satisfythis property.

Then we interpolate the polynomial withK(ε, X0, T ) respect toε and eval-uate the coefficients inX0 = γ0, Y1 = γ1, . . . , Yn = γn, T = γT to determinewhich is the least nonzero coefficient. From the bounds of Lemma 13 we deducethe complexity bounds of the statement of Lemma 19.

We remark that the algorithm of Lemma 19 requires the manipulation ofmatrices of size(nd)O(n) × (nd)O(n), instead of thedO(n2) × dO(n2)-matricesconsidered in [53].

Now we are in conditions to treat the general problem of the determinationof the isolated points of a given algebraic variety.

Proposition 20 Let{Cm,n,s,d,h}m,n,s,d,h∈N be a family of boolean circuits whichevaluates the coefficients inR of polynomialsF1, . . . , Fs of R[X1, . . . , Xn] ofdegreed on integers of logarithmic height bounded byh. Assume further thatthese coefficients have degree boundedd. LetLh := L(m, n, s, d, h) and`h :=`(m, n, s, d, h) be the size and depth ofCm,n,s,d,h respectively, and assume thatthe family{Cm,n,s,d,h}m,n,s,d,h∈N is uniform in spaceSh := S(m, n, s, d, h). LetHh := H(m, n, s, d, h) be the worst-case logarithmic height of the output ofCm,n,s,d,h.

LetV ⊂ Kn

be the algebraic variety defined by the polynomialsF1, . . . , Fs .Then, there exists a family of boolean circuits which, given randomintegersγε, γ0, . . . , γn, γT , γ1, . . . , γn, λ1, . . . , λn of logarithmic heightN :=O

(n log(nsd)

), computes the coefficients of a geometric solution of a zero-

dimensional variety containing the isolated points ofV . This family of booleancircuit has sizeLmax{N,h} + (

s(nd)nHmax{N,h})O(1)

, depth O(`max{N,h} +

n2 log(nd) log(nsd Hmax{N,h}))

and is uniform in spaceSmax{N,h} +O

(n log(nsdHmax{N,h} Lmax{N,h})

).

Proof. First of all we replace the varietyV by another variety containing be-tween its isolated points all the isolated points ofV and such that this new varietyis given byn equations. In this sense, for a givenn-tupleγ := (γ1, . . . , γn) ∈ Zn

we define fori = 1, . . . , n the followingn polynomials:

Fγi:= F1 + γiF2 + · · · + γ s−1

i Fs .

From [28, Lemme 3.4.1] it follows that there exists a polynomialQ(T1, . . . , Tn)

of degree bounded bysdn such that for anyn-tupleγ ∈ Zn withQ(γ1, . . . , γn) 6=0 the varietyV (γ ) defined byFγ1, . . . , Fγn

contains all the isolated points ofV

as isolated points. Hence, any polynomialP ∈ R[T ] that annihilates a givenlinear formU over the isolated points ofV (γ ) annihilatesU over the isolatedpoints ofV .

494 G. Matera

Therefore applying Schwartz–Zippel test we deduce that there exists anintegern-tuple (γ1, . . . , γn) of logarithmic height bounded byO

(n log(sd)

)which does not annihilateQ. We assume that our input parametersγ1, . . . , γn

satisfy this property.In order to compute the polynomialsFγ1, . . . , Fγn

we first compute thenumbersγ j

i for 1 ≤ i ≤ n and 1≤ j ≤ s. The coefficient corresponding to amonomial, saym, of Fγi

is computed as the inner product of the vector of thecoefficients ofF1, . . . , Fs corresponding tom with the vector(1, γ1, . . . , γ

s−11 ).

We observe that the resulting coefficients have logarithmic height bounded byHh + O

(sn log(sd)

).

From Lemma 17 and the Schwartz–Zippel test we see that there exist inte-gersλ1, . . . , λn of logarithmic heightO

(n log(nd)

)such that the linear form

U := λ1X1 + · · · + λnXn separates the isolated points ofV . We assume thatour input parametersλ1, . . . , λn satisfy this property.

Then we apply Lemma 19 in order to produce polynomials that annihilatethe linear formsX1, . . . , Xn, Zj := λ1X1 + · · · + λjXj + · · · + λnXn forj = 1, . . . , n andZ := λ1X1 + · · · + λnXn on the isolated points ofV . Letus observe that the coefficients of the computed polynomials have logarithmicheight bounded by(sdnHh)

O(1). Finally we apply Lemma 18 in order to producea geometric solution of the isolated points ofV .

Let us analyze the size, depth and space-uniformity of the algorithm we havejust described. For this purpose we apply the Macro expansion lemma using thepolynomial maps defined by the coefficients of the polynomialsFγ1, . . . , Fγn

and the maps given by the application of Lemma 19 and Lemma 18. Fromthe bounds of Lemmata 19 and 18 we deduce that the whole procedure canbe performed by means of a family of boolean circuits with sizeLmax{N,h} +(s(nd)nHmax{N,h}

)O(1)and depthO

(`max{N,h}+n2 log(nd) log(nsdHmax{N,h})

)which is uniform in spaceSmax{N,h} + O

(n log(nsdHmax{N,h}Lmax{N,h}

).

4 The Division Modulo a Reduced Complete Intersection Ideal

In this section, following the development made in [45], we treat a crucialproblem for the algorithmic division modulo a given polynomial ideal: thecomputation of the quotient of two polynomials modulo a radical completeintersection ideal.

This quotient will be the result of the action of a matrix whose entries canbe computed in suitable nonscalar depth. The computation of this matrix willbe essential in order to obtain the complexity bounds we are looking for.

The main ingredient for our division theorem is a duality theory based onthe existence of traces in Gorenstein algebras, whose basic facts we explainbelow. For proofs we refer to [48], Appendices E and F.


4.1 General trace theory

Let R be a polynomial ringZ[T1, . . . , Tm], let K be its quotient field and letR[X1, . . . , Xn] be the polynomial ring inn variables with coefficients inR. LetF1, . . . , Fn be a regular sequence of polynomials inR[X1, . . . , Xn] of degreeat mostd in the variablesX1, . . . , Xn, generating a radical ideal(F1, . . . , Fn).

We consider theR-algebraB given as the quotient ofR[X1, . . . , Xn] forthis ideal:

B := R[X1, . . . , Xn]/(F1, . . . , Fn) .

We assume that the ring extensionR → B represents a Noether normalizationof the varietyV (F1, . . . , Fn)defined by the polynomialsF1, . . . , Fn in a suitableaffine space. Thus,B is a freeR-module of rank bounded by the degree of thevariety V (F1, . . . , Fn). Furthermore, theR-algebraB is Gorenstein and thefollowing statements are based on this fact.

We considerB∗ := HomR(B, R) with theB-module structure defined bymeans of the scalar product

B × B∗ −→ B∗

that associates to each(b, τ ) in B ×B∗ theR-linear morphismb · τ : B −→ R

defined by(b · τ)(x) := τ(bx) for every elementx ∈ B.Since theR-algebraB is Gorenstein, its dualB∗ is a freeB-module of rank

1. Any elementσ of B∗ that generatesB∗ asB-module is called atraceof B.There exist two relevant elements ofB∗ which are denotedTr andσ . The firstone,Tr, is called thecanonical traceof B and is defined in the following way:given an elementb ∈ B, let us denote byηb : B −→ B the homothesy inducedby the multiplication byb. The imageTr(b) by Tr is defined as the ordinarytrace of the endomorphismηb of B (notice that this definition makes sense sinceB is a freeR-module). In order to introduceσ (which will be called a trace ofB), we need some additional notations. For any elementG ∈ R[X1, . . . , Xn]we denote byG its image inB, i.e. the residue class ofG modulo the ideal(F1, . . . , Fn). Let Y1, . . . , Yn be new variables and letY := (Y1, . . . , Yn). Let1 ≤ j ≤ n and letF (Y)

j := Fj(Y1, . . . , Yn) be the polynomial ofR[Y1, . . . , Yn]obtained substituting inFj the variablesX1, . . . , Xn byY1, . . . , Yn. We considerthe polynomial

F(Y)j − Fj =

n∑k=1

ljk(Yk − Xk) ∈ R[X1, . . . , Xn, Y1, . . . , Yn] (4)

where theljk are polynomials belonging toR[X1, . . . , Xn, Y1, . . . , Yn] of totaldegree bounded by(d − 1) (observe that theljk ’s are not uniquely determinedby the sequenceF1, . . . , Fn). We consider now the determinant1 of the matrix(ljk)1≤j,k≤n which can be written (non uniquely) as

1 =∑m

am(X1, . . . , Xn)bm(Y1, . . . , Yn) ∈ R[X1, . . . , Xn, Y1, . . . , Yn] ,

496 G. Matera

where theam’s are elements ofR[X1, . . . , Xn] and thebm’s are elements ofR[Y1, . . . , Yn]. The polynomial1 is called a pseudo-jacobian determinant ofthe regular sequenceF1, . . . , Fn. Observe that the polynomialsam andbm canbe chosen having degree bounded byn(d − 1) in the variablesX1, . . . , Xn andY1, . . . , Yn respectively.

