EFFECTIVE DIFFERENCE ELIMINATION AND NULLSTELLENSATZ

ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON

Abstract. We prove effective Nullstellensatz and elimination theorems for difference equa-tions in sequence rings. More precisely, we compute an explicit function of geometric quantitiesassociated to a system of difference equations (and these geometric quantities may themselvesbe bounded by a function of the number of variables, the order of the equations, and the degreesof the equations) so that for any system of difference equations in variables x = (x1, . . . , xn)and y = (y1, . . . , ym), if these equations have any nontrivial consequences in the x variables,then such a consequence may be seen algebraically considering transforms up to the order ofour bound. Specializing to the case of n = 0, we obtain an effective method to test whether agiven system of difference equations is consistent.

1. Introduction

We say that a sequence of complex numbers (aj)∞j=0 satisfies a difference equation with

constant coefficients if there is a nonzero polynomial F (x0, . . . , xe) ∈ C[x0, . . . , xe] such that,for every natural number j, the equation F (aj, aj+1, . . . , aj+e) = 0 holds. This can also bedefined for systems of difference equations in several variables. Such difference equations andthe sequences that solve them are ubiquitous throughout mathematics and in its applications tothe sciences, including such areas as combinatorics, number theory, and epidemiology, amongstmany others (see Section 4 for some of the examples).

In this paper we resolve some fundamental problems about difference equations. The ques-tions we answer include the following (for precise statements, including the way non-constantcoefficients can appear, see Section 3):

(1) Under what conditions does a system of difference equations have a sequence solution?(2) Can these conditions be made sufficiently transparent to allow for efficient computation?(3) Given a system of difference equations on (n + m)-tuples of sequences, how does one

eliminate some of the variables so as to deduce the consequences of these equations on thefirst n variables?

Our solution to the first question is a conceptual difference Nullstellensatz, to the second, aneffective difference Nullstellensatz, and to the third, an effective difference elimination algo-rithm. Even though the abstract Nullstellensatz is intellectually satisfying in that conditionsof different kinds are shown to be equivalent, namely the existential condition that there isa sequence solution to a system of difference equations and the universal condition that thedifference ideal generated by the equations is proper, the difficult work and applications, boththeoretical and practical, comes with our main effective theorems.

Effective elimination theorems and methods have a long history and play central roles incomputational algebra. Row reduction, or Gaussian elimination, is a fundamental techniquein linear algebra. Elimination for polynomial equations is substantially more complicated andhas been the subject of intensive and sophisticated work [5, 23, 22]. In recent work of thefirst two authors joined by Vo [28], effective elimination theorems were obtained for algebraicdifferential equations through a reduction to the polynomial case through the decomposition-elimination-prolongation method.

1

2 ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON

While these questions are important and difference equations have been studied intensivelyboth for their applications and theory, to our knowledge, none of these questions has receiveda satisfactory answer in the literature. We explain below how some known results, bothpositive and negative, may help explain the existence of this lacuna. In particular, in someessential ways, the effective Nullstellensatz and elimination problems for difference equationsare substantially more difficult than the corresponding problems for differential equations andthe methods of [28] do not routinely transpose to this context.

The foundational work on difference algebra, that is, the study of the theory of differencerings and of difference equations as encoded through the algebraic properties of rings of differ-ence polynomials, was initiated by Cohn in [7], following the tradition of Ritt and Kolchin indifferential algebra. Deep results have been obtained in this subject, but their relevance to theproblems at hand is hampered by their restrictions, for the Nullstellensatz and elimination the-orems, to the case in which solutions are sought in difference fields, and thus have little bearingon the structures used in practice, namely difference rings presented as rings of sequences, suchas CN given with the shift operator σ : (ai)

∞i=0 7→ (ai+1)∞i=0. Moreover, even if restricted to

difference fields, the known elimination theorems are at best theoretically effective.Chatzidakis and Hrushovski studied difference fields from the perspective of mathematical

logic in [6]. There, they established a recursive axiomatization for the theory of existentiallyclosed difference fields and proved a quantifier simplification theorem. From this it followsthat in principle there are effective procedures to check the consistency of difference equa-tions in difference fields and to perform difference elimination in difference fields. More recentwork of Tomasic [33, 34] geometrizes the quantifier simplification theorem and brings the com-plexity of these algorithms to primitive recursive, though this effectivity is still theoretical— to call the implicit bounds astronomical would be a gross understatement — and a prac-tical implementation of this work is infeasible. In symbolic computation, steps have beentaken towards extending the characteristic set method from differential algebra to the studyof difference and difference-differential equations in works of Gao, van der Hoeven, Li, Yuan,Zhang [14, 15, 27, 26]. These methods are more efficient than those coming from logic, but asthey are restricted to the study of inversive prime difference ideals, they, too, are fundamen-tally results about solutions to difference equations in difference fields and the constructionsof difference resultants depend on restrictive hypotheses.

