Computer algebra, integrability, involutivity and all that

Computer algebra, integrability, involutivity and allthat

Vladimir Gerdt

Laboratory of Information TechnologiesJoint Institute for Nuclear Research

141980, DubnaRussia

February 20, 2007, Turku

Gerdt (JINR,Dubna,Russia) Computer algebra, integrability, involutivity and all that CADE 2007 1 / 33

Contents

1 Introduction

2 Differential Systems in Involution

3 Involutive Division

4 Gröbner and Involutive Bases

5 Difference AlgebraRings of Difference PolynomialsRankingGröbner Bases

6 Some Implementations of GB/IB

7 Summary


IntroductionMost of real-world problems are described by systems of equationsrather then by an isolated single equation. Moreover, most oftenequations which arise in natural sciences and engineering are ofpolynomial type with respect to unknowns.Thus, in development of all three computer-aided approaches

1 symbolic algebraic analysis,2 numerical solving,3 visualization,

one has to pay a primary attention to systems of polynomially-nonlinear equations.

Goal of symbolic algebraic analysis: given equations, extract fromthem as much information on solutions as possible without (generallyimpossible) explicit integration/solving and/or "simplify/rewrite" theequations for the further numerical solving.


But what can we hope to do algorithmically in the general(polynomially-nonlinear) case of equation systems?

Check compatibility, i.e., consistency.Detect dimension of the solution space("arbitrariness" in general analytical solution for DEs).Eliminate a subset of variables.Reduce the problem to (a finite set) of "simpler" problems.Check satisfiability of an extra equation on the solutions.Find Lie symmetries (DEs).Formulate a well-posed initial value problem (PDEs).Compute "hidden constraints" for dependent variables ornumerical indices (ADEs).Rewrite into another form more appropriate for numerical solving.Generate finite difference schemes (for PDEs).........................................................................................


Is there a "universal" algorithmic tool for the listedsubproblems?

If the system has polynomial nonlinearity in unknowns with"algorithmically computable" coefficients, then such a tool exists andbased on transformation of the system into another set of equationswith certain "nice" properties.

For the conventional polynomial systems and some theirgeneralizations to noncommutative polynomials, for linear PDEs andlinear finite difference equations (FDEs) / recurrence relations (RRs)such a form is canonical, i.e., uniquely defined by the initial systemsand an order on the variables, and called reduced Gröbner basis (GB)(Buchberger’65).

Another "nice" canonical form is called Involutive Basis (IB) (Gerdt,Blinkov’98). IB is also GB, although (in most cases) redundant as aGröbner one.


Simple ExamplesCompatibility{

ux + 1 = 0,uy + u = 0

cross−derivation=⇒

{∂y (ux + 1) = 0,∂x(uy + u) = 0

=⇒ ux = 0 =⇒ 1=0 (!)

contradictionSolution space{

uxxy = 0,uxyy = 0

=⇒ u = C xy + f1(x) + f2(y)

arbitrary constant and two functionsElimination {

uxxy − u = 0,uxyy − u = 0

ux�uy

=⇒

{ux − uy = 0,

uyyy − u = 0

⇑ ux eliminated


ApplicabilityThe method of Gröbner / Involutive bases has been applied to:

commutative algebra and algebraic geometryinvariant theoryLie symmetry analysis of differential equationsdynamical systemspartial differential equationssymbolic summation and integrationnon-commutative algebraroboticsnumerics (e.g. wavelets construction and difference schemesgeneration)systems theory (e.g. control theory)constrained dynamics (Dirac’s formalism)reduction of loop Feynman integrals.....................................


Basic Idea

The general strategy of the Gröbner basis approach is toTransform a set F of equations (that describes the problem athand) another set G of polynomials with certain "nice" properties(called a Gröbner basis) such thatF and G are "equivalent" and G is "simple" than F .

From the theory and practice of Gröbner bases it is known:Because of some "nice" special properties of Gröbner bases,many problems that are difficult for general F are "easy" forGröbner basis GThere are algorithms and their implementations for transformingan F into G.The solution of the problem for G can often be easily translatedback into a solution of the problem for F .


