lnventiones math. 51,189-203 (1979) Inventione$ mathematicae �9 by Springer-Verlag 1979

Ultraproducts and Approximation in Local Rings I

Joseph Becker 1., J. Denef 2.*, L. Lipshitz 3, and L. van den Dries 4 .**

Purdue University, West Lafayette, Indiana 47907 2 Princeton University, Princeton, New Jersey 08540 3 Institute for Advanced Study, Princeton, New Jersey 08540, USA 4 University of Utrecht, Utrecht, Netherlands

In this paper we give new and quite elementary proofs of the strong approxima- tion theorems of M. Greenberg (Theorem 2.1), M. Artin (Theorem 3.2) and D. Popescu (Theorem 3.3). Our proofs use the ultraproduct construction. In w ! we give a brief outline of this construction and derive the properties we shall use.

Our method of proof gives a quite general way of deriving a strong approximation theorem from the corresponding approximation theorem. (By an "approximation theorem" we mean a theorem of the type of Theorem 3.1 and by a "strong approximation theorem" we mean a theorem of the type of Theorem 3.2 or 3.3.) For example if we have an approximation theorem with some of the Y's constrained to depend only on some of the X's then we can immediately derive the strong approximation theorem with the same con- straints. In w 4 we give a pair of such theorems with the constraints that some of the Y's depend only on X 1.

In w 5 we present some counterexamples: Let k be the algebraic closure of the field with p elements. There is a system of polynomials f ( X t, X2, Yt . . . . . Yv) over k such that the system f (X1, X2, Yt .... ,Y v)=0; Y1, Y6ek[[_X2~; Y2, Y3, Y6ek[[XI~ has a solution mod (X1, X2) J, for every j e N but has no solution in k[[_Xz, X2~. We also give a similar counterexample over the rationals.

In w 6 we discuss effectivity. In Ultraproducts and Approximation in Local Rings II we shall extend the

methods of this paper to give proofs of some further strong approximation theorems, e.g. the Theorem of Pfister and Popescu [17] as well as some new results.

The usefulness of the ultraproduct construction (or equivalently an appropri- ate form of the G~Sdel Completeness and Compactness Theorems) in analyzing algebraic questions about valued fields is due to Ax and Kochen [3] (cf. also

* Supported in part by the N.S.F. ** Supported by the Belgian Nationaal Fonds voor Wetenschappelijk Onderzoek *** The proofs in Sects. 2, 3 and 6 of this paper were found by L. van den Dries and the other three authors independently. Van den Dries would like to thank W. Baur for a helpful communication in connection with the proof of Theorem 2.1 below

0020-9910/79/0051/0189/$03.00

190 J. Becker et al.

Er~ov [9]). In [3] the similarity between the fields •p of p-adic numbers and the fields Fp((t)) of formal power series over the p-element field is made precise by the theorem (under the continuum hypothesis) that if D is a nonprincipal ultrafilter on the set of primes (see the definitions in w below) then the ultraproducts (I~ff).p)/D and (]-IFp((t)))/D are isomorphic. (The proof of this

P P

theorem is entirely algebraic.) This has the immediate consequence that for any first order statement q~ about valued fields, (o is true in all but a finite number of the Qp'S if and only if (p is true in all but a finite number of the fields Fr((t)).

Our use of the ultraproduct construction is somewhat similar.1 Classically the additional difficulty in proving strong approximation theorems, over and above that of proving approximation theorems, seems to be that one does not have a point at which to apply the algebraic and geometric methods - one only has an "approximate point". The ultraproduct construction eliminates this difficulty by providing an appropriate point and hence allows one to deduce the strong approximation theorems from corresponding approximation theorems.

As a simple illustration of these ideas consider the following well-known result. (This example does not quite fit the pattern given above but illustrates the role played by the ultraproduct construction.) Let f (X ,Y)eC[X, Y3 (X =(X1,. . . ,X,) , Y=(YI,...,YN)) and suppose that for every j e N there is a yjOl2[X] such that f (X, y j ) - 0 rood(X) j. Then there is a y~(l~[[X]] such that f(X,y)=O. One can give a very simple proof of this as follows. Let * denote the ultrapower with respect to a nonprincipal ultrafilter on N, (see w 1). Then from the assumption that for every j ~ N there exists a Yi such that f(X, yi)-O rood(X) J we know that in I/~[X]* there is a ~ such that f(X,y,)-=O mod(XV-, where (X)~= ~ (X)3cII?[X] *. But ff;[X]*/(X) ~176 ~-C*[[X]] and hence there is a

j e N

~ellS* [I-X~ such that f(X, y;)=0. Also ~* ~ (over the subfield generated by the coefficients of f, since they are both algebraically closed and of transcendence degree 2~~ Hence the required solution in I1~ []-X]] exists. Notice that the same proof works if we have constraints that some of the Y~'s depend only on some of the Xj's.

A. Robinson in [19] gave a proof of a special case of Greenberg's Theorem, using the results of Ax and Kochen [3]. Motivation has also come from the paper of Nerode [16].

w 1. Ultraproducts. The reader who is familiar with the ultraproduct construction can skip this section. For the reader who is unfamiliar with this construction we shall give a brief survey of the facts we shall use. More information about ultraproducts can be found in w 1 of [33 or any standard model theory text such as [73 .

In the cases we shall consider the index set will always be N, the natural numbers. A filter on N is a nonempty family D of subsets of N satisfying (i) qSCD, (ii) if s, tED then scTteD and (iii) ifseD and sc_tclN then teD. A filter D on N is principal if there is an reD such that D={tlr~_tc_lN}, otherwise it is

1 It is well-known that the results of Ax and Kochen and Er~ov don't generalise to local rings of dimension greater than one, see for example [8]

Ultraproducts and Approximation in Local Rings I 191

nonprincipal. An ultrafilter D on N is a filter on N which is maximal with respect to inclusion in the class of all filters on N. It is an exercise to show that a filter D on N is an ultrafilter if and only if for every r ~ N exactly one of r or N - r belongs to D. It is also easy to show that an ultrafilter D is nonprincipal if and only if it contains the Fr6chet filter F of cofinite subsets of N, i.e. D ~ F = { s _ N l N - s is finite}. The existence of nonprincipal ultrafilters follows by applying Zorn's Lemma to the class of filters on N which contain F.