Let cm ∈ R[X1, . . . , Xn] be the polynomial obtained substituting the vari-ablesY1, . . . , Yn by X1, . . . , Xn in bm. If J is the residue class of the JacobiandeterminantJ (F1, . . . , Fn) in B we have the following identity:

J =∑m

am · cm .

Since the ideal(F1, . . . , Fn) is radical we see thatJ is not a zero divisorof B. Furthermore, the image of the polynomial1 in the residue class ringR[X1, . . . , Xn, Y1, . . . , Yn] modulo the ideal(F1, . . . , Fn, F

(Y )1 , . . . , F (Y )

n ) isindependent of the particular choice of the matrix(lkj )1≤k,j≤n. This justifiesthe name “pseudo-jacobian” for the polynomial1. With these notations thereexists a unique traceσ ∈ B∗ such that the following identity holds inB:

1 =∑m

σ(am) · cm .

The main property of the traceσ is known as the “trace formula” (“Tate’s traceformula” of [48, Appendix F] or [39] being special case of it). The trace formulais the following statement: for anyG ∈ R[X1, . . . , Xn] the identity

G =∑m

σ(G · am) · cm (5)

holds true inB. Notice that the polynomial∑

m σ(G · am) · cm ∈ R[X1, . . . , Xn]of the identity (5) has degree in the variablesX1, . . . , Xn bounded byn(d −1).

We shall apply this trace formula in order to solve the lifting problem: givena polynomialG ∈ R[X1, . . . , Xn], find a polynomialG1 ∈ R[X1, . . . , Xn] ofdegree in the variablesX1, . . . , Xn bounded byn(d − 1), such thatG1 = G

holds inB.As one sees easily, the trace formula (5) solves this problem since it allows

us to choose forG1 the polynomial

G1 :=∑m

σ(G · am) · cm . (6)

By means of the element1 it is possible to describe the relation between the

traceσ just introduced and the canonical traceTr (see [48], Corollary E.19 andexample F.19), namely:

J · σ = Tr . (7)


4.2 The computation of the matrix that performs the division

Under the assumptions made in the previous subsection, we deduce from Propo-sition 20 that we may assume that we are given a geometric solution for theK-algebraK ⊗R B. This means that we dispose over the following items:

• a linear formU := λ1X1 + · · · + λnXn ∈ Z[X1, . . . , Xn] such thatu := U

is a primitive element forK ⊗R B.• a polynomialp ∈ R[T ] of degree bounded bydn such thatp(U) belongs

to the ideal(F1, . . . , Fn).• a nonzero elementρ ∈ R and polynomialsv1, . . . , vn ∈ R[T ] of degree

bounded bydn such that for any 1≤ j ≤ n ρXj − vj (U) belongs to(F1, . . . , Fn).

We can now state the main result of this section:

Proposition 21 Let notations and assumptions be as above. Let{Cm,n,d,h}m,n,d,h∈N be a family of boolean circuits which evaluates the coef-ficients inR of polynomialsF1, . . . , Fn, F of R[X1, . . . , Xn] of degreed onintegers of logarithmic height bounded byh. Assume further that this fam-ily of boolean circuits also computes the coefficients inR of polynomialsp, u, ρ, v1, . . . , vn which form the geometric solution for theK-algebraK⊗RB

mentioned above. LetLh := L(m, n, d, h) and`h := `(m, n, d, h) be the sizeand depth ofCm,n,d,h respectively, and assume that the family{Cm,n,d,h}m,n,d,h∈N

is uniform in spaceSh := S(m, n, d, h). LetHh := H(m, n, d, h) be the worst-case logarithmic height of the output ofCm,n,d,h.

Assume thatF is not a zero divisor inB. Let G ∈ R[X1, . . . , Xn] be apolynomial of degreedO(n) such thatF dividesG in B. Then, there exists afamily of boolean circuits which, given the degree ofG, computes the entriesin R of a matrixA and an elementθ ∈ R on integers of logarithmic heighthsuch that ifQ ∈ R[X1, . . . , Xn] denotes the polynomial whose coefficients areobtained applyingA to the coefficients ofG with respect to a suitable monomialbasis of theR-algebraB, then the following conditions are satisfied:

i) θG = QF

ii) The degree ofQ with respect to the variablesX1, . . . , Xn is bounded byn(d − 1).

This family of boolean circuits has sizeLmax{h1,h} + (dnHmax{h1,h}

)O(1), depth

`max{h1,h} + O(n2 log(d) log(dHmax{h1,h})

)and is uniform in spaceSmax{h1,h}

+O(n log(dHmax{h1,h})

), whereh1 = O

(n log(nd)

).

Proof. Following [45], letχF := T D + aD−1TD−1 + · · · + a0 ∈ K[T ] be the

characteristic polynomial of the homothesyηF . Define

F ∗ := FD−1 + aD−1FD−2 + · · · + a2F + a1

498 G. Matera

and notice thatFF ∗ = (−1)D+1a0 holds. SinceF is not a zero divisor ofB wehavea0 6= 0. Therefore let us callF ∗ the pseudo inverse ofF in B.

From Formula (5) we deduce thatF ∗G = ∑m σ(F ∗G · am) · cm and

following Formula (7) we haveJσ = Tr. Let us defineJ ∗ in a similar way toF ∗ as the pseudo inverse ofJ in B (notice that by assumptionJ is not a zerodivisor ofB). With this notation we have:

(J ∗J )F ∗G =∑m

Tr(J ∗F ∗Gam)cm .

In order to keep all the computations inR[X1, . . . , Xn] we defineτ := αDρ,whereα is the leading coefficient ofp, andN as the maximum of the degrees ofF , J , am andG. SettingF ′ := τNF , J ′ := τNJ , a′

m := τNam andG′ := τNG,we see that the matricesMF ′ , MJ ′ , Ma′

m, MG′ corresponding to the homoth-

esiesηF ′ , ηJ ′ , ηa′m, ηG′ in K ⊗R B with respect to the basis{1, u, . . . , uD−1}

(given by the primitive elementu of K ⊗R B) have all their entries inR.Observe also thatF ′∗ = τN(D−1)F ∗ andMF ′∗ = (−1)D+1Adj (MF ′) holds.

Then we have thatJ′is not a zero divisor ofB and that the identity

(J ′∗J ′)F ′∗G′ =∑m

Tr(J ′∗F ′∗G′a′m)bm

holds.In consequence, defining

Q :=∑m

Tr(J ′∗F ′∗G′a′m)cm =

∑m

Tr(Adj (MJ ′)Adj (MF ′)MG′Ma′

m

)bm

(8)and

θ := (J ′∗J ′)(F ′∗F ′) = det (MJ ′)det (MF ′)

we obtain the polynomialsQ andθ required.The matrixA can easily be extracted from the formula that defines the poly-

nomialQ as follows: observe that by Formula (5) we may assume without lossof generality that the degree ofG with respect to the variablesX1, . . . , Xn isbounded byn(d − 1) (using the fact that the trace isR-linear). Then the entriesof MG′ dependR-linearly on the coefficient vector ofG and so do the coeffi-cients ofAdj (M1′)Adj (MF ′)MG′Ma′

m. Taking into account that the canonical

trace isR-linear we see that the polynomialT r(Adj (MJ ′)Adj (MF ′)MG′Ma′

m

)can be expressed as anR-linear combination of the coefficients ofG. Finally,combining this expression with the coefficients ofbm with respect to the vari-ablesX1, . . . , Xn shows that the coefficients ofQ dependR-linearly on thecoefficient vector ofG.

Let us observe that the polynomialsθ andQ have been constructed in such away thatθ divides the polynomialQ in R[X1, . . . , Xn]. To prove this statement


we observe that there existsQ1 ∈ R[X1, . . . , Xn] such thatG ≡ Q1F holdsmodulo(F1, . . . , Fn)). This implies that

T r(J ′∗F ′∗G′a′m) = T r(J ′∗F ′∗F ′Q1a

′m)

= F ′∗F ′σ(J ′J ′∗Q1a′m) = (J ′∗J ′)(F ′∗F ′)σ (Q1a

′m)

holds.We now analyze the size, depth, and space-uniformity of the algorithm that

computes the matrixA of Proposition 21. Letn, m, d be fixed and let us denoteLh := L(n, m, d, h), `h := `(n, m, d, h) andHh := H(n, m, d, h). First of allwe observe that the dense representation of the polynomialslkj that occur inEquation (4) can be computed using spaceO(n logd). For example, using thefollowing simple identity:

Xα11 · · · Xαn

n − Yα11 · · · Yαn

n =n∑

i=1

(Xi − Yi) ·αi−1∑j=0

Yα11 · · · Y

αi−1

i−1 Xj

i Yαi−1−j

i Xαi+1

i+1 · · · Xαn

n

we deduce that all the coefficients appearing in the dense representation ofevery lkj can be described as a sum of some suitably chosen coefficients ofthe polynomialsFj and that this description can be made in spaceO(n logd).The additions required can be computed, using the three-for-two trick thatwe have mentioned in Lemma 3, by means of a family of boolean circuitswith sizeLh + O(dnHh) and depth h + O

(n log(dHh)


spaceO(Sh +n log(dHhLh)

). We observe that the polynomials computed have

logarithmic heightHh + O(n logd) on integers of logarithmic heighth. Thenwe compute the determinant of the matrix(lkj ) and the dense representationof the polynomialsK, am, bm by a process of recursive interpolation applyingLemma 13.

