Elimination of unknowns for systems of algebraic differential-difference equations

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1 Eimination of unknowns for systems of agebraic differentia-difference equations Wei Li, Aexey Ovchinnikov, Geb Pogudin, and Thomas Scanon KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences No.55 Zhongguancun East Road, , Beijing, China CUNY Queens Coege, Department of Mathematics, Kissena Bvd, Queens, NY 11367, USA CUNY Graduate Center, Mathematics and Computer Science, 365 Fifth Avenue, New York, NY 10016, USA Courant Institute of Mathematica Sciences, New York University, New York, NY 10012, USA University of Caifornia at Berkeey, Department of Mathematics, Berkeey, CA 94720, Abstract We estabish effective eimination theorems for differentia-difference equations. Specificay, we find a computabe function Br, s of the natura number parameters r and s so that for any system of agebraic differentia-difference equations in the variabes x = x 1,..., x q and y = y 1,..., y r each of which has order and degree in y bounded by s over a differentia-difference fied, there is a nontrivia consequence of this system invoving just the x variabes if and ony if such a consequence may be constructed agebraicay by appying no more than Br, s iterations of the basic difference and derivation operators to the equations in the system. We reate this finiteness theorem to the probem of finding soutions to such systems of differentia-difference equations in rings of functions showing that a system of differentia-difference equations over C is agebraicay consistent if and ony if it has soutions in a certain ring of germs of meromorphic functions. 1 Introduction Differentia-difference equations, or what are sometimes caed deay differentia equations, especiay when the independent variabe represents time, are ubiquitous in appications. See for instance [23] and the coection it introduces for a discussion of appications of deay differentia equations in bioogy, the discussion of the foow-the-eader mode in [19] for the use of differentia-difference equations to mode crowd behavior, and [1] for a thorough discussion of theory of deay differentia equations and their appications to popuation dynamics and other fieds. Much work has been undertaken in the anaysis of the behavior of the soutions of these equations. We take up and sove parae probems. First, we address the probem of determining the consistency of a systems of agebraic differentia-difference equations, and more generay, of eiminating variabes for such a system of equations. Secondy, we ask and answer the question of what structures shoud serve as the universa differentia-difference rings in which we seek our soutions to these equations. Our soution to the first probem, that is, of performing effective eimination for systems of differentiadifference equations, is achieved by reducing the probem for differentia-difference equations to one for ordinary poynomia equations to which standard methods in computationa agebra may be appied. Let us 1

2 state our main theorem, Theorem 3.1, now. Precise definitions are given in Section 2. We show that there is a computabe function Br, s of the natura number parameters r and s so that whenever one is given tupes of variabes x = x 1,..., x q and y = y 1,..., y r and a set F of differentia-difference poynomias in these variabes each of which has order and degree in y bounded by s over some differentia-difference fied, then the differentia-difference idea generated by F, that is, the idea generated by the eements of F and a of their transforms under iterated appications of the distinguished difference and derivation operators, contains a nontrivia differentia-difference poynomia in just the x variabes if and ony if the ordinary idea generated by the transforms of eements of F of order at most B = Br, s aready contains such a nontrivia differentia-difference poynomia in x. In particuar, taking q = 0, this gives a procedure to test the consistency of a system of differentia-difference equations. The reader may righty object that rather than giving a method for determining consistency of such a system of equations, what we have reay done is to give a method for testing whether there is an expicit agebraic obstruction to the existence of a soution. In what sense must a soution actuay exist if there is no such agebraic obstruction? This brings us to our second question of where to find the soutions. We address this probem in Section 5, in which we begin by proving an abstract Nusteensatz theorem to the effect that soutions may aways be found in differentia-difference rings of sequences constructed from differentiadifference fieds. Of course, in practice, one might expect that the differentia-difference equations describe functions for which the difference or deay operator takes the form σft = ft τ for some fixed parameter τ and the derivation operator is given by usua differentiation so that δf = df dt. We estabish with Proposition 5.7 that certain rings of germs of meromorphic functions serve as universa differentiadifference rings in the sense that every agebraicay consistent system of differentia-difference equations over C has soutions in these rings of germs. As a compement to this positive resut, we show that there are agebraicay consistent differentia-difference equations that cannot be soved in any ring of meromorphic functions as opposed to germs. The method of proof of our main theorem is modeed on the approach taken by three of the present authors in [21] for agebraic difference equations in that we modify and extend the decomposition-eiminationproongation DEP method. However, we encounter some very substantia obstaces in extending these arguments to the differentia-difference context. First of a, we argue by reducing from differentia-difference equations to differentia equations and then compete the reduction to agebraic equations using methods in computationa differentia agebra. Differentia agebra in the sense of Ritt and Kochin, especiay the Noetherianity of the Kochin topoogy, substitute for cassica commutative agebra and properties of the Zariski topoogy, but there are essentia distinctions preventing a smooth substitution. Most notaby, in the computation of a bound for the ength of a possibe skew-periodic train in [21], one argues by induction on the codimension of a certain subvariety. In that agebraic case, since the ambient dimension is finite, such an inductive argument is we-founded. This woud fai in the case at hand with differentia agebraic varieties. To dea with this difficuty, we must argue with a much subter induction stepping through a decreasing and hence finite chain of Kochin poynomias. With other steps of the argument, we must invoke or prove deicate theorems on computationa differentia agebra for which the corresponding resuts for ordinary poynomia rings are fairy routine. For instance, a key step in the cacuation of our bounds invoves computing upper bounds on the number of irreducibe components of a differentia agebraic variety given bounds on such associated parameters as the degrees of defining equations and the dimensions of certain ambient varieties. In the agebraic case, these bounds are provided by Bézout-type theorems. Here, we work to estabish such bounds with Proposition For the most part, the methods we empoy coud be used to prove anaogous theorems for partia differentia-difference equations. That is, we woud work with a ring R equipped with a ring endomorphism σ : R R and finitey many commuting derivations δ 1,..., δ n : R R each of which commutes with σ. Our arguments go through verbatim in this case up to the point of the computation of bounds on the number of irreducibe components of a differentia variety and to our knowedge it is an open probem 2