Let A i, i e N be a family of algebraic structures (e.g. rings, valuation rings, local rings, algebras kl IX] where k i is a field). Let D be an ultrafilter on N. We form the ultraproduct A* =(1-[ A~)/D (or the ultrapower A~/D if all the A~=A) as

i e N

follows. On the cartesian product 1-1 A~ (an element a of the cartesian product is iEN

a sequence iF--,a(i)) define an equivalence relation ~ by a~a' if and only if {ila(i)=a'(i)}eD. We denote the equivalence class of a by [a]. We define A* =(1-IAi)/D as the set of equivalence classes [a] with the operations and

relations on A* defined componentwise e.g. [a] + [ b ] = [ a + b ] and [a]- [b] = [a-b] , where (a +b)(i)=a(i) +b(i) and (a. b)(i) =a(i) . b(i).

The zero of A* is the equivalence class of the constant sequence 0(i)=0 and the unity of A* is the equivalence class of the constant sequence 1 (i)= 1. If each Ai has a total order relation < (e.g. A~ =Z), then A* has a total order relation < defined by [a] =<[a'] ~{ i l a ( i )<a ' ( i ) }eD. If each A i has a valuation ordi: A i

- { 0 } ~ then A* has a valuation ord*: A * - { O } ~ * = T Z , N/D defined by ord*([a])=[ordi(a(i)) ]. It is easy to see that these operations are well defined, using the fact that D is an ultrafilter. If all the A~=A then there is a natural embedding #: A--~ A* defined by #(a)= [a(i)] where a(i)= a for all i. In such cases we shall identify A with its image in A*.

The fundamental property of ultraproducts is the following

Theorem (Log, cf. [7, Theorem 4.1.9, p. 170]). Let L be afirst order language, let A i, i~1 be structures .[br L and let D be an ultrafilter on I. Then .for any [a]~(1-IAi)/D=A* and any.first order formula q~(x), q~([a]) is true in A* if and

i e l

only if {i[qo(a(i)) is true in Ai}~O.

The proof of this is quite straightforward (cf. [7, Theorem 4.1.9, p. 170]) but we shall not give it. We shall indicate below the proofs in a few special cases which we shall use later.

(i) If k is a field and D is an ultrafilter on N then it is easy to see that k* = kN/D is a field. For example if [a]~k* and [ a ] # 0 then there is an seD such that a(i)+O for all ies. Define b(i)=a(i) -1 if i ts and b( i )=0 if i(ss. Then for ies a(i). b(i) = 1 and hence [a]- [b] = 1.

Let R be a discrete valuation ring with valuation ord: R-{0}- -*Z , and prime p. Let R*=R•/D for some nonprincipal ultrafilter D. Let 7g*=7~/D, k =R/(p) and k*= k~/D. Then R* is a valuation ring with valuation ord* (defined as above) into 71", with prime p, (we identify R with # (R)~R* defined as above) and residue class field k* (i.e. R*/(p)= k*). If R is Henselian then so is R*. Z is a convex subgroup of Z* (under the embedding #) i.e., if [ a ]eZ* , m, neZ and

192 J. B e c k e r et al.

m < [a] < n then [ a ] t Z (more precisely/~(Z)). Moreover Z is properly contained in Z*, since D is nonprincipal. All of this follows directly from the definitions. We shall prove some of these assertions:

One can see that R* is a valuation ring as follows. Suppose ord*([a]) <ord*([b]) . Then, for some s tD, ord(a(i))<ord(b(i)) for all its. Hence for i t s there exists a ci~R such that a(i)q=b(i). Let c( i )=q for i t s and c(i)=0 for i e N - s . Then a(i)c(i)=b(i) for i ts, and [a] [c] = [b], since seD.

Suppose that R is Henselian (cf. [15, w 30, p. 103]). Let

f ( x ) = x " + [a,_ 1] x"- 1 + ... + [ao] '

go(x) = xk + [bk- 1] xk- 1 +.. . + [bo] and

ho(x) =x" -k+ [C,-k- 1] X"-*- 1 + ... + [Co] t R . Ix]

satisfy (1)f (x)- -go(x)ho(x) tp R* [x] and (2) go modulo p and h o modulo p have no common root (i.e. go R* Ix] + h o R* Ix] + p R* Ix] = R* Ix]). Then, there is an s t D such that for all i t s we have f i (x ) -go i (x )ho i (X) tpR[x] and goi(X) modp and hoi(X ) modp have no common root (where f i ( x )=x"+a , l ( i ) x " - l + . . . +ao(i ) etc.). Hence (since R is Henselian) for i t s there exist polynomials

x +bk_l( i )xk- l+. . .+b 'o( i ) and g i ( x ) = k ,

hi(x)_ = , �9 , -k C._k(OX + ... +C'o(i)~R[x]

such that gi(x)-g0i(x) and hi(x ) =-hoi(X ) modp and f/(x)=gi(x)hi(x ). Let

g(x)=x* + [b' k_ 1] xk- 1 + ... + [b~] and

h(x)= [C',_k] x"-k + ... + [c;] tR* [X].

Then g(x) --- go(x), h ( x ) - ho(x) mod p (in R* [x]) and f ( x ) = g(x) h(x) (since these statements are true for all its). Hence R* is Henselian.

To see that 7Z is a convex subgroup of Z*, suppose that m < [ a ] < n for m, nt2g and [ a ] tZ* . Then, for some s tD, m<a( i )<n for all its. Let s j={i ts la( i )

=j} for rn<j<n. Then s = 0 st and hence, for some jo (m<jo<n), S~otD. (For )=m

suppose not. Then N - s f i D for j = m .... ,n and hence (~ ( N - s j ) = N - s t D ) . j = m

Thus [a] =joEZ. Moreover let [7] t Z * be the equivalence class of the sequence 7(i)=i. Then [7] #:z, for all zeTZ, since D is nonprincipal.

(ii) For each i t N let A i be a quasi-local ring, i.e. a ring with only one maximal ideal JCi. Let D be a nonprincipal ultrafilter on N and let A * = ( H

i sN

AI)/D. Then A* is again a quasi-local ring with maximal ideal ~ ' , defined by

[a] t . g ~ {i[a(i)tJti} tD.

If the residue class field ofA i is kl, then the residue class field of A* is k* =(1-[ ieN

kt)/D. If the ideal J/t' i is generated by Xo(i ) . . . . . x,(i) for every i, then ~ ' is

Ultraproducts and Approximation in Local Rings I 193

generated by x 0 . . . . ,x , where xj denotes the equivalence class of the sequence xj(i).