Constructive Theory of Involutivity

Cauchy & Kovalevskaya (1875): Normal system of PDEsCartan (1899, 1901): Involutivity of exterior PDEs.Riquier (1910), Janet (1920), Thomas (1937): Involutivity of PDEs.Quillen (1964), Spencer (1965), Kuranishi (1967), Goldschmidt (1969):Formal Theory of differential systems (cf. Pommaret, 1978).Reid (1991): Standard Form of linear PDEs.Wu (1991): Relation of Riquier-Janet theory to Gröbner bases.Zharkov, Blinkov (1993): Pommaret Bases of polynomial ideals.Reid, Wittkopf, Boulton (1996): Reduced Involutive Form (Rif) of PDEs.Gerdt, Blinkov (1995-1998): Involutive Division =⇒ general Involutive Bases.Apel (1998): Admissible Involutive Division on a monomial set.Gerdt (1999): Involutive Systems of Linear PDEs.Seiler (2002): Combinatorial Aspects of Involutivity.Chen, Gao (2002): Involutive Characteristic Sets for PDEs.Hemmecke (2003): Sliced Involutive Division.Evans (2004): Noncommutative Involutive Bases.Gerdt, Blinkov (2005): Janet-like Division.


ImplementationArais, Shapeev, Yanenko (1974): Cartan algorithm in Auto-Analytik.Schwarz (1984): Riquier-Janet theory in Reduce.Hartley, Tucker (1991): Cartan algorithm in Reduce.Schwarz (1992): Linear differential Janet bases (DJB) in Reduce.Reid, Wittkopf, Boulton (1993): Standard Form and Rif (2000) in Maple.Seiler (1994): Formal theory in Axiom.Zharkov, Blinkov (1993); G., Blinkov (1995): Pol. Pommaret bases (PPB) inReduce.Kredel (1996): PPB in MAS.Nischke (1996): Polynomial JB (PJB) and PPB in C++ (PoSSoLib).Berth (1999): Polynomial and differential involutive bases in Mathematica.Cid (2000)-Robertz (2002-2005): PJB, DJB and Difference JB in Maple.Blinkov (2000-2005): PJB in Reduce, C++.Yanovich (2001-2004): PJB in C, Singular.Hausdorf, Seiler (2000-2002): DJB and DPB in MuPAD.Chen, Gao (2002): Involutive Extended Characteristic Sets in Maple.Hemmecke (2002): Sliced Division Algorithm in Aldor.Evans (2005): Noncommutative Involutive Bases in C.


Cauchy-Kovalevskaya TheoremA normal system of PDEs

∂mj uj

∂xmj1

= fj

(x , u, . . . ,

∂µ1+···+µn u∂xµ1

1 · · · ∂xµnn

)(1 ≤ j ≤ k)

x = (x1, . . . , xn), u = (u1, . . . , uk ),∑n

i=1 µi ≤ mj , µ1 < mj ≥ 1 which isanalytic at

xi = xoi , uj = uo

j ,∂µ1+···+µn uj

∂xµ11 · · · ∂xµn

n= po

j;µ1···µn, (1 ≤ i ≤ n, 1 ≤ j ≤ k)

has a unique analytic solution at (x01 , . . . , xo

n ) satisfying the initial data

uj = φj(x2, . . . , xn),∂uj

∂x1= φ

(1)j (x2, . . . , xn),

..................................... (1 ≤ j ≤ k)∂mj−1uj

∂xmj−11

= φ(mj−1)j (x2, . . . , xn),

for x1 = xo1 with functions φj , . . . , φ

(mj−1)j analytic at {xo

2 , . . . , xon }.

Normal systems are particular cases of involutive systems.Gerdt (JINR,Dubna,Russia) Computer algebra, integrability, involutivity and all that CADE 2007 11 / 33

Differential Systems in InvolutionLet Rq be a system of PDEs of order q in n independent variables xi(1 ≤ i ≤ n) and m dependent variables uα (1 ≤ α ≤ m)

Rq :{Φj(xi , uα, pα

µ) = 0 (1 ≤ j ≤ k) manifold in {pα|µ|≤q }

where µ = {µ1, . . . , µn} is multi-index, |µ| =∑n

i=1 µi ≤ q and

pαµ =

∂|µ|uα

∂xµ≡ ∂µ1+···+µnuα

∂xµ11 · · · ∂xµn

n, pα

|µ|=0 = uα

Definition An integrability condition for Rq is an equation of order ≤ qwhich is differential but not pure algebraic consequence of Rq.Example ( Seiler )

R1 :

{uz + y ux = 0uy = 0

=⇒{

uyz + y uxy + ux = 0uxy = uyz = 0

=⇒ ux = 0

=⇒ R1 : {ux = uy = uz = 0


Example (ODEs)

R2 :