3 whether such bounds exist, much ess what the bounds might be. For the bounds we compute, it heps that for ordinary differentia fieds, the coefficients of the Kochin poynomia are geometricay meaningfu. This is not so for partia differentia fieds, but the bounds on these coefficients obtained in [14] shoud make it possibe to extract expicit bounds anaogous to Br, s in the case of partia differentia-difference equations once the issue of bounding the number of irreducibe components has been resoved. Other potentia generaizations present themseves, but they, too, ie outside the reach of our present methods. For exampe, one might wish for an eimination theorem for differentia-difference equations in positive characteristic, especiay as there is no restriction on the characteristic for the eimination theorem for agebraic difference equations, but many difficuties arise in positive characteristic, starting with the non- Noetherianity of the corresponding differentia agebraic topoogy. In another direction, one might wish to aow for severa commuting difference operators, corresponding, for exampe, to aowing deays of various scaes in the deay differentia equations. Based on our preiminary investigations, we expect the utimate theorems to have a fundamentay different character for two or more difference operators. In particuar, even for agebraic difference equations, we know of exampes of consistent systems of difference equations with two commuting difference operators for which there are no skew-periodic soutions. On its own, this does not rue out the possibiity of an effective eimination theorem, but it does mean that the approach we take here cannot be appied. Finay, as a matter of proof technique, our approach is to reduce from equations in severa operators to equations in fewer operators and so on unti we reach purey agebraic equations. We expect that it may be possibe to compute better bounds by making use of integrabiity conditions in the DEP method to reduce directy from equations with operators to agebraic equations. We do not pursue this idea here. Some work on eimination theory for differentia-difference equations appears in the iterature, though the known resuts do not cover the probems we consider. In [17] agorithms for computing anaogues of Gröbner bases in certain differentia-difference agebras are deveoped. These agebras are rings of inear differentia-difference operators. So, the resuting eimination theorems are appropriate for inear equations, but not for the noninear differentia-difference equations we consider. Gröbner bases of a different kind for rings of differentia-difference poynomia rings are considered in [15, 27] with the aim of computing generaized Kochin poynomias. In these papers, the invariants are computed in fieds, but as one sees in appications and as we wi show in Section 5, one must consider possibe soutions in rings of sequences or of functions in which there are many zero divisors. In the papers [4] and [5], characteristic set methods for differentia-difference rings are deveoped. Whie one might imagine that these techniques may be reevant to the probems we consider, it is not cear how to appy them directy as once again, generay, characteristic set methods are best adapted to studying soutions of such equations in fieds. In [2] the mode theory of differentia-difference fieds of characteristic zero is worked out. The resuts incude a strong quantifier simpification theorem from which one coud deduce an effective eimination theorem in our sense. In [20] such quantifier simpification theorems were proven for fieds equipped with severa operators. An overview of the mode theory of fieds with operators may be found in [3]. A of these quantifier eimination theorems for difference and differentia fieds very strongy use the hypothesis that the soutions are sought in a fied. Aready at the eve of agebraic difference equations, the resuts of [8] show that if we aow for soving our equations in rings of sequences or their ike, then the corresponding ogica theory wi be undecidabe. In particuar, no quantifier eimination theorem of the kind known for differentia-difference fieds can hod. This makes our effective eimination theorem a that more remarkabe. This paper is organized as foows. We start in Section 2 by introducing the technica definitions we require to state our main theorems. These theorems are then announced in Section 3. With Section 4 we give the definitions of the technica concepts used in our proofs. We dea with the question of where we shoud seek the soutions to our differentia-difference equations in Section 5. The proof of our main theorem occupies Section 6. 3

4 2 Basic notation to state the main resut Definition 2.1 Differentia-difference rings. A differentia-difference ring R, δ, σ is a commutative ring R endowed with a derivation δ and an endomorphism σ such that δσ = σδ. For simpicity of the notation, we say R is a δ-σ-ring. When R is additionay a fied, it is caed a δ-σ-fied. If σ is an automorphism of R, R is caed an inversive δ-σ-ring, or simpy a δ-σ -ring. If σ = id, R is caed a δ-ring or differentia ring. Given two δ-σ-rings R 1 and R 2, a homomorphism φ : R 1 R 2 is caed a δ-σ-homomorphism, if φ commutes with δ and σ, i.e., φδ = δφ and φσ = σφ. For a commutative ring R, the idea generated by F R in R is denoted by F. For a δ-ring R, the differentia idea generated by F R in R is denoted by F ; for a non-negative integer B, the idea in R generated by the set {δ i F 0 i B} in R is denoted by F B. Definition 2.2 Differentia-difference poynomias. Let R be a δ-σ-ring. The differentia-difference poynomia ring over R in y = y 1,..., y n, denoted by R[y ], is the δ-σ ring R[δ i σ j y k i, j 0; 1 k n], δ, σ, σδ i σ j y k := δ i σ j+1 y k, δδ i σ j y k := δ i+1 σ j y k. A δ-σ poynomia is an eement of R[y ]. Given B N, et R[y B ] denote the poynomia ring R[δ i σ j y k 0 i, j B; 1 k n]. Given f R[y ], the order of f is defined to be the maxima i + j such that δ i σ j y k effectivey appears in f for some k, denoted by ordf. The reative order of f with respect to δ resp. σ, denoted by ord δ f resp. ord σ f, is defined as the maxima i resp. j such that δ i σ j y k effectivey appears in f for some k. Let R be a δ-σ-ring containing a δ-σ-fied k. Given a point a = a 1,..., a n R n, there exists a unique δ-σ-homomorphism over k, φ a : k[y ] R with φ a y i = a i and φ a k = id. Given f k[y ], a is caed a soution of f in R if f kerφ a. Definition 2.3 Sequence rings and soutions. For a δ-σ-k-agebra R and I = N or Z, the sequence ring R I has the foowing structure of a δ-σ-ring δ-σ -ring for I = Z with σ and δ defined by σ x i i I := xi+1 i I and δ x i i I := δxi i I. For a k-δ-σ-agebra R, R I can be considered a k-δ-σ-agebra by embedding k into R I in the foowing way: a σ i a i I, a k. For f k[y ], a soution of f with components in R I is caed a sequence soution of f in R. 4

5 3 Main resut Theorem 3.1 Effective eimination. For a non-negative integers r, s, there exists a computabe B = Br, s such that, for a: non-negative integers q, δ-σ-fieds k with char k = 0, and sets of δ-σ-poynomias F k[x s, y s ], where x = x 1,..., x q, y = y 1,..., y r, and deg y F s, we have σ i F i Z 0 k[x ] = {0} σ i F i [0, B] B k[xb ] = {0}. By setting q = 0 in Theorem 3.1 and using Proposition 5.3, we obtain: Coroary 3.2 Effective Nusteensatz. For a non-negative integers r, s, there exists a computabe B = Br, s such that, for a: δ-σ-fieds k with char k = 0, and sets of δ-σ-poynomias F k[y s ], where y = y 1,..., y r and deg y F s, the foowing statements are equivaent: 1. There exists a δ-fied L such that F = 0 has a sequence soution in L / σ i F i [0, B] B. 3. There exists a fied L such that the poynomia system { σ i F j = 0 i, j [0, B] } in the finitey many unknowns x B+s has a soution in L. 4 Definitions and notation used in the proofs Let R be a δ-σ-ring. For r, s N, et R[y r,s ] and R[y,s ] denote the poynomia ring and the δ-ring R[δ i σ j y k 0 i r, 0 j s; 1 k n] R[δ i σ j y k i 0, 0 j s; 1 k n], respectivey. Additionay, Ry r,s and Ry,s denote their fieds of fractions. The radica of an idea I in a commutative ring R is denoted by I. Let k L be two δ-fieds. A subset S L is said to be δ-independent over k, if the set {δ k s k 0, s S} is agebraicay independent over k. The cardinaity of any maxima subset of L that is δ-independent over k is denoted by δ-tr.deg L/k. In what foows, we wi consider every δ-fied k, δ as a δ-σ -fied with respect to δ and the identity automorphism. From this standpoint, the ring of differentia poynomias over k in y see [11, Chapter I, 6] can be reaized as k[y,0 ] k[y ]. We use ka,0 to denote the differentia fied extension of k generated by a tupe a. 5