As above, if A is Henselian, then A* is Henselian too.

(iii) Ultraproducts have certain saturation properties (cf. [7, w 6.1, p. 305]) of which we shall use only the following special case. Let Ai, ~ i ,D ,A* and ~ ' be as in (ii) above. Suppose ajeA* and aj+l=a ~ mod J/c'J for all j e N . Then there is a zeA* satisfying z==--aj rood J//J for all j~N.

Proof. Let aj(i) be a representative of aj. For each j there is an sj6D such that aj(i)=-a~(i) rood ~{zi for l = 0 . . . . , j and iESj. For any i~N let j(i) be the largest integer in

{j[j<=i and i~sj} w{O},

and set z(i)=aj,l(i ). Then z(i)-az(i ) mod Jr for l=0, 1 . . . . . j(i) and i~sjll). For all j and all i sa t i s fy ingj<i and i~sj we havej<j(i) and iesj(i~, and hence z(i)-=aj(i) mod ~'~. Thus, since D is nonprincipal, z = [z(i)] =-aj rood ~'J , for all j.

A similar argument, combined with Log' theorem, proves that A* is co 1- saturated, i.e. a countable set q~ of first order formulas is satisfiable in A*, provided that every finite subset of q~ is satisfiable in A*. (We say that a set of first order formulas q~j(z) is satisfiable in A* if there exists a zeA* satisfying ~o~(z) for all j.) We shall however not need this.

Remark. For our purposes it will be sufficient to use ultraproducts with respect to any nonprincipal ultrafilter. For other uses it can be necessary to choose the ultrafilter more carefully or to use the more abstract notion of enlargements (due to A. Robinson). For a discussion of enlargements see Chapter 1 of [14].

{}2. In this section we shall give a new proof of a theorem of Greenberg (Theorem 2.1). Recall that a discrete valuation ring R is called excellent if the fraction field of the completion of R is separable over the fraction field of R. (Cf. [12 Chapter 10, w 6].)

Theorem 2.1 (Greenberg [10, Theorem 1]). Let R be an excellent Henselian discrete valuation ring, with prime p. Let Y=(Y1 ..... Y,,) and f =(f l , ' " , fm) ~R [Y]" Then there is an integer N eN, depending on f, such that for any nonzero c~eN and any ~eR satisfying f(~)--O rood pN~, there is a yER satisfying f ( y ) = 0 and y=-f~ mod p'.

Let D be a nonprincipal ultrafilter on N and let R*=R~/D. Then (cf. w l(i)) R* is a Henselian valuation ring with valuation ord* into Z*=77N/D. Let k = R/(p), then R*/(p)= k* = ka~/D. Let V be any convex subgroup of 77* containing 77 (i.e. O<c~<fl, aE77*, flEV imply that c~V). Set lv={x~R*lord*(x)r Since V is convex I v is a prime ideal of R*. Set RV=R*/Iv . It is easy to see that R v is a valuation ring with value group V and residue class field k*. We denote the natural projection of R* onto R v by re. The natural embedding i of R in R* induces an embedding of R into R v, and we consider R as a subring of R v. The algebraic part of our proof of Greenberg's Theorem is contained in Lemmas 2.2 and 2.3. We shall find it convenient to use Lemma2.2, in the case V=7Z, in w167 and 4. Lemma 2.2 is an easy logical consequence of Greenberg's Theorem 2.1

194 J. Becket et al.

(see Remark (ii) at the end of w 2), and hence for our uses in w167 3 and 4 we could have omitted the proofs below of 2.2 and 2.3.

L e m m a 2.2. Let R, R*, V, R v, ~, i be as above. Let R' be any subring of R v containing R and finitely generated over R. Then R' can be lifted into R*, i.e. there is a homomorphism t): R ' ~ R* such that the following diagram commutes

R v ~ ~ - - R*

R'

R

Proof In Lemma 2.3 we shall show that the fraction field of R v is separable over the fraction field of R. Hence we have R ' = R [ y l , ...,Yr, w, , . . . ,ws] with the Yl, wi eRr , Yl . . . . ,Yr algebraically independent over R and w I . . . . . w s separably algebraic over R [ y l . . . . . Yr]. It is obvious that R[y 1 .. . . ,yr] can be lifted to R* - just send Yl to any element of ~ - l (y ) . Hence let R o be any subring of R v containing R, let 00 be a lifting of R o into R* and let w e R v be separably algebraic over R o. It is sufficient to prove that 0o can be extended to a lifting of Ro[w ] into R*. Let f be an irreducible polynomial for w over R o. Thus f ( w ) = 0 and f ' ( w ) + 0 (where f ' denotes the derivative of f ) . Choose O~R* such that rc(~) =w. Let g be the polynomial obtained by applying 0o to the coefficients of f. Hence rt(g(~)) = 0 and n(g'(#)) + 0, i.e. ord* g(#)6 V and ord* (g'(~))~ V. Since V is a convex subgroup of Z* we obtain ord*(g(ff)/(g'(~)) 2) >0. Hence by the Hensel-

lemma for g(uT) (which follows from Hensel's lemma applied to the /

Rychlik

1 [u5 g (~ ) h \ ) polynomial G ( h ) = g ~ g I + g ~ ] - l + h m o d p , there is a ~eR* such that

= _ . g ( ~ ) g(v~)=0 and w = w m o o ~ . This congruence implies that ~(#)=w. Hence by

mapping w to if, q*o extends to a lifting of R o [w] into R*.

Remark. From the above proof it is easy to see that R v can be lifted into R* if the characteristic of R is zero.

L e m m a 2.3. Assume the hypotheses of Lemma 2.2. Then Frac(R v) is separable over Frac(R) (Frac denotes the fraction field).