{x + y2 = 0x + x y + t x = 0

=⇒ 2 y y − x y − t x = 0

Definition The 1st prolongation Rq+1 of Rq

Rq+1 :

{Φj(xi , uα, pα

µ) = 0 (1 ≤ j ≤ k)DiΦj = 0 (1 ≤ i ≤ n)

where Di is the total derivative operator w.r.t. xi . Similarly, Rq+r isobtained by r prolongations of R.Definition R(1)

q = πq+1q

(Rq+1

)is the projection of Rq+1 in {pα

|µ|≤q }.Similarly, R(s)

q+r = πq+r+sq+r (Rq+r+s) is obtained from Rq by r + s

prolongations and s projections.Generally,

R(1)q+r ⊆ Rq+r =⇒ dim R(1)

q+r ≤ dim Rq+r

and the number of (algebraically independent) integrability conditionswhich arise at the (r + 1) prolongation step is

dim Rq+r − dim R(1)q+r


Definition A formally integrable system Rq has all the integrabilityconditions incorporated in it, that is,

(∀ r , s) [ R(s)q+r+s = Rq+r ]

Involutive system Rq is a formally integrable one with the complete(involutive) set of the leading derivatives (symbol of Rq).

Definition Given a system Rq, its transformation into an involutive formis called completion.

Theorem ( Cartan-Kuranishi ) For every differential system Rq thereexist integers r , s such that R(s)

q+r is involutive with the same solutionspace.


Involutive Division

Definition: An involutive separation L of variables is defined on M if forany finite monomial set U ⊂ M and for any u ∈ U there is defined asubset M(u, U) ⊆ X = {x1, . . . , xn} of variables generating monoidL(u, U) ≡ MM(u,U) such that

1 u, v ∈ U, uL(u, U) ∩ vL(v , U) 6= ∅ ⇐⇒ u ∈ vL(v , U)or v ∈ uL(u, U).

2 v ∈ U, v ∈ uL(u, U) ⇐⇒ L(v , U) ⊆ L(u, U).3 V ⊆ U =⇒ L(u, U) ⊆ L(u, V ) ∀u ∈ V .

Variables in M(u, U) are called (L−)multiplicative for u and those inNM(u, U) ≡ X \M(u, U) are (L−)nonmultiplicative for u, respectively.

involutive separation ⇐⇒ involutive division (G., Blinkov’98)

If w ∈ uL(u, U) then u is involutive divisor of w : u |L w =⇒ involutivereduction and involutive normal form NFL(f , F ) where f ∈ R andF ⊂ R.


Janet and Pommaret Division

Definition: (Janet’20, G.,Blinkov’98) For each finite monomial setU ⊂ M and 0 ≤ i ≤ n partition U into groups labeled byd0, . . . , di ∈ N≥0 (U = [0])

[d0, d1, . . . , di ] := {u ∈ U | d0 = 0, d1 = deg1(u), · · · , di = degi(u)}.

Variable xi is J-multiplicative for u ∈ U if u ∈ [d0, . . . , di−1] and

degi(u) = max{degi(v) | v ∈ [d0, . . . , di−1]} .

Notation: degi ≡ degxi, u @ v ⇐⇒ u | v ∧ u 6= v

Definition: ( Pommaret division ). For v = xd11 · · · xdk

k with dk > 0(k ≤ n) the variables xj , j ≥ k are multiplicative and the other variablesare nonmultiplicative. For v = 1 all the variables are multiplicative.


Example: U = {x21 x3, x1x2, x1x2

3}

Element Separation of variablesin U Janet Pommaret

MJ NMJ MP NMPx2

1 x3 x1, x2, x3 − x3 x1, x2x1x2 x2, x3 x1 x2, x3 x1x1x2

3 x3 x1, x2 x3 x1, x2

Definition: A monomial set U ∈M is L-complete or L-involutive if

(∀w ∈M) (∀u ∈ U) (∃v ∈ U) [ v |L u · w ]

The corresponding J-and P−completion of U = {x21 x3, x1x2, x1x2

3}

Janet : {x21 x3, x1x2, x1x2

3 , x21 x2},

Pommaret : {x21 x3, x1x2, x1x2

3 , x21 x2, . . . , x i+2

1 x2, . . . , x j+21 x3, . . .}

The Pommaret division is non-Noetherian


Gröbner and Involutive BasesA finite set F = {f1, . . . , fm} ∈ R := K[x1, . . . , xn] of multivariatepolynomials is a basis of the ideal I