6 Definition 4.1. A δ-fied K is caed differentiay cosed if, for a F K[y,0 ] and δ-fieds L containing K, the existence of a soution to F = 0 in L impies the existence of a soution to F = 0 in K. Definition 4.2 Differentia varieties and diffspec. Let K, δ be a differentiay cosed fied containing a differentia fied k, δ and y = y 1,..., y n. For F k[y,0 ], we write VF = {a K n f F fa = 0}. A subset X K n is caed a differentia variety over k if there exists F k[y,0 ] such that X = VF. For a subset X K n, we aso write X = diffspec R if there exists F K[y,0 ] such that X = VF and R = K[y,0 ]/ F note that R is not assumed to be reduced. We define R X := K[y,0 ] / F. A differentia variety VF is caed irreducibe if F is a prime idea. The generic point a 1,..., a n of an irreducibe δ-variety X = VF is the image of the y under the homomorphism K[y,0 ] K[y,0 ] / F. Taking differentia varieties as the basic cosed sets, we define the Kochin topoogy on K n. For a subset S K n, we define the Kochin cosure of S denoted by S Ko to be the intersection of a differentia subvarieties of K n containing S. Let X be an irreducibe δ-variety with the generic point a = a 1,..., a n. The differentia dimension of X, denoted by δ-dimx, is defined as δ-tr.deg Ka,0 /K. A parametric set of X is a subset {y i i I} {y 1,..., y n } such that {a i i I} is a differentia transcendence basis of Ka,0 over K. The reative order of X with respect to a parametric set U = {y i i I}, denoted by ord U X, is defined as ord U X = tr.deg Ka,0 / K a i, i I,0. The order of X is the maximum of a the reative orders of X [6, Theorem 2.11], that is, ordx = max{ord U X U is a parametric set of X}. 5 What is a universa δ-σ-ring for soving equations? This section is devoted to answering the question Question. In what rings is it natura to ook for soutions of differentia-difference equations? We wi show that rings of sequences are universa soution rings in the abstract mathematica sense. More precisey, we prove an anaogue of the Hibert Nusteensatz, Proposition 5.3. On the other hand, from the appications standpoint, it woud be natura if soutions of deay-differentia equations were functions defined on a subset of the compex pane or rea ine. It turns out that these two seemingy contradictory standpoints can be viewed as cosey reated via the construction described beow. 6

7 Definition 5.1 Rings of meromorphic functions. Let U C be an open nonempty set. We denote the ring of meromorphic functions on U by MU. MU is a fied if and ony if U is connected. Let D C be a nonempty discrete set. We define a ring MD of germs of meromorphic functions on D as the quotient MD := {f, U U is open such that D U, f MU}/, where the equivaence reation is defined by f1, U 1 f 2, U 2 z U 1 U 2, f 1 z = f 2 z. For every open nonempty U C, MU is a δ-ring with respect to the standard derivative. If U = U + {1}, then MU can be considered as a δ-σ -ring with respect to the shift automorphism σfz = fz 1. Simiary, for a nonempty discrete D C, MD is a δ-ring and. If additionay D = D + {1}, then MD is a δ-σ -ring with σ sending the equivaence cass of fz, U to the equivaence cass of fz 1, U + {1}. Definition 5.2 Transforms between functions and sequences. Dirty hack We define S := {z C 0.5 < Re z < 0.5} C. Consider a nonempty open subset U C such that U = U + {1}. Then we define a map ϕ U : MU MU S Z as foows. For every f MU and every j Z, we define f j MU S by f j z := fz + j. Then we set ϕ U f :=..., f 1, f 0, f 1,.... One can check that ϕ U defines an injective homomorphism of δ-σ -rings, where MU S Z bears a δ-σ -ring structure as descibed in Definition 2.3. The same can be done for a nonempty discrete D C such that D = D + {1} and D S =. Consider a nonempty open subset U 0 S. We define a map ψ U0 : MU 0 Z MU 0 + Z as foows. For every {f j } j Z MU 0 Z, we define a function f MU 0 + Z by setting fz U0 +{j} := f j z j for every j Z. Then we define ψ U0 {f j } j Z := f. One can check that ψ U0 defines an isomorphism of δ-σ -rings. The same can be done for a nonempty discrete D S. In Section 5.2, we show that MZ is a universa soution ring for δ-σ-equations over C Proposition 5.7 and derive a version of our effective eimination theorem for this case Coroary 5.8. Moreover, in Section 5.3, we show that there exists a system of δ-σ-equations that has a soution in MZ but, for every open U C such that U = U + {1}, does not have a soution in MU. 5.1 Soutions in sequences over δ-fieds Proposition 5.3. Let n Z 0, k be a δ-σ -fied. Then, for every F k[y ] and f k[y ] with y = y 1,..., y n, the foowing statements are equivaent: 1 for every δ-σ-fied extension k K, f vanishes on a soutions of F = 0 in K Z. 7