Proof Suppose that the characteristic of R is p 4= 0. Le t /~ be the completion of R. The natural embedding of R ~--./~ induces embeddings R* ~-~/~* (=R~/D) and Rv~--~R v. Since Frac(/~) is separable over Frac(R) it is sufficient to prove that Frac(R v) is separable over Frac(/~). Thus we may suppose that R is complete. Let K=Frac(R) , L=Frac(RV). Let K x be any finite field extension of K contained in K lip. We fix an extension of the valuation on L to L K 1. Let k~ be the residue class field of K~ and let F be the order group of K 1. We have (cf. [12, Ch. 12, w 4, Prop. 13]) that

[LK t :L] = e (LK ,/L) f (LK ~/L)

Ultraproducts and Approximation in Local Rings ! 195

where e is the ramification index and f the residue class degree. But

e(LKI/L)> IV+ F: V] > [Z* + F:;g*] = IF :7/] =e(K~/K) (1)

f(LK~/L) > [k* k, :k*] = [k, :k] =f(K~/K). (2)

In (1) only [Z*+F:2g*]=[F:7Z] requires proof. Let "leFc~2g*. Set e=e(K1/K ), then e)'e2~. Since 2g is a convex subgroup of 7/.* (cf. w 1 (i)), we deduce ),e2g. This proves (1). In (2) only the equality [k* k l:k*] = [k 1 :k] required proof. Notice that k*klck~(=k~/D ). Let a~ . . . . . a, ek~ be linearly independent over k. Sup- pose ~a~ej=O in k*, with aj=[aj(i)]ek*. Thus for some seD, we have ~aj(i)~j

J J = 0 in kl, for all ies. Hence aj(i)=O for all ies. Thus aj=0, and the c~ are linearly independent over k*. This proves (2).

Since R is complete we have (cf. [12, Ch. 12, w Prop. 18]) that [K~:K] =e(K1/K)f(K1/K ). Hence (LK 1 :L)=> [K 1 :K]. Thus L is linearly disjoint from K lip over K and hence L is separable over K (cf. [12, Ch. 10, w

Proof of Theorem 2.1. Suppose the theorem is false. Thus VN~N, 3aeN, 3xeR [c~,0 and ord(f(x))>N~ and -1 3yeR(f(y)=O and o r d ( y - x ) > c 0 ] , where -7 denotes "not". This gives sequences e~eN, x~eR for N = 0 , 1 , 2 . . . . which de- termine elements seN*, xeR* ( e= [(e~)], x = [(x~)]) satisfying ~ , 0 and

ord*(f(x))>mc~ for all m~N (3)

~3yeR*(f(y)=O and o r d * ( y - x ) > ~ ) . (4)

Inequality (3) follows from the definitions, since D contains every cofinite subset of N. Also (4) is easy to see directly from the definition of R*, since if there were a yeR* satisfying f ( y ) = 0 and o r d * ( y - x ) > ~ then for infinitely many i eN we would have f(y(i))= 0 and ord(y( i ) - xi)> cq (with y = [y(i)]). But there is no such y(i) by the definition of cq, xl. Set V = { f l e Z * l - m ~ < = [ l < m ~ for some meN}. Then V is a convex subgroup of 7/*. Let ~z: R*--~R v be as above. From (3) it follows that f(rc(x))=~f(x)=O. Set R'=R[~z(x)]cR v. By Lemma 2.2 there is a lifting 0: R ' ~ R*. Set y = 0(~(x)). Then f(y)= 0 and re(y)= re(x) and hence ord* (y -x) > e. This contradicts (4) and completes the proof.

Remarks. (i) Using similar methods one easily obtains a new proof of the result of Greenberg [11], (cf. van den Dries [20, p. 1463). Effectivity can be proved by the methods of Sect. 6. For an explicit algorithm see Birch and McCann [63.

(ii) Lemma 2.2 is an easy logical consequence of Greenberg's theorem 2.1.

Proof. Let R'=R[~(y)]cR v, with Y=(Yl . . . . . y,)eR*. Let f = ( f l . . . . . fm)eR[Y] generate the ideal of all feR[Y] vanishing on n(y). To prove Lemma 2.2 it suffices to find a yeR* such that f ( y ) = 0 and n(y)=n(y). Let N e N be as in Greenberg's theorem 2.1. There is a positive ~ in •* such that 7~V and ord *f(y)=> N oz. Indeed, since f(n(~))=0, we have ord*f(~)~ ~ and moreover we can write ord*f(y)=N~+r, with O<r<N. Thus by Greenberg's theorem 2.1 and the definition of ultraproducts (or Log' Theorem), there is a yeR* satisfying f(y) = 0 and ord* (y - ~) > ~. And since c~ V, this implies ~(y) = n(y).

196 j. Becker et al.

w 3. Artin [2] has proved the following theorems (3.1 and 3.2). Related theorems are proved in [1].

Theorem 3.1 ([2, Thm. 1.101). Let R be afield or an excellent discrete valuation ring with prime p. Let A be the henselization of R I X 1 . . . . . Xn] at the maximal ideal J/l = (p, X 1 . . . . . X,) (c f [15, w 43, p. 180]). Given an arbitrary system of polynomial equations f (Y) = O, Y = (I11 . . . . . YN) with coefficients from A, a solution ~= (Yl . . . . . Yu) in the J/g-adic completion A of A and an integer c, there exists a solution y= (Yl . . . . . yu)~A with y i - ~ i m o d J r 'c for i=1 . . . . . N.

Theorem 3.2 ([2, Thm. 6.1]). There is an integer valued function fl=[J(n,N,d, c~) defined for non-negative values of n, N, d, ~ with the following property: Let k be any field, let f =( f l . . . . . f,,) be polynomials in k[X, Y] where X =(X I . . . . . X,) and Y=(Yx . . . . . YN) with degree(f)<d. Suppose given polynomials Y=(Yl . . . . . YN) e k [ X ] such that f ( X , ~ ) - O mod(X) ~. Then there are elements Y=(Yl . . . . . YN) s k [ X ] ~ (the Henselization of k [X] at the maximal ideal (X)) solving the equations f ( X , Y)= 0 and such that y-y~ mod (Xy.

D. Popescu has extended the methods of Artin to prove the following theorem. (See also [17].)

Theorem 3.3 ([18, Thm. 1.4]). Let R be an excellent Henselian discrete valuation ring with prime p and let f=( fa , " ' , fm) be polynomials in R[X , Y] where X = (X 1 . . . . . X,) and Y = (Y1,..., YN). There is an integer valued function v(cO, depend- ing on f, such that, given polynomials y = (Ya .. . . . ~N)~R IX] satisfying f ( X , y:) - 0 mod(p,X) ~'), there are elements Y=(Yl .. . . ,yN)6R[X] ~ (the Henselization of R [X] at the maximal ideal (p, X) ) solving the equations f ( X , Y)= 0 and such that y - y mod(p, X)L

Theorem 3.2 is proved in [2, w by a careful analysis of the proof of Theorem 3.1. The proof of Theorem 3.3 given in [18] is very intricate and entails a substantial extension of the N6ron p-desingularization. In this section we show that Theorems 3.2 and 3.3 can be obtained as quite easy logical consequences of Theorem 3.1.