In a similar way we compute the jacobianJ and its dense representa-tion. First we compute the polynomials∂Fi

∂Xjon integers of logarithmic height

h1 := O(n log(nd)

)from the dense representation ofF1, . . . , Fn. Notice that

the resulting integers have logarithmic heightO(n log(d)Hh1

)and that the com-

putation can be performed by means of a family of boolean circuits with sizeLh + (dnHh1)

O(1) and depth h +O(n log(n) log(ndHh1)


spaceSh + O(

log(ndHh1)). Then we compute the determinant of the jacobian

matrix and finally we compute the dense representation ofJ by a process of re-cursive interpolation applying Lemma 13. The resulting integers have logarith-mic heightO

(nd log(dHmax{h1,h})

)O(1)and the computation can be performed

by means of a family of boolean circuits with sizeLmax{h1,h}+(dnHmax{h1,h})O(1)

and depth max{h1,h}+O(n log(n) log(ndHmax{h1,h})


Smax{h1,h} + O(

log(ndHmax{h1,h})).

500 G. Matera

Then we compute the matricesMF ′, MJ ′, Ma′m

andMG′ of the homothe-siesηF ′, ηJ ′, ηa′

mandηG′ induced by the multiplication byF ′, J ′, a′

m andG′

respectively. These matrices have the formMτNH whereH is a given polyno-mial H ∈ R[X1, . . . , Xn] of degree with respect to the variablesX1, . . . , Xn

bounded byN . The polynomialH is given by its dense representationH :=∑|µ|≤N aµX

µ11 · · ·Xµn

n with aµ ∈ R. From the identity:

MτNH =∑

|µ|≤N

aµτN−|µ|Mµ1τX1

· · ·Mµn

τXn(9)

we deduce that the computation ofMτNH can be reduced to matrix computa-tions.

In order to compute the matricesMτXifor 1 ≤ i ≤ n we observe that

MτXi= αDvi(

1αMαu), whereMαu is the matrix which is obtained by multiply-

ing the companion matrix of the polynomial1αp by α. Therefore, ifvi(T ) :=∑D

k=0 bkiTk we defineVi(T ) := ∑D

k=0 bkiαD−kT k. Then we have thatMτXi

=Vi(Mαu). The mapping which yields the entries of the matrixMαu from thecoefficients of the polynomialp can be easily described in spaceO(n logd).Then we compute the matricesMτXi

for 1 ≤ i ≤ n in the following way: wefirst compute the matricesM2

αu, . . . , MDαu applying Lemma 5, then we compute

α2, . . . , αD with D multiplications in depthO(logD). Finally we compute si-multaneously the productbkiα

D−k for 1 ≤ k ≤ D and 1 ≤ i ≤ n and thematricesMτXi

for 1 ≤ i ≤ n as some suitable linear combinations of the ma-tricesI, . . . , MD

αu. We observe that the productsbkiαD−k and the entries of the

matricesMkαu have logarithmic height bounded byO(dnHh)

O(1). Therefore,combining the Macro expansion lemma with Lemma 5 we deduce there existsa family of boolean circuits computing the matricesMτXi

for 1 ≤ i ≤ n withsizeLh + (dnHh)

O(1) and depth h +O(n2 log(d) log(dHh)

), which is uniform

in spaceSh +O(n log(dHh)

). Notice that the entries of the matricesMτXi

havelogarithmic height bounded by(dnHh)

O(1).Then we compute the matricesMF ′, MJ ′, Ma′

musing equation (9), the ma-

tricesAdj (MJ ′), Adj (MF ′) andθ = det (MJ ′)det (MF ′). LetG′ = ∑|µ|≤N gµ

Xµ11 · · ·Xµn

n . In order to compute the matrixAof the statement of the propositionwe observe that, since the canonical trace isR-linear, we have that

Tr(Adj (MJ ′), Adj (MF ′)MG′Mam

)=

∑|µ|≤N

cµgµ (10)

holds, where the coefficientscµ are given by the formula:

cµ := τN−|µ|Tr(Adj (MJ ′), Adj (MF ′)M

µ1τX1

· · ·Mµn

τXn

).

Observe that the polynomialscµ have logarithmic height(dnHmax{h1,h})O(1)

on integers of logarithmic heighth. Therefore, applying the Macro expansionlemma and Lemmata 4, 5 and 8 we obtain a family of boolean circuits com-puting the polynomialscµ with sizeLmax{h1,h} + (

dnHmax{h1,h})O(1)

and depth


`max{h1,h}+O(n2 log(d) log(dHmax{h1,h}

), which is uniform in spaceSmax{h1,h}+

O(n log(dHmax{h1,h})

).

Let bm := ∑|µ|≤nd b

(m)

µX

µ11 · · ·Xµn

n . Then applying equations (8) and (10)we have that

Q = ∑m Tr

(Adj (MJ ′)Adj (MF ′)MG′Ma′

m

)bm

= ∑m

∑µ(

∑µ cµgµ)b

(m)

µX

µ11 · · ·Xµn

n

= ∑µ

( ∑m b

(m)

µ

( ∑µ cµgµ

))X

µ11 · · ·Xµn

n

= ∑µ

( ∑µ

((∑

m b(m)

µ)cµ

)gµ

)X

µ11 · · ·Xµn

n

holds. Therefore, the entry(µ, µ) of the matrixA is given by(∑

m b(m)

µ

)cµ.

These entries can be computed by a family of boolean circuits whose size, depthand space-uniformity satisfies the statement of the proposition.

5 Applications to Geometric Elimination Problems

In this section we apply the techniques developed in the previous sections toconcrete elimination problems. We are going to study the consistency problemfor polynomial equation systems and the representation of the unity 1 in casethat the system under consideration does not have any solution (the effectiveNullstellensatz problem), the membership of a polynomial to a complete inter-section ideal and the corresponding representation problem and an algorithmicversion of Quillen–Suslin Theorem.

5.1 The consistency of polynomial equation systemsand the effective Nullstellensatz

Let F1, . . . , Fs be polynomials inZ[X1, . . . , Xn] of degrees bounded byd andlogarithmic height bounded byh. The problems we are going to solve in thissubsection are the following:

i) Decide whether the system defined byF1, . . . , Fs is inconsistent, that is,whether the algebraic varietyV ⊂ C n that consists of the common zerosof F1, . . . , Fs is empty.

ii) If this is the case, find a representation of the unity 1 as follows:

1 = P1F1 + · · · + PsFs

where the polynomialsP1, . . . , Ps belong toQ[X1, . . . , Xn].Let r be the dimension ofV and letV = Vr ∪· · ·∪V0 be the decomposition

of V in equidimensional components, whereVi is empty or an equidimensionalvariety of dimensioni for i = 0, . . . , r.

502 G. Matera

Let 0 ≤ i ≤ r and suppose thatVi is nonempty. As proved in [34], ifwe choosei generic hyperplanesH(i)

1 , . . . , H(i)i , the following conditions are

satisfied:

i) Vi ∩ H(i)1 ∩ · · · ∩ H

(i)i is a zero-dimensional variety of cardinality deg(Vi).

ii) Vj ∩ H(i)1 ∩ · · · ∩ H

(i)i = ∅ if j < i.

iii) Vj ∩ H(i)1 ∩ · · · ∩ H

(i)i is an equidimensional variety of dimensionj − i if

j > i.

Hence, the isolated points of the varietyV ∩ H(i)1 ∩ · · · ∩ H

(i)i are the points of

Vi ∩H(i)1 ∩· · ·∩H

(i)i . Thus, by deciding whether the varietyV ∩H

(i)1 ∩· · ·∩H

(i)i

has a positive number of isolated points one determines ifVi is empty or not.The greatesti such thatVi is not empty gives us the dimension ofV and in casethatVi is empty fori = 0, . . . , n we have thatV is empty.

In order to decide algorithmically the emptiness of the varietiesVi we needa bound on the degree of the polynomials which represent the condition ofgenericity required for the coefficients of the hyperplanesH

(i)1 , . . . , H

(i)i . In

this direction, we have the following result:

Lemma 22 There exist a polynomialPi ∈ Z[Ajk, Bj ; 1 ≤ j ≤ i, 1 ≤ k ≤ n]which is computable by an arithmetic circuit of nonscalar depthO(n2 logd),and which has the following property: any vectorγ ∈ Zn(i+1) with Pi(γ ) 6= 0yields the coefficients ofi hyperplanes satisfying conditionsi), ii) andiii) above.