8 2 There exists m N such that σ m f m σ j F j Z 0 k[y ]. Moreover, if σ = id k, then 2 is equivaent to: for every δ-σ-fied extension k K with σ = id K, f vanishes on a the sequence soutions of F in K Z. Proof. The impication 2 = 1 is straightforward because σ is injective. It remains to show 1 = 2. Suppose that 2 does not hod. Let I := σ j F j Z k[σ j y,0 j Z]. By [9, Theorem 2.1], I is an intersection of prime δ-ideas maybe, an infinite intersection. Assume that f I. Then there exists m N such that f m σ j F j [ m, m]. Appying σ m, we have σ m f m σ j F j Z 0, and this contradicts to the assumption that 2 does not hod. Thus, f / I, so there exists a prime δ-idea P with I P and f / P. Let U 0 be the quotient fied of the δ-domain k[σ j y,0 j Z]/P that has a natura structure of δ-fied. Let U be a differentiay cosed fied containing U 0. [18, Lemma 2.3] together with Zorn s emma impies that σ can be extended from k to U so that U is a δ-σ-fied. Note that if σ k = id, then we can set σ U = id. Let η = σ j y 1 j Z,..., σ j y n j Z U Z U Z, where σ j y k is the canonica image of σ j y k. Ceary, η is a soution of F = 0 in U Z but f does not vanish at it. Thus, 1 does not hod. So, 1 impies 2. Remark 5.4. The proof of Proposition 5.3 can be modified to show that the foowing conditions are aso equivaent 1 for every δ-σ-fied extension k K, f vanishes on a soutions of F = 0 in K N. 2 There exists m Z 0 such that f m σ j F j Z 0 k[y ]. Remark 5.5. In the case f = 1 so-caed weak Nusteensatz, the second condition of Proposition 5.3 is equivaent to the second condition in Remark 5.4. Thus, for f = 1, a the conditions of Proposition 5.3 and Remark 5.4 are equivaent. However, they are not equivaent for genera f as the foowing exampe shows. Exampe 5.6. Let k = Q. Consider F = {y 2 σy, y 2 σ 2 y} and f = yy 1. Let k K be an extension of δ-σ-fieds and a =..., a 1, a 0, a 1, a 2,... K Z a soution of F. For every i Z, we have a 2 i 1 a i = a 2 i 1 a i+1 = 0 = a i = a i+1. Combining with a 2 i = a i+1, we have a 2 i = a i. Thus, f vanishes at a. However, f does not vanish on the soution 1, 1, 1,... of F = 0 in Q N. 8

9 5.2 Soutions in germs Proposition 5.7. For every n Z 0, F C[y ], and f C[y ] with y = y 1,..., y n, the foowing statements are equivaent: 1 f vanishes on a the soutions of F = 0 in MZ. 2 There exists m N such that σ m f m σ j F j Z 0 C[y ]. Proof. The impication 2 = 1 is straightforward. It remains to show 1 = 2. Suppose that 2 does not hod. Let E be the subfied of C generated by the coefficients of F and f over Q. Proposition 5.3 impies that there exists a δ-fied K E such that F = 0 has a soution a = {a j } j Z in K Z such that fa 0. Repacing K by its δ-subfied generated by E and {a j } j Z, we can further assume that K is an at most countaby generated δ-fied extension of E. Hence K is at most countabe. [18, Lemma A.1] impies that there exists a homomorphism of δ-fieds θ : K M0 that maps E K isomorphicay to E C M0. This homomorphism can be extended to an injective homomorphism θ : K Z M0 Z of δ-σ -agebras over E. Then the composition of θ with the isomorphism ψ 0 : M0 Z MZ see Definition 5.2 is an injective homomorphism of δ-σ agebras over E. We set b := ψ 0 θa MZ. Then b is a soution of F = 0 and, since ψ 0 θ is injective, f does not vanish at b. This contradicts 1. Combining Proposition 5.7 with Theorem 3.1, we obtain: Coroary 5.8. For a non-negative integers r, s, there exists a computabe B = Br, s such that, for a: non-negative integer q, a set of δ-σ-poynomias F C[x, y s ], where x = x 1,..., x q, y = y 1,..., y r, and deg y F s, the foowing statements are equivaent there exists a nonzero g C[x ] that vanishes on every soution of F = 0 in MZ; σ i F i [0, B] B C[x ] {0}. 5.3 Soutions in meromorphic functions on open subsets of C In this section, we wi present a specific system of δ-σ-equations 3 that has a soution in MZ but, for any open U C such that U = U + {1}, does not have a soution in MU see Proposition 5.9. We reca some reevant facts about the Weierstrass -function: Let g 2, g 3 C be the compex numbers such that the Weierstrass function z with periods 1 and i the imaginary unit is a soution of x 2 = 4x 3 g 2 x g 3. 1 We wi use the fact that every nonconstant soution of 1 is of the form z + z 0 for some z 0 C, see [10, page 39, Koroar F]. Reca that the fied of douby periodic meromorphic functions on C with periods 1 and i is generated by z and z [13, page 8, Theorem 4]. Let ω := 1 + 2i, and consider a rationa function Rx 1, x 2 Cx 1, x 2 such that z + ω = R z, z. 2 9

10 Proposition 5.9. Consider the foowing system of agebraic differentia-difference equations in the unknowns x, y, w: x 2 = 4x 3 g 2 x g 3, σx = Rx, x, y 3 = 1 x, 3 x w = 1. 1 System 3 has a soution in MZ. 2 For every nonempty open subset U C such that U = U + {1}, system 3 does not have a soution in MU. Proof. Proof of 1. Let K be the agebraic cosure of the fied MC. We set x j = z + jω, 1 y j = 3 z + jω, and w j = 1 z + jω. The first equation in 3 hods for these sequences because every shift of z is its soution being an equation with constant coefficients. The second equation in 3 hods because x j+1 = z + j + 1ω = R z + jω, z + jω = Rx j, x j due to 2. A direct computation shows that the ast two equations in 3 aso hod. Thus, the system 3 has a soution in K Z. Combining Propositions 5.3 and 5.7, we see that 3 has a soution in MZ. Proof of 2. Assume the contrary, et U C be such a subset and xz, yz, wz be such a soution. Since 3 is autonomous, we can assume that 0 U by shifting U and the soution if necessary. We denote the connected component of U containing 0 by U 0. The ast equation of 3 impies that xz U0 is nonconstant. Then the first equation of 3 impies that there exists z 0 C such that We wi prove that, for every z U 0 and s Z 0, xz D0 = z + z 0. 4 xz + s = z + z 0 + sω 5 by induction on s. The base case s = 0 foows from 4. Assume that 5 hods for s 0. Then, using the second equation in 3, the inductive hypothesis, and 2, we have xz + s + 1 = Rxz + s, x z + s = R z + z 0 + sω, z + z 0 + sω = z + z 0 + s + 1ω. This proves 5. Let ε > 0 be a rea number such that U contains the ε-neighbourhood of 0. Kronecker s theorem impies that there exist s Z 0 and m, n Z which we fix such that z 0 + sω n mi < ε. We set z 1 = n + mi z 0 sω. Then, since z 1 < ε, we have z 1 U 0, and so z 1 + s U. 5 impies xz 1 + s = z 1 + z 0 + sω = n + mi =. Then yz 1 + s = 0. Let d 1 be the order of zero of y at z 1 + s. Then x = 1 has a poe of order 3d at y 3 z 1 + s. We arrive at a contradiction with the fact that a the poes of are of order two [13, page 8]. 10