Proof of Theorem 3.2. Suppose that Theorem 3.2 is false. Then there exist n, N, d, c~N such that for each j ~ N there is a field kj and polynomials fj =(f~l , . . . , f jm)ek j[X, Y] of degree<d ( X = ( X I , . . . , X , ) , Y=(Y1 . . . . . YN)) and yj =(yjl , . . . ,yjN)ek~[X] satisfying fj(X,y~)---0 mod(X) j but there is no yj =(Yjl .. . . ,yjN)ek~[X] ~ satisfying f~(X,y.i)=0 and y~-yj mod(X)L Since the number of linearly independent polynomials over kj in X and Y of degree_-<d is uniformly bounded in terms of n, N and d, the mj can be taken to be bounded and hence without loss of generality we may assume that m~=m V~eN. Let D be a nonprincipal ultrafilter on N and let A*=(1- I kj[X]~)/D, k*=( l ~ kj)/D. We

jeN j~N

consider k* [X] as a subring of A*, identifying X with [(X, X .. . . . X . . . . )1. Let Yi(J) =~ji~kj[X] for j~N, let ~i=[~i(j)j~N]eA * for i=1 . . . . . N and let ~ = ( ~ . . . . . YN)' Similarly let f be the element of k* [X, Y] which corresponds to the sequence f j e k j [X , Y]. (The coefficients of f correspond to the sequences of coefficients of

Ultraproducts and Approximation in Local Rings I 197

the fj). Notice that f has degree <d. Since D contains all cofinite subsets of N we have f (X, ~)-0 mod (X) j for all j e N but there is no yeA* such that f(X, y)=0 and y - y ' mod(X) ~. (This follows immediately from the definition of the ultra- product A* as in our proof of Theorem2.1). Let (X)~=~(X) icA *, let A 1

i e N

=A*/(X) ~, and let rc be the projection of A* onto A 1. Since g is injective on k* [X] cA*, we consider k* IX] as a subring of A 1. We need the following

Lemma 3.4. With the above notation, A 1 ~k*[[X]] (as k*[X] algebras).

Proof. We define a map 2: A*~k*[[_X~ as follows. Let a=[a(j)]eA*, a(j)ekj[X]~ckj[[_X]]. Write aO')=~ah(j)X h, where h is a multi-index and

h

ah(j)ek ~. Set 2(a)=~[ah(j)]Xhek*[[X]]. It is easy to verify that 2: a~--~2(a) is a h

homomorphism with kernel (X) ~. Moreover 2 maps (I-I kj[X])/DcA* onto jeN

k*[[X]]. Indeed let b=~[bh( j ) ]Xh~k *[[X] with bh(j)ek i. Set a(j) h

= ~ bh(j)xhekj[X], then 2([a(j)])=b. This finishes the proof of Lemma 3.4. h<=j

From now on we identify A 1 with k*l[X]]. We have thatf(X,~(yO)=rcf(X,y') = 0 since f(X,~)=-O mod(X) j for all j e N . Hence by Theorem3.1 there is a yek*[X] ~ such that f ( X , y ) = 0 and y=-~(~) mod(X) ~. Since A* is Henselian (cf. w l(ii)) there is a k*[X]-homomorphism ~: k*[X] ~ ~ A * (cf. [15, Thm. 43.5, p. 181]). By the uniqueness of the Henselization, rcoO is the identity on k*[X] ~. Thus we have f(X, O(y))=0 (in A*) and 0(Y)-Y rood(X) ", since rt tp(y)=y-rc(~) mod(X) ". But we know there is no such O(y)eA*. This contradiction proves Theorem 3.2.

Remarks. (i) Using similar methods one can also show that the degree of Yl . . . . . Y~. over k[X] can be bounded by a function of n, N, d and ~. (If P(X, Y) is the minimal polynomial of yek[X]~ over k[X] we define the degree of y to be the total degree of P(X, Y).)

(ii) The results of Hermann and Stolzenberg (i.e. [2, Thm. 6.5]) used in Artin's proof of Theorem 3.2 can also be established quite efficiently using model theoretic techniques. (See [20] and the references there in.)

Proof of Theorem 3.3 Suppose that Theorem 3.3 is false for the excellent Henselian discrete valuation ring R. Then there exist f = (fl, ..., f~)e R [X, Y] and ~eN such that for each j e N there is a y j e R [ X ] satisfying f(X, y j )=0 mod(p, X) j, but there is no yeR[X] ~ satisfying f(X,y)=O and Y-Y1 m~ (X=(X1 ..... X,), Y=(Y1 . . . . . YN)). Again let D be any nonprincipal ultrafilter on N, let A* =(R [X]~)N/D and R* =RN/D. Notice that R IX] cA*, and R* cA* .

Let ~ =(Yi . . . . , ~ ) e A * where Yi corresponds to the sequence {Y~i}j~. Then as above f(X, ~v)==-0 rood (p, X) j for all j e N , but there is no yeA* such that f (X, y) = 0 and y - ~ mod(p,X) ". Let fi~176 ; c R * , . JIC~=j~O (P,X) jcA*. Let R~

=R*/fi ~ and AI =A*/~l ~. We denote both the projection of A* onto Aa and the projection of R* onto R~ by n. Notice that R ~ R~ and R IX] c A 1. We need the following

198 J. B e c k e r et al.

Lemma 3.5. With the above notation, R 1 is a complete discrete valuation ring and A 1 ~ R 1 [[X]] (as R [ X ] algebras).

Proof R 1 is a discrete valuation ring (cf. w moreover R 1 is complete by w l(iii). We define a map 2: A*~RI[[X]] as follows. Let a=Ea(j)]~A*, a ( j ) ~ R [ X ] - ~ R [ [ X ~ . Write a( j )=~ah( j )X h, where h is a multi-index and

h

ah(j)ER. Set 2(a)=~n([ah(j)])Xh~Rl[[X~. As in the proof of Lemma 3.4, it is h

easy to verify that 2: a~--~2(a) is a surjective homomorphism with kernel ~///~. This proves Lemma 3.5.