I =< F >= {m∑

i=1

hi fi | hj ∈ R }

Given a polynomial set F and a linear monomial order � such that

(i) m 6= 1 =⇒ m � 1, (ii) m1 � m2 ⇐⇒ m1m � m2m

∀m, m1, m2, one can select the leading monomial lm(f ) of any f ∈ Rand define a Gröbner basis G ⊂ R of ideal I =< G >:

(∀f ∈ I) (∃g ∈ G) [ lm(g) | lm(f ) ]

Similarly, given an involutive division L, an involutive basis H of I isdefined as

(∀f ∈ I) (∃h ∈ H) [ lm(g) |L lm(f ) ]

An involutive L-basis is a (generally redundant) Gröbner basis with theL-complete set of leading monomials


Definition:Given a finite set F ⊂ R, a polynomial p ∈ R, and a monomial order�, a normal form NF (p, F ) of p modulo F is defined as

NF (p, F ) = p′ = p −∑

ij

αijmij fj

where αij ∈ K, fj ∈ F , mij ∈M, lm(mijgj) � lm(p) and there are nomonomial in p′ multiple of any leading monomial of elements in F .

Similarly, an (L−)involutive normal form NFL(p, F ) is defined. Theonly distinction is that in the latter case all the monomial factors mijmust be L−multiplicative for fj , i.e. mij ∈ L(fj , F ), and p′ cannot containmonomials L−multiple of any leading monomials in F .

This yields another definition of a Gröbner (GB) G and an involutive(IB) basis H of ideal I:

GB : p ∈ I ⇐⇒ NF (p, G) = 0 ,

IB : p ∈ I ⇐⇒ NFL(p, H) = 0 .


Algorithms

Gröbner bases can be computed by Buchberger’s algorithm(Buchberger’85) which implemented in most of moderngeneral-purpose computer algebra systems such as Maple,Mathematica, Reduce, MuPAD, etc, or by more efficient algorithms: F4(Faugère’98), involutive algorithm (Gerdt’05).

The fastest implementations are inMaple (library FGb implementing F4)Magma (F4)JB and GINV (Involutive algorithm) http://invo.jinr.ruSingular (Buchberger’s and involutive algorithms)


Simplest Form of Buchberger’s Algorithm

Algorithm: Gröbner Basis(F ,�)

Input: F ∈ K[x1, . . . , xn] \ {0}, a finite set; ≺, an orderOutput: G, a Gröbner basis of Id(F )1: G := F ;2: do3: choose a pair f1, f2 ∈ G and compute S(f1, f2)4: h := NF (S(f1, f2), G)5: if h = 0 then6: goto 3 and choose the next pair7: else8: G := G ∪ {h}9: fi

10: od while h 6= 011: return G

S−polynomial S(f1, f2) := c1t1f1 − c2t2f2. Here c1, c2 ∈ K and t1, t2 aremonomials such that c1t1lm(f1) = c2t2lm(f2).


Algorithm: Involutive Basis (F ,≺,L)

Input: F , a polynomial set; ≺, a monomial order; L, an involutive divisionOutput: G, a minimal involutive basis of Id(F )1: choose f ∈ F without g ∈ F \ {f} : lm(g) @ lm(f )2: G := {f}; Q := F \ G3: do4: h := 05: while Q 6= ∅ and h = 0 do6: choose p ∈ Q without q ∈ Q \ {f} : lm(q) @ lm(p)7: Q := Q \ {p}; h := NFL(p, T )8: od9: if h 6= 0 then

10: for all {g ∈ G | lm(g) A lm(h)} do11: Q := Q ∪ {g}; G := G \ {g}12: od13: G := G ∪ {h}; Q := Q ∪ { g · x | g ∈ G, x ∈ NML(g, G) }14: fi15: od while Q 6= ∅ return G


Example

Table: Computation of Janet basis for F = {x2y − 1, xy2 − 1}, x � y

Steps of Sets G and Qalgorithm elements in G NMJ Q

initialization xy2 − 1 − {x2y − 1}iteration x2y − 1 −

xy2 − 1 x {x2y2 − x}x − y − {xy2 − 1, x2y − 1}x − y −y3 − 1 x {x2y − 1, xy3 − x}x − y −y3 − 1 x { }


Some Properties of (minimal) IB

Uniqueness of monic IB. Ideal(F ) = Ideal(G) ⇐⇒ IB(F )=IB(G).Idempotency of IB.G := IB =⇒ IB(G)=GTrivial Ideal. Ideal(F ) = K [x1, . . . , xN ] ⇐⇒ 1 ∈ IB(F ).Solvability. A system of equations F is solvable ⇐⇒ 1/∈ G.Finite Solvability. Polynomial system F has only finite manysolutions ⇐⇒ ∀ 1 ≤ i ≤ n | ∃f ∈IB(F ) such that lm(f ) is a power ofxi .Number of Solutions. The number of solutions of F (withmultiplicities) = cardinality of {u |u /∈ "set of multiples oflm(IB(F ))"}.