11 6 Proof of the main resut The proofs are structured as foows. In Section 6.1, we embed the ground δ-σ-fied k to a differentiay cosed δ-σ -fied K. In Section 6.2, we extend the technique of trains deveoped in [21] for difference equations to the differentia-difference case. Section 6.3 begins with Section 6.3.1, in which we estabish a bound Coroary 6.13 for the number of components of a differentia-agebraic variety. This bound repaces the Bézout bound, extensivey used in [21] but acking in the differentia-agebraic setting. In Section 6.3.2, we show that the existence of a sufficienty ong train impies the existence of a soution in K Z. Finay, in Section 6.4, we use these ingredients to prove the main resut, Theorem Constructing big enough fied K k Throughout Section 6 k is the δ-σ-fied from Theorem 3.1, K is a fixed differentiay cosed δ-σ -fied containing k. The existence of such fied foows from Lemma 6.1. Lemma 6.1. For every δ-σ fied k of characteristic zero, there exists an extension k K of δ-σ-fieds, where K is a differentiay cosed δ-σ -fied. Proof. We wi show that there exists a δ-σ fied K 0 containing k. The proof of [16, Proposition 2.1.7] impies that one can buid an ascending chain of σ-fieds k 0 k 1 k such that, for every i N, there exists an isomorphism ϕ i : k k i of σ-fieds, σk i+1 = k i, and ϕ i = σ ϕ i+1 for every i N. We transfer the δ-σ-structure from k to k i s via ϕ i s. Then ϕ i = σ ϕ i+1 impies that the restriction of δ on k i+1 to k i coincides with the action of δ on k i. We set K 0 := k i. Since the action δ and σ is consistent with the ascending chain 6, K 0 is a δ-σ-extension of k 0 = k. It is shown in [16, Proposition 2.1.7] that the action of σ on K 0 is surjective. [2, Theorem 3.15] impies that K 1 can be embedded in a differentiay cosed δ-σ -fied K. 6.2 Partia soutions and trains Definition 6.2. Let F k[y ], where y = y 1,..., y n, be a set of δ-σ-poynomias. Suppose h = max{ord σ f f F }. A sequence of tupes a 1,..., a n K +h K +h is caed a partia soution of F of ength if a 1,..., a n is a δ-soution of the system in y,+h 1 : {σ i F = 0 0 i 1}. Exampe 6.3. Let k = Qt be the δ-σ -fied with δ = d dt and σft = ft + 1 for each ft Qt. Let f = t δy + σy k[y ]. So h = 1. A partia soution of f of ength is a sequence a 0, a 1,..., a K +1 such that t + i δa i + a i+1 = 0, i = 0, 1,..., 1. A soution of f in K N is a sequence a i i N K N such that for each i N, t + i δa i + a i+1 = 0. With the above set F of δ-σ-poynomias, we associate the foowing geometric data anaogousy to [21]: i N 11

12 the δ-variety X A H defined by f 1 = 0,..., f N = 0 regarded as δ-equations in k[y,h ] with H = nh + 1, and so X = diffspec R X, R X := K[y,h ] / f 1,..., f N ; two projections π 1, π 2 : A H A H n defined by π 1 a 1,..., σ h a 1 ;... ; a n,..., σ h a n := a 1, σa 1,..., σ h 1 a 1 ;... ; a n,..., σ h 1 a n, π 2 a 1,..., σ h a 1 ;... ; a n,..., σ h a n := σa 1,..., σ h a 1 ;... ; σa n,..., σ h a n. By σx, we mean the δ-variety in A H defined by f1 σ,..., f N σ, where f i σ the coefficients of f i. is the resut by appying σ to Definition 6.4. A sequence p 1,..., p A H is a partia soution of the tripe X, π 1, π 2 if 1 for a i, 1 i, we have p i σ i 1 X and 2 for a i, 1 i <, we have π 1 p i+1 = π 2 p i. A two-sided infinite sequence with such a property is caed a soution of the tripe X, π 1, π 2. Lemma 6.5. For every positive integer, F has a partia soution of ength if and ony if the tripe X, π 1, π 2 has a partia soution of ength. System F has a soution in K Z if and ony if the tripe X, π 1, π 2 has a soution. Proof. It suffices to show that the first assertion hods. Suppose that a 0, a 1,..., a h+ 1 is a partia soution of F. Let p i = a i 1, a i,..., a i 1+h for i = 1,...,. Then p 1,..., p is a partia soution of X, π 1, π 2 of ength. For the other direction, et p 1,..., p be a partia soution of X, π 1, π 2 of ength. Since π 1 p i+1 = π 2 p i, there exist a 0,..., a h+ 1 K such that, for each i, p i = a i 1, a i,..., a i 1+h. Thus, a 0, a 1,..., a h+ 1 is a partia soution of F of ength. Definition 6.6 cf. [21]. For N or +, a sequence of irreducibe δ-subvarieties Y 1,..., Y in A H is said to be a train of ength in X if 1 for a i, 1 i, we have Y i σ i 1 X and 2 for a i, 1 i <, we have π 1 Y i+1 Ko = π 2 Y i Ko. Lemma 6.7. For every train Y 1,..., Y in X, there exists a partia soution p 1,..., p of X, π 1, π 2 such that for a i, we have p i Y i. In particuar, if there is an infinite train in X, then there is a soution of the tripe X, π 1, π 2. Proof. The proof is simiar to that of [21, Lemma 6.8]. To make the paper sef-contained, we wi give the detais beow. To prove the existence of a partia soution of X, π 1, π 2 with the desired property, it suffices to prove the foowing: Caim. There exists a nonempty open in the sense of the Kochin topoogy subset U Y such that for each p U, p can be extended to a partia soution p 1,..., p of X, π 1, π 2 with p i Y i i. 12