Proof. From [27] we deduce that each componentVi can be described as theset of common zeros of certain polynomialsG

(i)1 , . . . , G

(i)ti in Z[X1, . . . , Xn]

of degree bounded bydn. We introduce new variablesAjk, Bj for 1 ≤ j ≤ i

and 1≤ k ≤ n and denote byH(i)j the following generic hyperplane:

H(i)j :=

n∑k=1

AjkXk + Bj

for k = 1, . . . , j .Following [34] there exists a nonempty Zariski open subset ofCn(i+1)

with the following property: the hyperplanes obtained by specializing the vari-ablesAjk, Bj in the coordinates of any point of this Zariski open set inter-sectVi in deg(Vi) points. Hence, the varietyW consisting of the solutions inC[Ajk, Bj ; 1 ≤ j ≤ i, 1 ≤ k ≤ n]

nof the system

G(i)1 = 0, . . . , G

(i)ti = 0, H

(i)1 = 0, . . . , H

(i)i = 0

is zero-dimensional and consists of deg(Vi) points. From Section 3 we deducethe existence of a well-parallelizable arithmetic circuit of nonscalarO(n2 logd)

that computes the coefficients inZ[Ajk, Bj ; 1 ≤ j ≤ i, 1 ≤ k ≤ n] of thefollowing polynomials:


• a linear formU ∈ Z[X1, . . . , Xn], and• a polynomialp ∈ Z[Ajk, Bj ; 1 ≤ j ≤ i, 1 ≤ k ≤ n][T ] of degree deg(Vi)

such thatU separates the points ofW andp is the primitive minimal polynomialthat annihilatesU overW .

Let γ be a vector ofZ n(i+1) such that if we replace the variablesAjk, Bj byγ in p, the resulting polynomial ofZ[T ] is squarefree and has degree deg(Vi).Then the hyperplanes obtained fromγ intersectVi in exactly deg(Vi) points.Hence, we impose to the coefficients of the hyperplanesH

(i)1 , . . . , H

(i)i the con-

dition of not annihilating the leading coefficientα of p and the discriminant1 ∈ Z[Ajk, Bj ; 1 ≤ j ≤ i, 1 ≤ k ≤ n] of p. The product of both polynomials,the generic condition we are looking for, can be computed with nonscalar depthO(n2 logd) (here the computation of1 relies on subresultant techniques as in[8]).

Now we are going to introduce a condition of genericity for the coeffi-cients of the hyperplanesH(i)

1 , . . . , H(i)i which guarantees that condition ii)

is fulfilled. For this purpose, we use a diophantine version of the effectiveNullstellensatz. Since generically the intersection(Vi−1 ∪ · · · ∪ V0) ∩ (H

(i)1 ∪

· · · ∪ H(i)i ) is empty, the same happens when the situation is considered in

C[Ajk, Bj ; 1 ≤ j ≤ i, 1 ≤ k ≤ n]n. Applying the effective Nullstellensatz in

the version of [45], we deduce the existence of a nonzero polynomiala ∈Z[Ajk, Bj ; 1 ≤ j ≤ i, 1 ≤ k ≤ n], computable by means of an arithmeticcircuit of nonscalar depthO(n2 logd), that belongs to the ideal generated bythe polynomials which defineVi−1 ∪ · · · ∪ V0 andH

(i)1 , . . . , H

(i)i .

Any evaluation of the variablesAjk, Bj in a vectorγ ∈ Z n(i+1) such thata(γ ) 6= 0 guarantees that the linear variety defined by the intersection of the hy-perplanesH(i)

1 (γ ), . . . , H(i)i (γ ) does not intersectVi−1∪· · ·∪V0. Thus, the con-

dition of genericity we are looking for is the non vanishing of the polynomiala.Let j > i. It remains to satisfy the third condition, namely thatWj :=

Vj ∩H(i)1 ∩· · ·∩H

(i)i does not contain isolated points. Using the same arguments

as before, we consider the situation overC[Ajk, Bj ; 1 ≤ j ≤ i, 1 ≤ k ≤ n]n.

In the same way as in Section 3 we construct an arithmetic circuit of non-scalar depthO(n2 logd) which represents a geometric solution of a suitablezero-dimensional varietyWj that contains the isolated points ofWj . Let U1 ∈Z[X1, . . . , Xn] be the linear form given by this geometric solution and letp1 ∈ Z[Ajk, Bj ; 1 ≤ j ≤ i, 1 ≤ k ≤ n][T ] be the minimal polynomial ofU1. Since the varietyWj does not have isolated points, the intersection of thevarietyWj with the varietyWj must be empty. In order to check this conditionwe replace the variablesX1, . . . , Xn in the polynomials which defineWj by theparameterizations obtained in the geometric solution ofWj . In this way we ob-tain some polynomials belonging toZ[Ajk, Bj ; 1 ≤ j ≤ i, 1 ≤ k ≤ n][T ] withthe property that their greatest common divisor withp(T ) equals the unity 1.

Following the strategy of Lemma 11, we compute a polynomialb inZ[Ajk, Bj ; 1 ≤ j ≤ i, 1 ≤ k ≤ n] which can be written as an arithmetical

504 G. Matera

expression of the involved polynomials. Any pointγ ∈ Zn(i+1) such thatb(γ ) 6= 0 yields the coefficients of some hyperplanesH

(i)1 , . . . , H

(i)i such

that the corresponding intersectionWj ∩ Wj is empty. Hence, the polynomialb represents the remaining condition of genericity.

Finally, multiplying theseO(n) conditions of genericity we obtain apolynomialPi ∈ Z[Ajk, Bj ; 1 ≤ j ≤ i, 1 ≤ k ≤ n], computable with non-scalar depthO(n2 logd), which verifies that any vectorγ ∈ Zn(i+1) such thatPi(γ ) 6= 0 yields the coefficients ofi hyperplanes satisfying all the requiredconditions.

Now we can solve the consistency problem for polynomial equationsystems:

Theorem 23 There exists a probabilistic Turing machineM which, on in-put the dense representation of polynomialsF1, . . . , Fs of degreed and log-arithmic heighth, computes the dimension of the algebraic varietyV :=V (F1, . . . , Fs) ⊆ Cn using spaceO

(n4 log(nd) log(nsdh)

)and time

(nd)O

(n2 log(nsdh)

).

Proof. As we have mentioned before, the computation of the dimension canbe easily reduced to determine whether the algebraic varietyV ∩ H

(i)1 ∩ · · · ∩

H(i)i has isolated points for 1≤ i ≤ n, where the hyperplanesH(i)

1 , . . . , H(i)i

satisfy the requirements of Lemma 22. For this purpose, applying Proposition20 with R = Z we compute a geometric solution of a zero-dimensionalWi

containing the isolated points ofV ∩ H(i)1 ∩ · · · ∩ H

(i)i . Then we replace the

parameterizations of the geometric solution ofWi in the equations that definethe varietyV ∩H

(i)1 ∩· · ·∩H

(i)i and obtain some univariate polynomials whose

greatest common divisor, saygi(u), equals the unity 1 if and only if there areno isolated points ofV ∩ H

(i)1 ∩ · · · ∩ H

(i)i (see e.g. [11]).

We observe that a nonzeroZ-multiple of the polynomialgi(u) can becomputed by a family of boolean circuits with size

(sh(nd)n

)O(1)and depth

O(n2 log(nd) log(nsdh)


(n2 log(nsdh)

). This

can be proved combining the Macro expansion lemma, Proposition 20 andLemma 11.

Combining [45, Lemma 14] with Lemma 22, we see that for 1≤ i ≤ n

the polynomialPi in the statement of Lemma 22 has degree bounded bydcn2,

wherec is a suitable constant not depending neither ond nor n. Therefore,applying Schwartz–Zippel test we deduce that a random choice of an(ni)-tupleon integers in the set{1, . . . , d(c+1)n2}ni does not annihilate the polynomialPi

with probability at least 1− 1dn3 .

The probabilistic Turing machineM in the statement of the theorem startsguessing an integer(ni)-tupleγ (i) in the set{1, . . . , d(c+1)n2}ni for 1 ≤ i ≤n. The vectorγ (i) provide the coefficients ofi hyperplanesH(i)

1 , . . . , H(i)i of

logarithmic heightO(n2 logn logd) that verify Conditions i), ii) and iii) of


Lemma 22 with probability at least 1− 1dn3 . The vectorsγ (1), . . . , γ (n) are

stored in working space.Then the machineM decides for 1≤ i ≤ n the existence of isolated points

of the algebraic varietyV ∩ H(i)1 ∩ · · · ∩ H

(i)i for 1 ≤ i ≤ n. For this purpose,

we have that there exists a deterministic Turing machineM1 computing thestandard encoding of a boolean circuitC which computes a nonzero integermultiple of the polynomialgi(u) which equals the unity 1 if and only if thevarietyV ∩ H

(i)1 ∩ · · · ∩ H

(i)i does not have isolated points.