From now on we identify A 1 with R 1 [[X~. We have f ( X , n(~))= ~ f (X , ~)= 0 (in A 0. Since R~ is complete it is excellent and hence by Theorem 3.1 there is a y ~ R I [ X ] - such that f ( X , y ) = O and y=n(~) mod(p,X) ~. For rings R 2 such that Rc_R2~_R1, let R2[X] ~ denote the intersection of all the Henselian rings dominated by RI[[X ~ which contain Rz[X ]. ( I f Rz[X ] is normal, then its Henselization is isomorphic with R2EX] ~, see [15, Thin. 43.5, p. 181].) For some R2~_R ~ which is finitely generated over R we have that y ~ R 2 [ X ] - . By Lemma 2.2 (with V=TZ) there is a lifting of R 2 into R* cA*. This lifting extends naturally to a lifting ip of R 2 IX] into A* (i.e. ~ is the identity on R I-X] and n o is the identity on R E I-X]). Since A* is Henselian (cf. w 1 (ii)), ~0 extends to a lifting ~t o f R z [ X ] ~ into A* (cf. [15, Thin. 43.5, p. 181]. Indeed ~p extends to the derived normal ring of R 2 IX]).

Hence f (X ,~(y ) )=O in A* and ~ ( y ) - ~ mod(p,X) ", since n ~ ( y ) = y ~ ( ~ , ) mod(p,X) ". But we know (above) that no such ~9(y)eA* exists. This con- tradiction proves Theorem 3.3.

w 4. In this section we prove some extensions of Theorems 3.1, 3.2 and 3.3. Theorem 4.1 is due to M. Artin (oral communication) and can also be proved quite easily without using ultrapowers. It seems that a proof of Theorem 4.2 using conventional methods would be rather difficult.

Theorem 4.1. Let k be a field, X = ( X 1 .. . . . X , ) and let A = k [ X ] ~ (the henseli- zation of k [X] at the maximal ideal (X)). Let Y=(Y1 ... . . YN), f = (fl . . . . ,fm)~k IX, r ] , ~ = ();1,..., YN) 6k [[X]] with Yl .. . . . ~ k ~_X1] I. Suppose that

f(X,~)---0, and let ~ IN . Then there exists Y=(YI . . . . . YN) 6A with Yl ..... y ~ k [X1] ~ such that f ( X , y) = 0 and Yi - Yi m~ ~ for i = 1 . . . . , N.

Proof First notice that the theorem as stated follows immediately from the special case where ~t = 0, i.e., the congruence condition y - y rood(X) ~ falls away. Indeed the congruence condition can be replaced by polynomial equations.

Let fi be given. Freeze Yl .. . . . y ~ k [ [ X ~ . Then by Theorem3.1 for R =k[[X~]] we know that there exist y~+~,...,yuek[[Xx]l [X2 , . . . ,X , ]~ such that

f ( X , ~ . . . . . fi~, y~+x,. . . ,ys)=0. Choose a subring R' of k[[X~]] which is finitely generated over k[X~]~ and such that ~ . . . . , ~ R ' and Y~+ 1 . . . . . YN ~R ' [X 2 . . . . . X,] ~ (the intersection of all the henselian rings dominated by k[[X-~ and containing R'[X 2 . . . . . X . ] ) . Now by Lemma2.2 (with R= k[X~] ~ and V=7/) there is a lifting ~b such that the following diagram commutes, where * denotes the ultrapower with respect to a nonprincipal ultrafilter D.

Uttraproducts and Approximation in Local Rings I 199

k* ~Xa ]1 ~ - ~ : , (k IX, ] ~)*

k [ X d - ~ /

(i is the natural embedding and ~ is the project ion of ( k [ X a ] - ) * onto k*[[Xl~ =(k[X~]~)*/(XO ~ where (X1)~'= (~ (XO i as in the proofs of Theorems 3.2 and

i e N

3.3.) Moreove r 0 extends natural ly to a lifting 0 of R'[X 2 . . . . . X . ] ~ into ( k [ X ] ~ ) * (cf. p roof of Theorem 3.3). Since Yl . . . . . yr~R' we have that O(Ya) . . . . . ~9(fr)~(k[Xx] ~)*. Also

f ( x . ~, (y 1) . . . . . ~' (Yr). ~' (y~ +1) . . . . . ~, (yN)) = ~, f ( x , Y l . . . . . Y~. y. + l . . . . . yN) = 0

(in (k[X]-)*) . The theorem (for ~ = 0 ) follows now from the definition of ul trapower.

Theorem 4.2. Using the notation of Theorem 3.2, there is an integer valued function f l '=fl ' (n,N,d,x) such that if k is an), .field and f = ( f l . . . . . fm)ek[X, Y] (X = (X l . . . . . X,) and Y = (Y1 . . . . . YN)) have degrees < d and there exists a y = (Ya . . . . . YN) e k [ X ] with Yl . . . . . y~ek[Xl] satisfying f ( X , ~)=-0 rood(X) ~', then there exists y =(y~ . . . . . yN)ek[X]~ with Yl . . . . . yrek[X~]~ satisfying f ( X , y ) = O and Yi =-Yl m ~ ~ for i= 1 . . . . . N.

Proof The proof of Theorem 4.2 is identical with our p roof of Theorem 3.2 above except that one uses Theorem 4.1 in place of T h e o r e m 3.1.

Theorem 4.3. Using the notation of Theorem 3.3, there exists an integer valued function v'(~), depending on f such that given polynomials Y=( -~ I . . . . . YN)eR[ X'] with ~ .. . . . ~ R such that f(X,F;)=-O m o d ( p , X ) ~'l~), there are elements y =(Yl . . . . . yu)6R[X] ~, with Yl . . . . . Yr~R, solving the equations f ( X , Y ) = O and such that y =- ~ rood (p, X)L

Proof We use the nota t ion of our proof of Theorem 3.3. Suppose 4.3 is false, then there is Y=(Yl . . . . . ~N)~A* with ~ . . . . . ~ E R * such that f ( X , ~ ) = O mod(p ,X) j for all j~IN, and such that there is no yeA*, with Yl .. . . . Yr ~R*, satisfying f ( X , y ) = O and y=-~ mod(p,X)L We have f ( X , ~ ( ~ ) ) = 0 (in A1). By Theorem3.1 (for R I [ X ] ~ and the variables Y~+~ . . . . . Yu) there are Y~+I . . . . . YN ~R~[X] ~ such that f ( X , ~ ( ~ l ) . . . . . ~(~), y~+~ . . . . . y u ) = 0 and y ~ - ~ ( ~ ) rood(p, X) ", i = r + 1 . . . . . N. Choose a ring R2 ~ R a, which is finitely generated over R and such that ~(~a) . . . . . E ( y r ) G R 2 and Yr+I .. . . , y u ~ R z [ X ] ~. Now proceed as in our p roof of 3.3.