13 We wi prove the Caim by induction on. For = 1, take U = Y 1. Since each point in Y 1 is a partia soution of X, π 1, π 2 of ength 1, the Caim hods for = 1. Now suppose we have proved the Caim for 1. So there exists a nonempty open subset U 0 Y 1 satisfying the desired property. Since Y 1 is irreducibe, U 0 is dense in Y 1. So, π 2 U 0 is dense in π 2 Y 1 Ko = π 1 Y Ko. Since U 0 is δ-constructibe, π 2 U 0 is δ-constructibe too. So, π 2 U 0 contains a nonempty open subset of π 1 Y Ko. Since π 1 Y is δ-constructibe and dense in π 1 Y Ko, π 2 U 0 π 1 Y is δ-constructibe and dense in π 1 Y Ko. Let U 1 be a nonempty open subset of π 1 Y Ko contained in π 2 U 0 π 1 Y and U 2 = π 1 1 U 1 Y. Then U 2 is a nonempty open subset of Y. We wi show that for each p U 2, there exists p i Y i for i = 1,..., 1 such that p 1,..., p is a partia soution of X, π 1, π 2. Since π 1 p U 1 π 2 U 0, there exists p 1 U 0 such that π 1 p = π 2 p 1. Since p 1 U 0, by the inductive hypothesis, there exists p i Y i for i = 1,..., 1 such that p 1,..., p 1 is a partia soution of X, π 1, π 2 of ength 1. So p 1,..., p is a partia soution of X, π 1, π 2 of ength. For two trains Y = Y 1,..., Y and Y = Y 1,..., Y, denote Y Y if Y i Y i increasing chain of trains Y i = Y i,1,..., Y i,, i Y i,1 Ko,..., i Y n,i Ko for each i. Given an is a train in X which is an upper bound for this chain. For each j, i Y i,j Ko is an irreducibe δ-variety in σ j 1 X. So by Zorn s emma, maxima trains of ength aways exist in X. Fix an N. Consider the product X := X σx σ 1 X, and denote the projection of X onto σ i 1 X by ϕ,i. Note that X = diffspec R X K R σx K... K R σ 1 X. Let Note that W X, π 1, π 2 := {p X : π 2 ϕ,i p = π 1 ϕ,i+1 p, i = 1,..., 1}. 7 W = diffspec R X Rπ2 X R σx Rπ2 σx... R π2 σ 2 X R σ 1 X, under the injective K, δ-agebra homomorphisms, for a i, 1 i 1, R π2 σ i 1 X R σ i 1 X and R π2 σ i 1 X R σ i X induced by π 2 and π 1, respectivey. Lemma 6.8. For every irreducibe δ-subvariety W W, ϕ,1 W Ko,..., ϕ, W Ko is a train in X of ength. Conversey, for each train Y 1,..., Y in X, there exists an irreducibe δ- subvariety W W such that Y i = ϕ,i W Ko for each i = 1,...,. 13

14 Proof. The first assertion is straightforward. We wi prove the second assertion by induction on. For = 1, W 1 = X, and we can set W = Y 1. Let > 1. Appy the inductive hypothesis to the train Y 1,..., Y 1 and obtain an irreducibe subvariety Y W 1 X 1. Then there is a natura embedding of Y Y into X. Denote Y Y W by Ỹ. Since Y W 1, Ỹ = {p Y Y π 2 ϕ, 1 p = π 1 ϕ, p}. Let Then we have a k, δ-isomorphism Z := π 2 ϕ 1, 1 Y = π 1 Y. 8 R Y RZ R Y RỸ under the k, δ-agebra homomorphisms i 1 : R Z R Y and i 2 : R Z R Y induced by π 2 ϕ 1, 1 and π 1, respectivey. Equaity 8 impies that i 1 and i 2 are injective. Denote the fieds of fractions of R Y, R Y, and R Z by E, F, and L, respectivey. Let p be any prime differentia idea in E L F, R := E L F /p, and π : E L F R be the canonica homomorphism. Consider the natura homomorphism i: R Y RZ R Y E L F. Since 1 ir Y RZ R Y, the composition π i is a nonzero homomorphism. Since i 1 and i 2 are injective, the natura homomorphisms i Y : R Y R Y RZ R Y and i Y : R Y R Y RZ R Y are injective as we. We wi show that the compositions π i i Y : R Y R and π i i Y : R Y R are injective. Introducing the natura embeddings i E : E E L F and j Y : R Y E, we can rewrite π i i Y = π i E j Y. The homomorphisms i E and j Y are injective. The restriction of π to i E E is aso injective since E is a fied. Hence, the whoe composition π i E j Y is injective. The argument for π i i Y is anaogous. Let S := R Y RZ R Y / p RY RZ R Y, which is a domain, and the homomorphisms π i i Y : R Y S and π i i Y : R Y S are injective. We et W := diffspec S. For every i, 1 i <, the homomorphism ϕ,i = π i i Y ϕ 1,i : R Y i R Y S is injective as a composition of two injective homomorphisms. Hence, the restriction ϕ,i : W Y i is dominant. 6.3 Technica bounds Number of prime components in differentia varieties In this section, we fix a δ-fied k and x = x 1,..., x n. For a commutative ring R and subsets I and S of R, we et I : S = {r R s S : rs I}. 14

15 Lemma 6.9. There exists a computabe function Gn, r, D such that, for every r Z 0 and a prime idea I k[x r,0 ] such that I = I k[x r,0 ] and degi D, there exists f k[x r,0 ] \ I such that I : f is a prime differentia idea; degf Gn, r, D. Proof. Compute a reguar decomposition of I using the Rosenfed-Gröbner agorithm with an ordery ranking: I = C 1 : H1... CN : HN. Since the Rosenfed-Gröbner agorithm with an ordery ranking does not increase the orders of the poynomias, C 1,..., C N k[x r,0 ]. Since I is prime, there exists i, say i = 1, such that k[x r,0 ] C 1 : H 1 = I. 9 We show that J := C 1 : H 1 is a prime differentia idea. Suppose P 1, P 2 k[x,0 ] with P 1 P 2 J. Let P i i = 1, 2 be the partia remainder of P i with respect to C 1 [25, p. 396]. Then P 1 P 2 J. Due to Rosenfed s emma [25, p. 397], P 1 P 2 k[x,0 ] C 1 : H 1 k[x,0 ] I : H 1 = k[x,0 ] I. Since k[x,0 ] I is prime, at east one of P i beongs to k[x,0 ] I J. So P 1 J or P 2 J. Thus, J is prime. Equaity 9 together with C 1 k[x r,0 ] impy that C 1 : H 1 = I : H 1, so I : H1 is a prime differentia idea. Since the differentia poynomias from C 1 together with some of their derivatives constitute a trianguar set for I, [26, Theorem 1] impies that the degree of every initia and every separant of C 1 is bounded by Since there are at most nr eements in C 1, setting 2nr nr+1 D 2nr D. Gn, r, D := 2nr + 12nr nr+1 D 2nr nr + 1D and f := H 1 finishes the proof of the emma. Lemma For every differentia ring R, subring S R, and ideas I, P 1,..., P S such that I = P 1... P, we have I = P 1... P Proof. [22, Lemma 8] impies that, for every s > 0, P s 1... P s P 1... P s. Since taking the radica commutes with intersections, we have P s 1... P s P 1... P s. We aso have P 1... P s P s 1... P s = P s 1... P s. Taking s =, we obtain P 1... P I P 1... P. 15