In our case the coefficients of the polynomialsF1, . . . , Fs, H(i)1 , . . . , H

(i)i

which are the inputs of the boolean circuitC are available either from the inputin the case ofF1, . . . , Fs or from working space in the case ofH

(i)1 , . . . , H

(i)i .

The 3n + 3 integer parameters required to apply Proposition 20 are obtainedas a consequence of the application of Schwartz-Zippel test. Therefore, theyare randomly chosen in a set{1, . . . , (nsd)c

′n}3n+3 with probability of successat least 1− 1

(nsd)3n2 and stored in working space. We recall that the circuitC

has size(sh(nd)

)O(1)and depthO(n2 log(nd) log(nsdh) and the machineM1

requires spaceO(n log(nsdh)

)to compute its standard encoding.

Then the machineM applies Borodin’s argument to evaluate the circuitC.It evaluates the circuitC by following a standard “depth first” traversal of thegraph ofC and generating the standard encoding of every node of the graph ofC by a call to the machineM1. A stack of depthO

(n2 log(nd) log(nsd)

)is used

to keep track of the path from the current output bit of the circuit to the nodewhich is being evaluated. For every cell of the stack to have sizeO(1), a specialencoding of the path is used in such a way that at every step in the computation,the stack is seen as a string on the alphabet{0, 1, `}. This string encodes thechoiceof the path at every level (node) between its left and right input. Atthe same time the result of the evaluation of the left subcircuit is stored, if theright subcircuit is being traversed. In this way, the machineM deterministicallyevaluates the circuitC using only additional spaceO

(n2 log(nd) log(nsdh)

)and time(nsdh)

O

(n2 log(nd)

).

Assuming now thatV is the empty set we solve the effective Nullstellensatzproblem.

Theorem 24 There exists a probabilistic Turing machine using space

O(n3 log(nh) log(nsdhh1)

)and time(nd)

O

(n2 log(sndhh1)

)which, on input the

dense representation of polynomialsF1, . . . , Fs of degreed and logarithmicheighth, computes an integera and polynomialsP1, . . . , Ps ∈ Z[X1, . . . , Xn](on integers of logarithmic heighth1) such that the following identity holds:

a = P1F1 + · · · + PsFs.

Proof. First of all, we perform a preprocessing of our polynomial data. For thispurpose let us observe that there exist a natural numbert ≤ n and(t +1)-linear

506 G. Matera

combinations ofF1, . . . , Fs , sayF ′1, . . . , F

′t+1, with the following properties:

• V (F ′1, . . . , F

′t+1) = V (F1, . . . , Fs).

• F ′1, . . . , F

′i form a regular sequence fori = 1, . . . , t .

• (F ′1, . . . , F

′i ) is a radical ideal ofQ[X1, . . . , Xn] for everyi = 1, . . . , t

• the coefficientsλij occurring in the linear combinations that define thepolynomialsF ′

1, . . . , F′t+1 verify the condition|λij | ≤ dn and can be chosen

at random from the set{1, . . . , d2n}n with probability of success greater than1 − 1

dn2 .

This is achieved by means of a combination of Schwartz–Zippel test and aneffective version of Bertini Theorem (cf. [40]) in the version of [45].

Furthermore, there exists a linear change of coordinates

(X1, . . . , Xn) → (Y1, . . . , Yn)

such that the variablesY1, . . . , Yn are in Noether position with respect to thevarietyVi := V (F ′

1, . . . , F′i ) for i = 1, . . . , t . Applying the arguments of [16]

or [45], we deduce that the entries of the matrix that performs this linear changeof variables can be generated by means of the Schwartz–Zippel test in such away that their logarithmic height is of orderdO(n).

The problem of finding a polynomial combination ofF1, . . . , Fs that equalsthe unity 1 will be solved by means of Proposition 21. This is done by performingdivisions modulo suitable complete intersection ideals.

The idea is the following: sinceF ′t+1 divides the unity 1 modulo the ideal

(F ′1, . . . , F

′t ), it is possible to compute by means of Proposition 21 an element

θt+1 of the polynomial ringRt := Q[Y1, . . . , Yn−t ] and a polynomialQt+1 ∈Q[Y1, . . . , Yn] of degree bounded bynd such thatθt+1 dividesQt+1 and suchthat the condition

θt+1 · 1 ≡ Qt+1F′t+1 modulo(F ′

1, . . . , F′t )

is satisfied.This means thatθt+1

(1 − F ′

t+1Qt+1

θt+1

)belongs to(F ′

1, . . . , F′t ), which is an

equidimensional ideal of dimensionn − t intersecting the polynomial ringRt

only in zero. Therefore 1− F ′t+1

Qt+1

θt+1belongs to the ideal(F ′

1, . . . , F′t ). Hence

F ′t divides 1− F ′

t+1Qt+1

θt+1modulo (F ′

1, . . . , F′t−1), which in turn implies that

θt+1F′t divides the polynomialθt+1 · 1− Qt+1F

′t+1 which belongs to the ideal

(F ′1, . . . , F

′t−1).

Now we divide the polynomialθt+1 · 1− Qt+1F′t+1 by θt+1F

′t modulo the

ideal(F ′1, . . . , F

′t−1). In this way we obtain an elementθt of the polynomial ring

Rt−1 := Q[Y1, . . . , Yn−t+1] and a polynomialQt ∈ Q[Y1, . . . , Yn] of degreebounded bynd such thatθt dividesQt in Rt−1 and such that

θtθt+1 · 1 − θtQt+1F′t+1 − Qtθt+1F

′t ∈ (F ′

1, . . . , F′t−1)

holds.


Applying this argument recursively, we compute polynomialsθ1, . . . , θt+1,Q1, . . . , Qt+1 of Q[Y1, . . . , Yn], such that the following conditions are satisfiedfor 1 ≤ i ≤ t + 1:

• the polynomialθi belongs toRi−1 := Q[Y1, . . . , Yn−i−1].• the polynomialQi belongs toRi−1[Yn−i , . . . , Yn] and its degree inYn−i , . . . ,

Yn is bounded bynd.• θi dividesQi in Q[Y1, . . . , Yn].• The polynomialθt−i · · · θt+1 · 1−Qt−i · · · θt+1F

′t−i−· · ·−θt−i · · · θtQt+1F

′t+1

belongs to the ideal(F ′1, . . . , F

′t−i−1).

As a consequence, at the end of the last recursive step we obtain the identity:

θ1 · · · θt+1 · 1 = Q1θ2 · · · θt+1F1+θ1Q2θ3 · · · θt+1F1+· · ·+θ1 · · · θtQt+1Ft+1

(11)where eachθi divides Qi in Q[Y1, . . . , Yn]. Thus, applying Lemma 15, weare able to compute without divisions a nonzero integera and polynomialsP1, . . . , Pt+1 ∈ Z[Y1, . . . , Yn] such thatPi = a Qi

θifor i = 1, . . . , t + 1.

Substituting this identity in (11) we obtain a representation:

a = P ′1F

′1 + · · · + P ′

t+1F′t+1. (12)

Taking into account equation (12) and the fact thatF ′i = ∑s

j=1 λijFj for 1 ≤i ≤ t + 1, we obtain the representation we are looking for:

a = P1F1 + · · · + PsFs (13)

where the polynomialPi is defined asPi := ∑t+1i=1 λijP

′i for 1 ≤ i ≤ s.

Let us analyze the size, depth and space-uniformity of the family of booleancircuits which computes the polynomialsQi+1 andθi+1. These polynomialsare obtained by a division modulo the ideal(F ′

1, . . . , F′i ), which is viewed as

a zero-dimensional ideal ofRi [Yn−i+1, . . . , Yn]. This requires the computationof a geometric solution of the variety defined byF ′

1, . . . , F′i considered as

zero-dimensional varieties overC(Y1, . . . , Yn−i)i, which is obtained applying

Proposition 20. For this purpose, we observe that the coefficients inRi of thepolynomialsF ′

1, . . . , F′i can be computed on integers of logarithmic heighth1

from the dense representation ofF1, . . . , Fs and the coefficients of the matricesof the preprocessings mentioned above by means of a family of boolean circuitswith sizes3dO(n)(hh1)

2 and depthO(n logdshh1) which is uniform in spaceO(n logdshh1). We remark that the resulting integers have logarithmic heightbounded byh + dh1 + O(n logsd).

Notice that the polynomialsF ′1, . . . , F

′i define a zero-dimensional variety

and that this variety is defined byas many equations as unknowns. Thereforeapplying Proposition 20 we obtain a family of boolean circuits computing ageometric solution of the variety defined byF ′

1, . . . , F′i overC(Y1, . . . , Yn−i)

i

with size(shh1(nd)n)O(1) and depthO(n2 log(nd) log(ndshh1)

)which is uni-

form in spaceO(n2 logdshh1). We observe that the resulting integers havelogarithmic height bounded by(sdnhh1)

O(1).