Contents

1 Introduction






7 Summary


Rings of Difference PolynomialsLet {y1, . . . , ym} be the set of difference indeterminates, e.g. functionsof n−variables {x1, . . . , xn}, and θ1, . . . , θn be the set of mutuallycommuting difference operators (differences), e.g.,

θi ◦ y j = y j(x1, . . . , xi + 1, . . . , xn).

A difference ring R with differences θ1, . . . , θn is a commutative ring Rwith a unity such that ∀f , g ∈ R, 1 ≤ i , j ≤ n θi ◦ f ∈ R and

θiθj = θjθi , θi ◦ (f + g) = θi ◦ f + θi ◦ g, θi ◦ (f g) = (θi ◦ f )(θi ◦ g)

Similarly one defines a difference field.

Remark. The above and below concepts are translated to differentialalgebra if θi are the partial derivations

θi ◦ y j = ∂iy j(x1, . . . , xi , . . . , xn).


Let K be a difference field. Denote by R := K{y1, . . . , ym} thedifference ring of polynomials over K in variables

{ θµ ◦ yk | µ ∈ Zn≥0, k = 1, . . . , m } .

Denote by RL the set of linear polynomials in R and use the notations

Θ = { θµ | µ ∈ Zn≥0 } .

A difference ideal I in R is an ideal I ∈ R close under the action of anyoperator from Θ. If F := {f1, . . . , fk} ⊂ R is a finite set, then thesmallest difference ideal containing F denoted by Id(F ). If F ⊂ RL,then Id(F ) is a linear difference ideal.


Contents

1 Introduction






7 Summary


Ranking

A total ordering ≺ over the set of θµy j is a ranking if it satisfies1 θiθ

µ ◦ y j � θµ ◦ y j

2 θµy j � θν ◦ yk ⇐⇒ θiθµ ◦ y j � θiθ

ν ◦ yk ∀ i , j , k , µ, ν.

If µ � ν =⇒ θµ ◦ y j � θν ◦ yk the ranking is orderly.If i � j =⇒ θµ ◦ y j � θν ◦ yk the ranking is elimination.

Given a ranking �,

every linear polynomial f ∈ RL \ {0} has the leading term a θ ◦ y j ,θ ∈ Θ;lc(f ) := a ∈ K \ {0} is the leading coefficient;lm(f ) := θ ◦ y j is the leading monomial.


Contents

1 Introduction






7 Summary


Gröbner Bases

Given nonzero linear difference ideal I = Id(G) and term order �, itsgenerating set G = {g1, . . . , gs} ⊂ RL is a Gröbner basis (GB)(Kondratieva,Levin,Mikhalev,Pankratiev’99) of I if

∀f ∈ I ∩ RL \ {0} ∃g ∈ G, θ ∈ Θ : lm(f ) = θ ◦ lm(g) .

It follows that f ∈ I is reducible modulo G

f −→g

f ′ := f − lc(f ) θ ◦ (g/lc(g)), f ′ ∈ I, . . . , f −→G

0 .


Some Computer Algebra Systems and Packages

Software Commutative PDE LFDE Languagealgebra

Maple + diffalg Ore_algebra MapleRif Maple

Gb CFGb C

Mathematica + − − CReduce + − − Lisp

OreModules − LPDE LFDE MapleJanet − LPDE − MapleLDA − − LFDE MapleGINV + − − Pyton/C++

JB + − − C


Summary

Completion of algebraic/differential/difference systems toinvolution is the most general and universal algorithmic techniquesfor study their algebraic properties.Normal systems of PDEs are particular cases of involutivesystems.Involutive systems have all their integrability conditionsincorporated in them that makes easier their qualitative andquantitative (numerical) analysis.There are efficient algorithms and implementing them software forcomputing polynomial, linear differential and difference GB/IB.Our publications, computer experiments and GINV software areavailable on the Web: http://invo.jinr.ru


Summary



Summary



Summary



Summary



Summary



Documents

Computer algebra, integrability, involutivity and all that