16 Lemma There exists a computabe function F n, r, m, D such that, for every r, m, D and radica idea J k[x r,0 ] of dimension m and degree D, deg k[x r,0 ] J F n, r, m, D. Proof. [22, Theorem 3] together with [7, Proposition 3] impy that k[x r,0 ] J = k[x r,0 ] J B, where B := D nr+12m+1. Thus, deg k[x r,0 ] J deg J B. The Bézout inequaity impies that deg J B D nr+1b. Thus, we can finish the proof of the emma by setting F n, r, m, D = D nr+1b. Proposition There is a computabe function Cn, r, m, D such that, for every nonnegative integers r, m, D and every radica idea I k[x r,0 ] such that degi D, dimi m, I = I k[x r,0 ], the number of prime components of I does not exceed Cn, r, m, D. Proof. We fix r for the proof and wi prove the proposition by constructing the function Cr, m, D by induction on a tupe m, D with respect to the exicographic ordering. Consider the base case m = 0. Then there are at most D possibe vaues for x 1,..., x n and every prime component of I is the maxima differentia idea corresponding to one of these vaues. Thus, the proposition is true for Cn, r, 0, D = D. Consider m > 0. If I is not prime, then Lemma 6.10 impies that the number of prime components of I does not exceed max max Cn, r, m, D Cn, r, m, D, 10 D D =D where a Cn, r, m, D i are aready defined by the inductive hypothesis. Consider the case of prime I. Lemma 6.9 impies that there exists f k[x r,0 ] \ I such that degf Gn, r, D and I : f is a prime differentia idea. Every prime component of I either is equa to I : f or contains f. In the atter case, the component is a component of I, f. Let J := I, f k[x r,0 ]. Then dimj m 1 and Lemma 6.11 impies that degj F r, m 1, Gn, r, D. Then the number of prime components of I does not exceed 1 + Cn, r, m 1, F r, m 1, Gn, r, D. 11 Thus, one can define Cn, r, m, D to be the maximum of 10 and 11. Proposition 6.12 and Lemma 6.11 impy the foowing coroary. Coroary For every radica idea I k[x r,0 ] of dimension at most m and degree at most D, the number of prime components of I does not exceed Cn, r, m, F n, r, m, D. 16

17 6.3.2 Bound for trains Now we try to give a bound so that the existence of a maxima train of certain ength in X wi definitey guarantee the existence of at east one infinite train in X. Definition 6.14 Kochin poynomias for δ-varieties and trains. The Kochin poynomia of an irreducibe δ-variety V = VF, where F K[y,0 ], where y = y 1,..., y n, is the unique numerica poynomia ω V t such that there exists t 0 0 such that, for a t t 0 and the generic point a of V see Definition 4.2, ω V t = tr.deg Ka t,0 /K. The Kochin poynomia of a δ-variety is defined to be the maxima Kochin poynomia of its irreducibe components. An irreducibe component X 1 of a δ-variety X is caed a generic component if ω X1 t = ω X t. We define the Kochin poynomia of a train Y = Y 1,..., Y in X as ω Y t := min ω Yi t. i Remark The Kochin poynomia of an irreducibe δ-variety V is of the form see [12, formua 2.2.6] and [11, Theorem II.12.6d] ω V t = δ-dimv t ordv. The foowing emma shows how the coefficients of a Kochin poynomia change under a projection. Lemma Let V A n be an irreducibe δ-variety and π 1 : A n A n 1 be the projection to the first n 1 coordinates. Then we have δ-dim π 1 V Ko δ-dimv and ord π1 V Ko ordv. Proof. Let a be a generic point of V. Then π 1 a is a generic point of W := π 1 V Ko. Ceary, So, we have It, therefore, suffices to show that Suppose δ-dimw < δ-dimv = d. Then we have ω W t ω V t and δ-dimw δ-dimv. δ-dimw = δ-dimv = ordw ordv. δ-dimw < δ-dimv = ordw ordv. δ-dimw = δ-dimv 1 = d 1. Since the order of an irreducibe δ-variety V is equa to the maxima reative order of V with respect to a parametric set, without oss of generaity, suppose ordw = tr.deg K π 1 a,0 / K πn d 1 a,0, where π n d 1 : A n A d 1 is the projection to the first d 1 coordinates. Since δ-tr.deg K a,0 / K = 1 + δ-tr.deg K π1 a,0 / K, a n is δ-transcendenta over Kπ 1 a,0, i.e., δ-tr.deg K a,0 / K π1 a,0 = 1. Therefore, we have ordw = tr.deg K π 1 a,0 / K πn d 1 a,0 = tr.deg K a n,0 π1 a,0 / K an,0 πn d 1 a,0 = ordy1,...,y d 1,y n V ordv. 17

18 Lemma For a s Z 0 and F k[y s,0 ], where y = y 1,..., y n, the order of each component of VF is bounded by ns. Proof. It foows directy by [24, p. 135] and [6, Theorem 2.11]. Definition For a non-negative integers n, s, h, d, Z 0 -vaued poynomias ω Z[t], we define Bω, n, s, h, d to be the smaest M N { } such that, for every tripe X, π 1, π 2 with X = VF A nh+1, F k[y s,h ], y = y 1,..., y n, degf d, if there exists a train in X of ength M and Kochin poynomia at east ω, then there exists an infinite train in X. Remark For a non-negative integers n, s, h, d and Z 0 -vaued poynomias ω Z[t], Bωt, n, s, h, d B0, n, s, h, d. In the foowing, we wi show that Bωt, n, s, h, d is finite for a ωt 0 and the numerica data n, s, h, d and is aso bounded by a computabe function in these numerica data. For the ease of notation, we denote Ln, r, d := C n, r, nr + 1, F n, r, nr + 1, d, which is computabe. So by Coroary 6.13, given a system S of δ-poynomias in n δ-variabes of order bounded by r and degree bounded by d, the number of components of the δ-variety VS is bounded by Ln, r, d. By Lemma 6.8, the number of maxima trains in X of ength is bounded by Lnh +, s, d. We now define two increasing sequences A i n, h, s, d i N and τ i n, h, s, d i N as foows: A 0 = Lnh + 1, s, d + 1, A i+1 = A i + Lnh + 1A i, s, d for i 0 τ 0 = nsh + 1, τ i+1 = τ i + nsh + 1A τi + 1 for i Lemma We have B0, n, s, h, d A τnh+1 n,h,s,dn, h, s, d, which is computabe. Proof. Temporariy, fix X. By Coroary 6.13, we know upper bounds for the number of irreducibe components of X and for the number of maxima trains in X of any fixed ength. The main idea of the proof is to construct a decreasing chain of Kochin poynomias ω 0 t > ω 1 t > and for each ω i t, give an upper bound B i for Bω i t, n, s, h, d. Since the Kochin poynomias are we-ordered, the decreasing chain wi stop at some ω J t = 0. Let ω 0 t = ω X t. Let B 0 be the number of generic components of the δ-variety X pus 1. Consider a train Y 1,..., Y B0 in X of Kochin poynomia at east ω 0 t. So for each i, σ i+1 Y i is a δ-subvariety of X with Kochin poynomia at east ω X t, so σ i+1 Y i must be a generic component of X. Since X has ony B 0 1 generic components, there exists a < b N such that σ a+1 Y a = σ b+1 Y b, which impies Y b = σ b a Y a. Thus, we can construct an infinite train..., Y a, Y a+1,..., Y b 1, σ b a Y a, σ b a Y a+1,..., σ b a Y b 1,