508 G. Matera

The procedure for the computation ofQi+1 andθi+1 depends recursively onthe computation ofQi+2 andθi+2. In order to parallelize this procedure, we ob-serve that Proposition 21 allows to compute a matrixA(i) which applied linearlyon the coefficients of the dividendGi+1 := θi+2 · · · θt+1 ·1− θi+2 · · · θtQt+1F

′t+1 − · · · − Qi+2 · · · θt+1F

′i+2 gives the coefficients of the

quotientQi+1, and this depending only on the degree of the dividendGi+1 andon the coefficients of the polynomialsF ′

1, . . . , F′i+1. We also observe that the

fact thatRi ⊆ Ri+1 implies that the polynomialsGi+k, Qi+k viewed as poly-nomials with coefficients inRi verifies that degYn−i+1,...,Yn

Gi+k ≤ (n − 1)d anddegYn−i+1,...,Yn

Qi+k ≤ (n − 1)d.Following [45] we compute independently ofGi+1, . . . , Gt+1 the following

items:

• All the coefficients inRi, . . . , Rt+1 of the polynomialsF ′1, . . . , F

′t+1. As we

have mentioned before there exists a family of boolean computing these co-efficients on integers of logarithmic heighth1 with sizes3dO(n)(hh1)

2 anddepthO(n logdshh1) which is uniform in spaceO(n logdshh1). We re-mark that the resulting integers have logarithmic height bounded byh + dh1 + O(n logsd).

• The valueθj ∈Rj−1 and the matrixA(j) with entries inRj−1[Yn−j+2, . . . , Yn]given by the application of Proposition 21 fori+2 ≤ j ≤ t+1. From Propo-sition 21 we see that there exists a family of boolean circuits computing theseitems on integers of logarithmic heighth1 with size(shh1(nd)n)O(1) anddepth O

(n2 log(nd) log(ndshh1)

)which is uniform in space

O(n2 logndshh1). We remark that the resulting integers have logarithmicheight bounded by(sdnhh1)

O(1).• MatricesTj for i + 1 ≤ j ≤ t + 1 which transform the coefficients of

a polynomialP ∈ Rj1[Yn−j+2, . . . , Yn] of degree bounded by(n − 1)d

which has worst-case logarithmic heightH(j, h1) on integers of logarith-mic heighth1, on the coefficients of the polynomialFjP of degree at mostnd on integers of logarithmic heighth1. Applying Lemma 3 we see thatthis can be done by means of a family of boolean circuits of size(nshh1H(j, h1)d

n)O(1) and depthO

(n logdshh1H(j, h1)

)which is uni-

form in spaceO(n logdshh1H(j, h1)

).

• MatricesBj for i + 1 ≤ j ≤ t + 1 which transform the coefficients inRj−1 of a polynomialP ∈ Rj1[Yn−j+2, . . . , Yn] of degree bounded bynd

which has worst-case logarithmic heightH(j, h1) on integers of logarith-mic heighth1 on its coefficients when viewed as a polynomial with coef-ficients inRj−2 on integers of logarithmic heighth1. Applying Lemma 13we see that this can be done by means of a family of boolean circuits of size(shh1H(j,O(n logd))(nd)n)O(1) and depthO(n2 log(nd) log(ndshh1 H(j,

O(n logd))) which is uniform in spaceO(n log(ndshh1H(j, O(n logd))).

If we define fori+1 ≤ j ≤ t+1 the matrixNj := Bj(θj Id−θj+1 · · · θt+1TjQj),we have that the coefficients of the polynomialQi+1 in Ri can be obtained by


multiplying the matrixNi+2 · · ·Nt+1 by the coefficient vector of the polynomial1 viewed as a polynomial ofRt . Therefore, applying Lemma 4 we conclude thatthe coefficients of the polynomialsQi+1 andθi+1 can be computed by a family ofboolean circuits of size(shh1(nd)n)O(1) and depthO


)which is uniform in spaceO(n2 logndshh1). We remark that the resulting in-tegers have logarithmic height bounded by(sdnhh1)

O(1).Then we apply Lemma 15 to obtain a family of boolean circuits of the

same size, depth and space-uniformity as above computing the integera andthe polynomialsP ′

1, . . . , P′t+1 of equation (12). Finally, we obtain the poly-

nomialsP1, . . . , Ps of equation (13) by computing the linear combinationsPj = ∑t+1

i=1 λijP′i for 1 ≤ j ≤ s.

Now we can describe the probabilistic Turing machineM which solves theeffective Nullstellensatz problem. This machineM starts guessing theO(n2)

integer parameters of logarithmic heightO(n log(nsd)

)required to perform the

preprocessing given by the effective versions of Bertini Theorem and Noethernormalization and to apply Propositions 20 and 21 and Lemma 15. Since allthese parameters are obtained by applying Schwartz–Zippel test, with the sameidea as in the proof of Theorem 23 we can conclude that they can be randomlychosen with probability of success� 1

2. These parameters are stored in workingspace.

Then we evaluate the boolean circuitC which computes the integeraand the polynomialsP1, . . . , Ps of equation (13) on input the dense repre-sentation of the polynomialsF1, . . . , Fs . Recall that the circuitC has size(shh1(nd)n)O(1) and depthO


), and that there exists

a deterministic Turing machineM1 computing the standard encoding ofC us-ing spaceO(n2 logndshh1).

Then the machineM evaluates the circuitC following Borodin’s argumentin the same way as in the proof of Theorem 23. Therefore, the machineM

deterministically evaluates the circuitC using additional spaceO(n2 log(nd)

log(nsdhh1))

and time(nd)O(n2 log(nsdhh1)).

5.2 The membership and representation problems in the caseof complete intersection ideals

LetF, F1, . . . , Ft be polynomials inZ[X1, . . . , Xn] such thatF1, . . . , Ft form aregular sequence inQ[X1, . . . , Xn]. We assume that the polynomialsF1, . . . , Ft

andF have degree and logarithmic height bounded by some constantsd andh

respectively.The problem we are going to deal with is the decision about the membership

of F to the ideal generated byF1, . . . , Ft and its representation in this ideal. Thatis, we want to determine whetherF belongs to the ideal generated byF1, . . . , Ft

in Q[X1, . . . , Xn] and, if this is the case, to find polynomialsP1, . . . , Pt ∈Z[X1, . . . , Xn] such that the following identity holds:

F = P1F1 + · · · + PtFt .

510 G. Matera

Theorem 25 There exists a probabilistic Turing machine which, on inputthe dense representation of polynomialsF, F1, . . . , Ft of degreed andlogarithmic heighth such thatF1, . . . , Ft generate a complete intersectionideal, solves the membership and representation problems using spaceO

(n3 log(nd) log(ndhh1)

)and time(nd)O(n2 log(ndhh1)).

Proof. First we produce a change of variables(X1, . . . , Xn) → (Y1, . . . , Yn)

such the variablesY1, . . . , Yn are in Noether position with respect to the varietyV (F1, . . . , Fi) for 1 ≤ i ≤ t . Applying the arguments of [16] or [45], we deducethat the entries of the matrix that performs this linear change of variables canbe generated by means of the Schwartz–Zippel test in such a way that theirlogarithmic height is of orderdO(n).

We study first the membership problem. We observe thatF ∈ (F1, . . . , Ft )

if and only if the homothesy defined byF in B := Q[Y1, . . . , Yn]/(F1, . . . , Ft )

is the zero endomorphism.Let R := Q[Y1, . . . , Yn−t ] and letK be the quotient field ofR. Applying

Proposition 20 we obtain a family of boolean circuits of size(hh1(nd)n)O(1) anddepthO

(n2 log(nd) log(ndhh1)

), which is uniform in spaceO(n2 logdhh1)

computing the following polynomials of integers of logarithmic heighth1:

• a linear formU := λ1Y1 + · · · + λnYn ∈ Z[Y1, . . . , Yn] such thatu := U

is a primitive element forK ⊗R B.• a polynomialp ∈ R[T ] of degreeD bounded bydn such thatp(U) belongs

to the ideal(F1, . . . , Ft ).• a nonzero elementρ ∈ R and polynomialsv1, . . . , vn ∈ R[T ] of degree

bounded byD − 1 such that for any 1≤ j ≤ n ρYj − vj (U) belongs to(F1, . . . , Ft ).

We observe that the resulting integers have logarithmic height bounded by(dnhh1)

O(1).Following the ideas of the proof of Proposition 21 we defineτ := αDρ

whereα is the leading coefficient ofp and compute the matrixMF ′ of the ho-mothesy byF ′ := τ dF with a family of boolean circuits of size

(hh1(nd)n

)O(1)

and depthO(n2 log(nd) log(ndhh1)

), which is uniform in spaceO(n2 logdhh1).