Remark. The implicat ion " t heo rem of type 3.1 for fields implies theorem of type 3.2" is true for any constraints on the y~, for example:

Yl . . . . , y~ 6k [[X~]], y~, + ,, ..., y~2~k~_X,, X2-~ . . . . .

y . . . . +a,... ,y~. ek[[Xa . . . . . X~-ll

and similar constraints on the y~. The same proofs works.

200 J. Becker et al.

w 5. In this section we give some counterexamptes to problems related to w 4. In [4], using the methods of [5], it was shown that there is a polynomial f(X,, X 2 , I:1 . . . . . Y4)E~[X, Y] such that there exist y l C ~ - X l ~ , y2e~[[X2~ and y3, y 4 ~ [ X 1 , X2-]] satisfying f (X 1, X 2, Yl . . . . . y4)=0 but there exist no y l eC{X,} , y2eC{X2}, and Y3, Y 4 ~ C { X I , X 2 } (the convergent power series rings) such that f(Xl, X2, Yl . . . . . Y4) =0.

The key idea of [5] is to notice that one can define composition of power series as follows: Let h(X1), g(X2), a(X2) be fixed. (The notation h(X 0 means that h depends only on X 1.) Then

h(~(X2))=g(X2) iff 3s(xl, X2)[h(X,)=g(X~)+ S(X,, X2)(X,-~(X~))].

One direction is immediate by substitution, the other follows by Taylor's theorem.

It is easy to write down a functional equation which only has a divergent solution e.g. f (X + X 2) = 2f(X) + X. The example in [4] is obtained by applying the above observation to translate such a functional equation into a system of polynomial equations f(X, Y)=0 with the constraints that some of the Y's depend only on X 1 and some only on X 2. This shows that the statement of the form 3.1 with such constraints on the Y's is false. We shall show that some weaker assertions are also false.

(i) Let k be a field and consider the equation

f ( c X ) - f ( X ) = ( 1 - X ) - ' - l , cek, c#=O.

X n This equation has only the solutions f(X)= a 0 +,~=1 c"-~' a~ ~k. Hence if we let

k be the algebraic closure of the field IFp with p elements, then this equation (with unknowns X,c) will have no solution in k[[X-B with cek, c~:0, but it will have solutions mod X J for all j~IN. The above device can be used to translate this equation into a system (1) of polynomial equations over k[XDX2] with constraints that some of the Y's depend only on X, and some only on X 2, which have a solution mod(X1,X J for al l je]N but have no solution in k~_X1, X2]]:

Y2 = Y, + Y~(G -x2)

Y3 = Y, + Ys ( Y6 X I - X 2)

(1:3 - 1:2)( 1 - X,) = 1 - (I - X,)

Y6Y7 =1 (1)

}:i, Y6ek~X2~

Y~, Y3, Y6~k~X,-B.

(Y6ek[~X1]]nk[FXS[]=k stands for c; Y6~=0, since Y6YT--I; Y,=f(x2), Y2 =fiX,) and 1:3 =f(c X,).)

(ii) In this example we give a system (3)-(7) of equations and constraints which has no solution in ~[[X~ but has a solution in Q[[X]]/(X) ~ for all jelN.

Ultraproducts and Approximation in Local Rings I 201

Let k be a field. We consider the differential equation

Xl ( ? f (Xl, X2) - X 2 ~ - - ( X 1, X2) C 2

= ~, X ] X { = [ ( I - X 1 ) - ' - I ] [ ( 1 - X 2 ) - ' - I ] . (2) i,j>= 1

For c6k, c4=0 this equation has only the solutions

i j XI X2 ao~k" f ( X t , X2)=ao+ ~ c~i_~ '

i , j> l

(This is easy to verify.) We use the above device to translate this into a system of equations (3)-(7) over the ring Q[X~ ... . . Xs] with some of the variables depend- ing on only X~ and X 2 and some on only X 3, X4, Xs:

YI(XI,X2) ~ Y2(X3, X4,Xs) mod(X1 - X 3 , X 2 - X 4 ) (3)

Y2(X~,X~,xd= r~(x,,x2)+x~ Y~(x~,x9 +X~ Y~ mod(X 3 - X 1 - X s , X 4 - X 2 ) (4)

Y2(X3, X 4, Xs)~ Y1(X ,,X2)-F X5 Yv(X l,X2)

+ X2 Y8 mod(X3 - X 1, X4 - X2 - X.s) (5)

Ygek (i.e. Yg(X,, X2)=Ylo(X 3, X 4, Xs)) and Y9 Y1t =1 (6)

rJx, Ys-x~ r ~ = [ ( l - X 0 - ~- 1] [(1 - x g - ~ - 1]. (7)

(The variables Yo and Ys depend on all the X's.) Suppose (3)-(7) are satisfied. Set f (X1, X2) = Yx (X1, X2).

Set X 1 = X 3, X 2 = X ~ in (3) to conclude that Y2(X3, X4, Xs)=f(X3, X4). Setting X 3 = X 1 + X 5 and X 4 = X 2 in (4) we get that f ( X 1 +X s , Xz)=f(Xx, X2)

+XsY~(X~,X2)+X2yo. From this it follows that Y s = ~ - ( X t , X 2 ) . I Simi-

larly YT= ~u-~Jv (X1, X2). Then Eq. (7) is just Eq. (2), with c= Yg. Finally (6) l Y A 2

guarantees that c4:0 and c~k. Thus it is clear that this system (3)-(7) has no solution over Q[[X]] but has a solution in Q[[X~/(X) j for all j~IN. Notice that the system has a solution in k[[X~ if k ~Q .

(iii) There is no v~N having the following property: if the system (3)-(7) has a solution ~ in G [[X]]/(X) v, then the system has a solution y in G [[X]] with y = rood(X). On the other hand it is well-known (cf. introduction) that any system (with any constraints), which has a solution in G[[X]]/(X) j for all j6]N, has a solution in G[[X]]. It is easy to see that there is not such a v. Indeed for every v there is a c , 6 ~ such that the system (3)-(7) has a solution mod(X) * with Yg=cv but there is no solution in (E[[X]] with Y9=-c~ rood(X) ( ~ Y9=c,).