19 Suppose ω i t and B i have been constructed. We now try to do it for i + 1. Let B i+1 = B i + D i, 13 where D i is the number of maxima trains in X of ength B i. Consider the fibered product W Bi X, π 1, π 2, as in 7, and, for each irreducibe component W of W Bi, denote to be the train corresponding to W. Let Y W = ϕ Bi,1W Ko,..., ϕ Bi,B i W Ko ω i+1 t := max { ω YW t ωyw t < ω i t, W is a component of W Bi }, and set max = 0. Consider a maxima train Y 1,..., Y Bi+1 in X with Kochin poynomia at east ω i+1 t. We wi show this B i+1 works. Introduce D i + 1 trains Z 1,..., Z Di+1 of ength B i in X, σx,..., σ D i X, respectivey, such that for each j, Z j = Z j 1,..., Zj B i := Yj,..., Y j+bi 1. Then for each j, consider a maxima train Z j of ength B i containing Z j. So σ j+1 Z j is a maxima train of ength B i in X. There are two cases to consider: { ωyw t ωyw t < ω i t, W is a component of W Bi } =. Case 1 In this case, ω i+1 t = 0, and for each j, ω σ j+1 Z j t ω it. By the construction of B i, we coud construct an infinite train through each σ j+1 Z j. { ωyw t ωyw t < ω i t, W is a component of W Bi }. Case 2 If there exists some j 0 such that ω σ j 0 +1 Z j 0 t ω it, then by the construction of B i, we coud construct an infinite train through this σ j 0+1 Z j 0. Suppose now that, for each j, ω σ j+1 Z j t = ω i+1t. Since there are ony D i number of maxima trains in X of ength B i, there exist a < b such that σ a+1 Z a = σ b+1 Z b. Since ω σ a+1 Z a t = ω i+1t, there exists such that Then for σ a+1 Z a σ a+1 have ω σ a+1 a Z t = ω i+1t. ω σ a+1 Z a t = ω i+1t, a Z and the Kochin poynomia of Y 1,..., Y Bi+1 is at east ω i+1 t. So we σ a+1 Z a = σ a+1 a Z. 19

20 Simiary, we can show So σ b+1 Z b = σ b+1 b Z. σ a+1 Y a+ 1 = σ a+1 Z a = σ a+1 Thus, we have a Z = σ b+1 Y b+ 1 = σ b a Y a+ 1. b Z = σ b+1 Z b = σ b+1 Y b+ 1. Therefore, we can construct an infinite sequence Y 1,..., Y a+ 1,,..., Y b+ 2, σ b a Y a+ 1,..., σ b a Y b+ 2,.... As we described in the first paragraph, as the process goes on, we has constructed a decreasing chain of Kochin poynomias ω 0 t = ω X t > ω 1 t > ω 2 t >. Since the Kochin poynomias are we-ordered, this chain is finite, so the above process wi stop at step J at which we coud get ω J t = 0, either in the case in which { ωyw t ωyw t < ω J 1 t, W is a component of W BJ 1 } =, or in the case in which the set is nonempty and the maxima Kochin poynomia in the set is 0. By Lemma 6.8 and Coroary 6.13, for the number D i of maxima trains in X of ength B i, we have By Coroary 6.13, we have D i Lnh + 1B i, s, d, so B i+1 B i + Lnh + 1B i, s, d. 14 For each i, 0 i J, et a i and b i be such that B 0 Lnh + 1, s, d + 1. ω i t = a i t b i. For i = 0, we have a 0 = δ-dimx and b 0 = ordx. For every j, 0 j a 0, we define i j to be the argest integer in [0, J] such that a 0 a ij j. Then J = i a0. The decreasing of the Kochin poynomias impies that, for a j, 0 j < a 0 : we have i 0 b 0 and i j+1 i j + b ij by the definition of ω ij +1t and Lemma 6.16, b ij +1 is bounded by the maxima order of the components of W Bij, so by Lemma 6.17, b ij +1 nsh + 1B ij. 16 Comparing the recursive formuas 12 with inequaities 14, 15, and 16, we see that B i A i for every i, 0 i J; i j τ j for every j, 0 j a 0 = δ-dimx. 20

21 Thus, B J = B iδ-dimx A iδ-dimx A τδ-dimx A τnh+1. As a consequence, we have the foowing resut. Coroary For a s, h Z 0 and F k[y s,h ], F = 0 has a soution in K Z if and ony if F = 0 has a partia soution of ength D := A τnh+1 n,h,s,dn, h, s, d. Proof. Let X A H be the δ-variety defined by F = 0 regarded as a system of δ-equations in y, σy,..., σ h y, where H = nh + 1. By Lemmas 6.5 and 6.7, F = 0 has a partia soution of ength D resp. F = 0 has a soution in K Z if and ony if there exists a train of ength D in X resp., there exists an infinite train in X. By Lemma 6.20, if there exists a train of ength D in X, then there exists a infinite train in X. So the assertion hods. 6.4 Proof of Theorem 3.1 We wi prove a more refined version of Theorem 3.1: Theorem For a non-negative integers r, s, h, d, there exists a computabe B = Br, s, h, d such that, for a: non-negative integers q, δ-σ-fieds k, sets of δ-σ-poynomias F k[x s,h, y s,h ], where x = x 1,..., x q, y = y 1,..., y r, and deg y F d, we have σ i F i Z 0 k[x ] {0} σ i F i [0, B] B k[xb ] {0}. Proof. The = impication is straightforward. We wi prove the = impication. For this, et A := A τrh+1 r,h,s,dr, h, s, d, B δ be the bound B from [22, Theorem 1] with and B := B δ + s. By assumption, Suppose that If α rs + 1A + h + 1, m rs + 1A + h + 1, and d d, 1 σ i F i Z 0 kx [y ]. 17 σ i F i [0, A] B δ k[xb ] = {0} σ i F i [0, A] B δ kxb [y,a+h ], then there woud exist c i,j kx B [y,a+h ] such that 1 = B δ A c i,j δ i σ j f. 19 i=0 j=0 f F Mutipying equation 19 by the common denominator in the variabes x B, we obtain a contradiction with 18. Hence, by [22, Theorem 1], 1 / σ i F i [0, A] kxb [y,a+h ]. 21