We remark that the resulting integers have logarithmic height bounded by(dnhh1)

O(1). Finally, applying Lemma 7 we obtain a family of boolean cir-cuits computing the rank of this matrix whose size, depth and space-uniformityare the same as above.

Now we can describe the probabilistic Turing machineM which solves themembership problem for complete intersection ideals. This machineM startsguessing theO(n2) integer parameters of logarithmic heightO

(n log(nd)

)re-

quired to perform the preprocessing given by the Noether normalization, theapplication of Proposition 20 and Lemma 7. Since all these parameters areobtained by applying Schwartz–Zippel test, we can conclude that they can be


randomly chosen with probability of success� 12. These parameters are stored

in working space.Then we evaluate the boolean circuitC which computes the rank of the ma-

trix MF ′ on input the dense representation of the polynomialsF, F1, . . . , Ft .Recall that the circuitC has size

(hh1(nd)n

)O(1)and depthO

(n2 log(nd)

log(ndhh1)), and that there exists a deterministic Turing machineM1 com-

puting the standard encoding ofC using spaceO(n2 logdhh1).Then the machineM evaluates the circuitC following Borodin’s argument

in the same way as in the proof of Theorem 23. Therefore, the machineM

deterministically evaluates the circuitC using additional spaceO(n2 log(nd)

log(ndhh1) and time(nd)O(n2 log(ndhh1)). FinallyM outputs 1 (=true) if and onlyif the rank ofMF ′ equals zero.

Once the membership ofF to the ideal(F1, . . . , Ft ) has been established,the representation ofF is performed by the procedure described in the proof ofTheorem 24, starting withF instead of the polynomial 1. Therefore, from thebounds of Theorem 24 we deduce the statement in the theorem.

5.3 An effective version of Quillen–Suslin Theorem

A matrix F of Z[X1, . . . , Xn]r×s with s ≥ r is calledunimodularif the idealgenerated inQ[X1, . . . , Xn] by all its minors of sizer × r is the trivial idealQ[X1, . . . , Xn].

LetF :=(Fij (X1, . . . , Xn)

)1≤i≤r,1≤j≤s

be a unimodular matrix ofZ[X1, . . . ,

Xn]r×s . Let us denote by deg(F ) the maximal degree of all its entries and letd = 1 + deg(F ). From the Quillen–Suslin theorem (see [50]) we deduce theexistence of a unimodular matrixM of sizes × s such thatFM = [Ir , 0] holds,where [Ir , 0] denotes ther × s-matrix obtained by adding to ther × r-identitymatrixIr s − r zero columns. The problem consists in finding such a matrixM.

Theorem 26 There exists a probabilistic Turing machine which, on input thedense representation of polynomialsFij for 1 ≤ i ≤ r, 1 ≤ j ≤ s of degreedand logarithmic heighthsuch that the matrixF :=(

Fij (X1, . . . , Xn))

1≤i≤r,1≤j≤s

is a unimodular matrix ofZ[X1, . . . , Xn]r×s , computes a unimodular matrixM of sizes × s such thatFM = [Ir , 0] holds. This Turing machine uses space

O(n4 log(rnd) log(sndhh1)

)and time(sndhh1)

O

(n2 log(rnd)

).

Proof. The procedure performed in order to computeM is divided inn steps,wheren unimodular matricesM1, . . . , Mn of sizess × s are constructed suchthat for any 1≤ i ≤ n the following condition is satisfied:

F · Mn · · ·Mi+1 = (Fij (X1, . . . , Xi, 0, . . . , 0)

)1≤i≤r,1≤j≤s

.

512 G. Matera

Since the matrixF · M1 · · ·Mn is unimodular and has its entries inQ, by meansof a standard triangularization procedure the latter can be reduced to the form[Ir , 0]. If M0 is the (unimodular) matrix that performs this triangularization,M := Mn · · ·M0 is the matrix we are looking for.

Let us remark that the matrices(Fij (X1, . . . , Xi, 0, . . . , 0)

)1≤i≤r,1≤j≤s

can

be produced simultaneously from the input. Hence the computation of the ma-tricesM0, . . . , Mn can be performed in parallel.

Following the scheme proposed in [18] (see also [19]), the procedure forthe computation of each matrixMi is divided in four steps. In order to simplifythe notations, we will only describe the computation ofMn.

In the first step, we construct a sequencec1, . . . , cN of polynomials of degreebounded by(rd)2 such that the following conditions are satisfied:

• 1 ∈ (c1, . . . , cN)

• For everyk ∈ {1, . . . , N}, there exists a nonsingular matrixKk ∈ GLs(Q)

such that

ck = Res(det [F (k)

1 , . . . , F (k)r ], det [F (k)

1 , . . . , F(k)r−1, F

(k)r+1]

)holds. HereF (k)

1 , . . . , F(k)r+1 are the columns of the matrixF (k) := F · Kk.

The matricesKk have the following form:

Kk =

1α1 1

α1β1 α2

α1β21 α2β2

. . ....

... 1αr−1 1

αr−1βr−1 0 1...

......

... αr

. . .

α1βs−21 α2β

s−32 . . . αr−1β

s−rr−1 0 αs−r−1

r . . . 1

(14)

whereα1, . . . , αr, β1, . . . , βr−1 are suitable integers.The key point is that the polynomialsc1, . . . , cN must generate the triv-

ial ideal. The problem is therefore to find a short sequence(α(k)1 , . . . , α(k)

r ,

β(k)1 , . . . , β

(k)r−1)k=1,...,N of (2r−1)-tuples of integers of small logarithmic height

such that the algebraic variety defined by the polynomialsc1, . . . , cN is empty.We consider indeterminatesS1, . . . , Sr, T1, . . . , Tr−1 and the matrixK ∈

Z[Si, Tj ; 1 ≤ i ≤ r, 1 ≤ j ≤ r − 1] obtained by replacing eachαi by Si andeachβj by Tj in the matrixKk of (14).

Likewise, we define:


• F ′ := [F ′1, . . . , F

′r+1] = F · K

• D′1 := det [F ′

1, . . . , F′r ]

• D′2 := det [F ′

1, . . . , F′r−1, F

′r+1]

• c := ResXn(D′

1, D′2)

Applying [18, Proposition 5.6] we see that for everyx ∈ Cn there exists a vector(α, β) ∈ N2r−1 such thatc(x, α, β) 6= 0 holds. Thusc(x, S1, . . . , Sr, T1, . . . ,

Tr−1) is a nonzero polynomial ofZ[Si, Tj ; 1 ≤ i ≤ r, 1 ≤ j ≤ r − 1]for everyx ∈ Cn. Furthermore,c(x, S1, . . . , Sr, T1, . . . , Tr−1) can be com-puted by means of an arithmetic circuit of sizeO(s3r4(rd)2n) and nonscalardepthO

(n log(sd)

). Applying the Heintz–Schnorr test we deduce the exis-

tence of a correct test sequence ofN := O(s7(rd)5n) elements(α(k), β(k)) ∈{1, . . . , m}2r−1 for k = 1, . . . , N , with m := (sd)O(n), such that for everyx ∈ Cn there exists an indexk ∈ {1, . . . , N} with c(x, α(k), β(k)) 6= 0. More-over, such a correct test sequence can be generated randomly with probabilityof success at least 1− m

N6 � 1

2.Let ck := c(X1, . . . , Xn, α

(k), β(k)) for k = 1, . . . , N . One sees easilythat the polynomialsc1, . . . , cN have degree bounded by(rd)2 and that theircomputation requires only the computation of the determinant ofO

(s7(rd)5n

)polynomialr × r-matrices.

In the second step we construct a representation

Xn = a1c1 + · · · + aNcN

of the variableXn in the ideal generated by the polynomialsc1, . . . , cN follow-ing the ideas of Subsection 5.1.

In the third step we computeN unimodular matricesM(1)n , . . . , M(N)

n ofQ[X1, . . . , Xn]s×s such that, defining fork = 1, . . . , N the polynomialbk :=∑k

h=1 ahch, the following identity holds:

F (k)(bk)M(k)n = F (k)(bk−1) . (15)

Here we writeF (k) := F · Kk andF (k)(bk) for the matrix obtained fromF (k)

by replacing the variableXn by bk.From identity (15) we deduce that the matrixEk := Kk · M(k) · K−1

k satisfiesthe following property:

F(bk)Ek = F(bk−1) . (16)

Since we haveck = ResXn(D

(k)1 , D

(k)2 ), there exist polynomialsgk, hk ∈

Q[X1, . . . , Xn] such thatck = gk · D

(k)1 + hk · D

(k)2 (17)

holds. Observing that the polynomialck does not depend onXn, we deduce that

ck = gk(bk) · D(k)1 (bk) + hk(bk) · D

(k)2 (bk) (18)

holds.