202 J. Becker et al.

Similar remarks are true for algebraically closed fields of infinite transcen- dence degree over lFp (cf. (i)).

Remark. A more interesting case is that of the example in the remark after Theorem 4.3. We don't know what happens in this case.

w 6. In this section we prove the computability of the function/3 of Theorem 3.2. The computability of/~ follows by examining Artin's proof of Theorem 3.2 (i.e. Thm. 6.1 of [2]) (cf. [5] and [13]). Although our proof of Theorem 3.2 is not at all effective one can still prove the computability of/~ using some concepts from logic, without analyzing the proof of 3.2 but using only the existence of/?. We shall assume that the reader is familiar with the concepts of a first order language and a first order theory. We shall also use the G6del completeness theorem for first order logic which says that if T is a first order theory and ~o a first order statement (in the language of T) then q~ is provable from T if and only if ~0 is true in every model of T.

Theorem 6.1. There is a computable function Theorem 3.2.

Proof We shall describe a first order theory structures are models of T. It will be apparent

which satisfies the condition of

T below by describing which from our description what the

appropriate first order language and axioms for T are. We shall not make these explicit. Fix n, N, d, c~ (as in Theorem 3.2). A model for T will be a local ring A, containing a field k, with maximal ideal ~ ' generated by X~ .. . . . X, satisfying: (i) The X~ are algebraically independent over k (this will be an infinite set of axioms - one for each degree). (ii) A/~r k [X, ..... X,]/(X~ ..... X,) i for each i = 1,2 . . . . . (iii) A is henselian.

Let q~ be the first order statement which says that the integer fl satisfies the conditions of Theorem 3.2 for A, i.e. if f (X , Y) is a system of polynomials of degrees<d in X = ( X 1 ..... X,) and Y=(Y1 . . . . . Y~) over k and .~=(-Yl . . . . . YN) ek IX] (of degrees <//) satisfy f ( X , ~ ) - 0 mod ,g~, then there is a y = (Yl .. . . , YN) eA such that f (X ,y )=O and y - y m o d . N ~. Let [Jo=P(n,N,d,~ ). (By this we mean the value shown to exist in Theorem 3.2). We shall show that if A is any model of T then ~b~o is true in A. Let A be a model of T and let f be any system of polynomials of degrees <d in X and Y o v e r k A (the field in A). Let ~ekA[X], and suppose that f ( X , ~ ) - O m o d ~ '~~ (in A). Now by (i) and (ii) A ~ k A[X] and f ( X , ~ ) - 0 mod(X) t~~ in k a IX]. Hence by Theorem 3.2 there exists a y E k a [ X ] ~ such that f (X ,y )=O and y = y mod(X) ". By (iii), A ~ k a [ X ] ~ and hence yeA, f (X , y ) = 0 and y - ~ mod ~ " . This shows that cbp0 is true in A.

Now by the GiSdel Completeness Theorem there is a proof of 4~o from T. We calculate a function/3 as follows. The axioms of T are effectively given so we can generate the (first order) theorems of T. Do this in some fixed way until one obtains a theorem of the form ~r (We know this eventually happens since q~po is a theorem of T). Let q~, be the first such to occur. Set/~(n, N, d, c~)=/~.

Remarks. (i) The function/3' of Theorem 4.2 can be shown to be computable by a similar proof.

Ultraproducts and Approximation in Local Rings 1 203

(ii) The above method of proof~ while establishing the computability of/~ gives a hopelessly inefficient algorithm.

References

1. Artin, M.: On the solutions of analytic equations, Inventiones Math. fi, 277-291 (1968) 2. Artin, M.: Algebraic approximation of structures over complete local rings, Puhl. Math. I.H.E.S.,

36, 23- 58 (1969) 3. Ax, J., Kochen, S.: Diophantine problems over local fields I, II Amer. J. Math, 87, 605--648

(1965); llI Ann. Math. (2), 83, 437~,56 (1966) 4. Becker, J.: A counterexample to Artin approximation with respect to subrings, Math. Ann., 230,

195-196 (1977) 5. Becket, J., gipshitz, L.: Remarks on the elementary theories of formal and convergent power

series, Fund. Math., (to appear) 6. Birch, B.J., McCann, K.: A criterion for the p-adic solubility of Diophantine equations, Quart. J.

Math., Oxford, (2) 18, 59~63 (1967) 7. Chang, C.C., Keisler, H.J.: Model Theory, North Holland, Amsterdam 8. Delon, F.: R6sultats D'ind6cidabilit6 Dans Les Anneaux de S6ries Formelles, (to appear) 9. Ergov, Yu.: On the elementary theory of maximal normed fields, Algebra i Logika, 4, 31-69

(1965); 5, 8-40 (1966); 6, 31-37 (1967) 10. Greenberg, M.J.: Rational points in henselian discrete valuation rings, Publ. Math. I.H.E.S. 31,

59~54 (1966) 11. Greenberg, M.J.: Strictly local solutions of diophantine equations, Pacific J. Math., 51, 143 153

(1974) 12. Lang, S.: Algebra, Addison-Wesley, 1965 13. Lascar, D.: Caract6re effectif des th+or6mes d'approximation de M. Artin, (to appear) 14. Luxemburg, W.AJ.: A General Theory of Monads. In: Applications of Model Theory to

Algebra, Analysis and Probability, pp. 18-86, New York: Holt Rinehart and Winston, 1969 15. Nagata, M.: Local rings, lnterscience, New York, 1962 16. Nerode, A.: A decision method for p-adic integral zeros of diophantine equations, Bull. A.M.S.

69, 513-517 (1963) 17. Pfister, G., Popescu, D.: Die strenge Approximationseigenschaft lokaler Ringe, Inventiones Math.,

30, 145-174 (1975) 18. Popescu, D.: A strong approximation theorem over discrete valuation rings, Rev. Roum. Math.

Pures et Appl. Tome XX, No 6, 659~92 (1975) 19. Robinson, A.: Elementary embeddings of fields of power series, Journal of Number Theory, 2,

237-247 (1970) 20. van den Dries, L.: Model Theory of Fields, Thesis, Utrecht, June 1978

Received July 19, 1978

Current addresses: J. Becker, State University of New York, Buffalo, New York 14214 L. Lipshitz, Purdue University, West Lafayette, Indiana, 47907, USA L. van den Dries, University of Utrecht, Utrecht, Netherlands