The situation for difference equations in sequence rings differs starkly. Simple examples showthat consistency checking in difference fields is not the same problem as consistency checkingfor sequences. For example, the system of difference equations xσ(x) = 0, x + σ(x) = 1 hasno solution in a difference field, but the sequence 0, 1, 0, 1, . . . is a solution in CN.

More seriously, theorems of Hrushovski and Point [21] show that the logical methods usedfor difference fields fail dramatically for sequence rings. In particular, they show that thefirst-order theory of CN regarded as a difference ring is undecidable. Thus, we cannot derivea consistency checking method from a recursive axiomatization of this theory nor can weproduce an elimination algorithm from an effective quantifier elimination theorem; no suchaxiomatization or quantifier elimination procedure exists. That we succeed in solving theeffective consistency checking and effective elimination problems for difference equations insequence rings is all the more surprising given these undecidability results.

Let us explain more precisely what we actually prove and where the new ideas appear in ourarguments. We have two main theorems: Theorem 1 an effective Nullstellensatz and Theorem 2an effective difference elimination theorem. Strictly speaking, the effective Nullstellensatz is

EFFECTIVE DIFFERENCE ELIMINATION AND NULLSTELLENSATZ 3

a special case of an effective elimination theorem, but we prove elimination by bootstrappingthrough the Nullstellensatz.

The key to our proofs is a new form of the decomposition-elimination-prolongation (DEP)method. As is completely standard, a system of difference equations may be regarded as asystem of algebraic equations in more variables together with specifications that certain coor-dinates should be obtained from others by the application of the distinguished endomorphismand the usual DEP methods allow for one to cleverly reduce questions about the original sys-tem of difference equations to questions entirely about algebraic equations. A version of theDEP method for difference equations in difference fields is employed in [19] for the purposeof computing explicit bounds in Diophantine geometric problems. This DEP method cannotwork for the problems at hand (see Section 5) but we overcome this obstacle by developinga strikingly more robust DEP method. With our theorems we explicitly bound the numberof prolongations required to solve the problems at hand, i.e. testing a system of differenceequations for consistency or computing a nontrivial element of the elimination ideal. Ourbounds are quite small. Indeed, roughly speaking, for balanced systems in which the numberof equations is equal to the number of variables, the bounds are very small and even for under-determined systems the bounds are humanly comprehensible, and small enough in many casesto permit efficient computation.

We draw an interesting theoretical conclusion from our work towards the explicit boundsfor the difference elimination problem in Section 7. Specifically, with Proposition 4 we showthat for (K, σ) any algebraically closed difference field, whenever a finite system of differenceequations over K is consistent in the sense that it has a solution in some difference ring, thenit already has a solution in the ring of sequences of elements of K. We give a soft proof of sucha difference Nullstellensatz under the hypothesis that K is uncountable with Proposition 1.In extending this result to general K we use the geometric analysis of our proof of our maintheorem and a remarkable theorem of Hrushovski on the first-order theory of the Frobeniusautomorphism.

The paper is organized as follows. We give the basic definitions in Section 2, and thenintroduce the notation and terminology specific to our paper. The main results, Theorem 1for the effective Nullstellensatz and Theorem 2 for the effective elimination, are expressedin Section 3. In Section 4, we illustrate our results in several practical examples. WithSection 5, we present counterexamples to an effective strong difference Nullstellensatz and tothe application of the usual DEP method to these problems. The proofs of the main theoremsare presented in Section 6. Finally, in Section 7, we strengthen the difference Nullstellensatzgiving equivalent criteria for the existence of sequence solutions to systems difference equationsover any algebraically closed field.

2. Preliminaries

A detailed introduction to difference rings can be found in [7, 25].

Definition 1. A difference ring is a pair (A, σ) where A is a commutative ring and σ : A→ Ais a ring endomorphism.