514 G. Matera

Let F(k)j denote thej -th column of the matrixF (k). From the proof of

[18, Lemma 4.5 ] and the fact that the congruence relationbk ≡ bk−1 modulock · Q[X1, . . . , Xn] holds, we deduce by means of the Taylor expansion ofthe entries ofF (k)

j the existence of a column vectorG(k)j ∈ Q[X1, . . . , Xn]r×1

which satisfies the condition

F(k)j (bk) − F

(k)j (bk−1) = ckG

(k)j . (19)

Combining (18) and (19) we obtain the following identity:

F(k)j (bk) − F

(k)j (bk−1) = D

(k)1

(gk(bk)G

(k)j

)+ D

(k)2

(hk(bk)G

(k)j

). (20)

Let B(k)1 andB

(k)2 be the following matrices:

B(k)1 := Adj [F (k)

1 (bk), . . . , F(k)r (bk)] (21)

B(k)2 := Adj [F (k)

1 (bk), . . . , F(k)r−1(bk), F

(k)r+1(bk)] . (22)

Then we have the following identities:

D(k)1 (bk)gk(bk)G

(k)j = [F (k)

1 (bk), . . . , F(k)r (bk)]B

(k)1 gk(bk)G

(k)j

:=r∑

j=1

ηjF(k)j (bk)

D(k)2 (bk)hk(bk)G

(k)j = [F (k)

1 (bk), . . . , F(k)r−1(bk), F

(k)r+1(bk)]B

(k)2 hk(bk)G

(k)j

:=∑j 6=r

ηjF(k)j (bk)

where (η1, . . . , ηr)t := B

(k)1 gk(bk)G

(k)j and (η1, . . . , ηr−1, ηr+1)

t := B(k)2 hk

(bk)G(k)j . Applying these identities to (20) we obtain the following equation for

j = r + 2, . . . , s:

F(k)j (bk) − F

(k)j (bk−1) = (η1 + η1)F

(k)1 (bk) + · · · + ηrF

(k)r (bk) + ηr+1F

(k)r+1(bk)

From these equations we construct a unimodular matrixM(k) such that

F (k)(bk)M(k) = [F (k)

1 (bk), . . . , F(k)r+1(bk), F

(k)r+2(bk−1), . . . , F

(k)s (bk−1)]

holds.In order to finish the construction of the matrixM(k), we define the following

(r + 1) × (r + 1) unimodular matrixT (k):

T (k) := 1

ck

Adj

(F

(k)1 (bk) . . . F (k)

r (bk) F(k)r+1(bk)

0 . . . −hk(bk) gk(bk)

)

·(

F(k)1 (bk−1) . . . F (k)

r (bk−1) F(k)r+1(bk−1)

0 . . . −hk(bk−1) gk(bk−1)

).


Then,M(k) := M(k) · (T (k) ⊕ Is−r−1) is the matrix we are looking for.

Finally in the fourth step we compute the productMn := ∏Nk=1 Ek =∏N

k=1 KkM(k)K−1

k . From the identities:F = F(Xn) = F(bN)

F (bN)En = F(bN−1)...

F (b1)E1 = F(b0) = F(0)

we deduce that the matrixMn satisfies the required conditions.After n parallel steps like the one described above, we obtain an algorithm

which computes the matrixM we are looking for.We now analyze the algorithm described above in order to prove that it can be

performed by a family of boolean circuits with size(shh1(rnd)n)O(1) and depthO

(n2 log(rnd) log(sndhh1)

), which is uniform in spaceO(n2 logndshh1).

We assume that we are given the dense representation of the polynomials(Fij )1≤i≤r,1≤j≤s and a sequence of(α(k), β(k))1≤k≤N of (2r − 1)-tuples of inte-gers of logarithmic height bounded byO(n logsd) which define a correct testsequence in the sense of Heintz–Schnorr of lengthN := (

s(rd)n)O(1)

elements.We first observe that there exists a family of boolean circuits computing the

polynomialsFij on integers of logarithmic heighth1 with sizers(hh1dn)O(1)

and depthO((n + logd) logdhh1


(n logdhh1

).

We remark that the resulting integers have logarithmic heightO(h + dh1 +n logd).

We also compute the matricesKk for 1 ≤ k ≤ N by means of a familyof boolean circuits of size

(sn(rd)n logd

)O(1)and depthO

(log(s) log(snd)

),

which is uniform in spaceO(n logsdn). The resulting integers have logarithmicheightO(sn logsd).

Then we compute the matrix productF (k) = [F (k)1 , . . . , F

(k)r+1] := F · Kk,

the determinantsD(k)1 := det [F (k)

1 , . . . , F (k)r ] andD

(k)2 := det [F (k)

1 , . . . , F(k)r−1,

F(k)r+1] and the resultantck := ResXn

(D(k)1 , D

(k)2 ) for 1 ≤ k ≤ N . Taking into

account that the entries of the matrixF (k) have logarithmic heightO(h+dh1+sn logsd), the polynomialsD(k)

1 , D(k)2 have degree(rd)2 and logarithmic height

O(rh + rdh1 + rsn logsd) on integers of logarithmic heighth1, and that thecoefficients which result from the interpolation ofD

(k)1 , D

(k)2 with respect to

Xn have logarithmic height(sdhh1n)O(1) on integers of logarithmic heighth1,combining the Macro expansion lemma with Lemmata 4, 6 and 13 we deducethat there exists a family of boolean circuits computing the polynomialck for 1 ≤k ≤ N on integers of logarithmic heighth1 with sizeshh1(rd)O(n) and depthO

(log(rd) log(sndhh1)

), which is uniform in spaceO(n logsndhh1). These

polynomials have logarithmic height(sdhh1n)O(1) on integers of logarithmicheighth1.

Applying the ideas of the proof of Theorem 24 we obtain a family of booleancircuits computing polynomialsa1, . . . , aN ∈ Z[X1, . . . , XN ] on integers of

516 G. Matera

logarithmic heighth1 and a nonzero integerγ such that the following equationholds:

γXn = a1c1 + · · · + aNcN

This family of boolean circuits has size(shh1(rnd)n)O(1) and depthO


)and is uniform in spaceO(n2 logndshh1). We re-

mark that the resulting integers have logarithmic height bounded by(shh1(rd)n

)O(1).

Then we compute the matrixM(k)n of equation (15) for 1≤ k ≤ N . For this

purpose we first compute the polynomialsgk, hk of equation (17) applying theideas of Lemma 11 and then we evaluate them inXn = bk applying Lemma 3.Then we compute the vectorsG

(k)j of equation (19) and the matricesB

(k)1 , B

(k)2

of equations (21) and (22) for 1≤ k ≤ N , applying Lemma 8. We compute theproductB(k)

1 gk(bk)G(k)j andB

(k)2 hk(bk)G

(k)j for 1 ≤ k ≤ N applying Lemma

4, the matrixT (k) for 1 ≤ k ≤ N applying Lemmata 8 and 4 and finally thematrixM(k)

n we are looking for. We observe that the integers involved in thesecomputations have logarithmic height bounded by

(shh1(rd)n

)O(1). Therefore,

combining the Macro expansion lemma with Lemmata 11, 3, 8 and 4 we obtaina family of boolean circuits computing the matrixM(k)

n for 1 ≤ k ≤ N with size(shh1(rnd)n)O(1) and depthO



spaceO(n2 logndshh1).Finally, a combination of Lemmata 8 and 4 yields a family of boolean cir-

cuits computing the matrixMn with size, depth and space-uniformity as above.A final multiplication of the matricesMn, . . . , M0 yields the unimodular matrixM we are looking for. As a consequence of the bounds for the complexity of thecomputation of the matricesM0, . . . , Mn, we deduce that there exists a family ofboolean circuits computing the matrixM with size(shh1(rnd)n)O(1) and depthO


), which is uniform in spaceO(n2 logndshh1).

Now we can describe the probabilistic Turing machineT of the statementof the theorem. This machineT starts guessing theO(n2) integer parametersof logarithmic heightO

(n2 log(nsd)

)required to apply the ideas of Theorem

24. Since all these parameters are obtained by applying Schwartz–Zippel test,we can conclude that they can be randomly chosen with probability of success� 1

2. These parameters are stored in working space.Then we evaluate the boolean circuitC which computes the matrixM on

input the dense representation of the polynomialsFij for 1 ≤ i ≤ r, 1 ≤ j ≤ s.Recall that the circuitC has size(shh1(rnd)n)O(1) and depthO

(n2 log(rnd)

log(sndhh1)), and that there exists a deterministic Turing machineM1 com-

puting the standard encoding ofC using spaceO(n2 logndshh1).Then the machineM evaluates the circuitC following Borodin’s argument

in the same way as in the proof of Theorem 23. Therefore, the machineM deter-ministically evaluates the circuitC using only additional spaceO

(n2 log(rnd)

log(sndhh1))

and time(sndhh1)O

(n2 log(rnd)

).


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Probabilistic Algorithms for Geometric Elimination