Example 1. If R is any commutative ring, then the sequence rings RN and RZ are differencerings with σ defined by σ((xi)i∈N) := (xi+1)i∈N (σ((xi)i∈Z) := (xi+1)i∈Z, respectively).

Definition 2. A map of difference rings ψ : (A, σ) → (B, τ) is given by a map of ringsψ : A→ B such that that τ ◦ ψ = ψ ◦ σ.

4 ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON

Remark 1. We often abuse notation saying that A is a difference ring when we mean the pair(A, σ).

Definition 3. If A is a difference ring, then the free difference A-algebra in one generator xover A, A{x}, also called the ring of difference polynomials in x over A, may be realized as theordinary polynomial ring A[{σj(x) : j ∈ N}] in the indeterminates {σj(x) : j ∈ N}. Iteratingthis procedure, one obtains the difference polynomial ring A{x1, . . . , xn} in n variables.

Definition 4. For P ∈ A{x1, . . . , xn} and 1 ≤ i ≤ n, we define the order of P with respect toxi, denoted ordxi(P ) to be the maximal h for which σh(xi) appears in P . If no σh(xi) appears,we set ordxi(P ) := −1.

Example 2. ordx3(σ3(x1) + x2 + σ(x3)2 + 1)=1.

Definition 5. If (A, σ) is a difference ring and S ⊆ A{x1, . . . , xn} is a set of difference poly-nomials over A, (A, σ) ⊆ (B, σ) is an extension of difference rings, and b = (b1, . . . , bn) ∈ Bn

is an n-tuple from B, then we say that b is a solution of the equations {f = 0 | f ∈ S} if,under the unique map of difference rings A{x1, . . . , xn} → B given by extending the given mapA→ B and sending xi 7→ bi for 1 ≤ i ≤ n, every element of S is sent to 0.

Example 3. Let (A, σ) = (Q, id) and (B, σ) = (QN, σ), where σ is the shift (to the left)operator. Then the tuple b = ((1, 0, 1, 0, . . .), (2017, 1, 0, 1, . . .)) ∈ B is a solution of theequations {

σ(x1) + x1 − 1 = 0,

σ(x2)− x1 = 0.

Definition 6. If (A, σ) is a difference ring and S ⊆ A{x1, . . . , xn} and B is a non-negativeinteger, the B-th transform of S is the set

{σB(f) | f ∈ S

}. So, the 0-th transform of S is S.

The B-th transform of a system of difference equations is defined similarly.

Example 4. The 2-nd transform of the system{σ(x1)5 = x1 + x2

2

x33 + x1 + 1 = 0

is the system {σ3(x1)5 = σ2(x1) + σ2(x2)2

σ2(x3)3 + σ2(x1) + 1 = 0.

Definition 7. A difference equation f(x1, . . . , xn) = 0 is said to be a consequence of a systemof difference equations S = 0 if there exists a non-negative integer B such that f belongs tothe polynomial ideal generated by the 0-th,. . . , B-th transforms of S, that is

f ∈(σi(f) | 0 6 i 6 B, f ∈ S

).

Example 5. Let S = 0 be the system{f1 = x2σ(x1)− x1 − 1 = 0

f2 = σ(x2)− x22 = 0.

The equation σ2(x1)x22 − σ(x1)− 1 is a consequence of S = 0 with B = 1, because

σ(f1)− σ2(x1)f2 = σ(x2σ(x1)− x1 − 1)− σ2(x1)(σ(x2)− x22) = σ2(x1)x2

2 − σ(x1)− 1.

EFFECTIVE DIFFERENCE ELIMINATION AND NULLSTELLENSATZ 5

3. Main results

For all d ∈ Z>0 and D ∈ Z>0 we define

B(d,D) =

D + 1 if d = 0,

2 +D2 + D(D−1)(D−2)6

if d = 1,

B(d− 1, D) +DB(d−1,D) if d > 1.

3.1. Effective difference Nullstellensatz.

Theorem 1. Let k be a difference field and F = {f1 = 0, . . . , fN = 0} a system of differenceequations, with f1, . . . , fN ∈ k{u1, . . . , ur}. We set

hi := maxj=1,...,N

ordui fj and H = h1 + . . .+ hr + r.

Let d(F ) and D(F ) denote the dimension and the sum of the degrees of the components ofthe affine variety defined by F over k in the affine H-space, respectively. Then system Fhas a solution in a difference ring containing k if and only if the system consisting of the0-th,. . . , (B(d,D)− 1)-th transforms of F is consistent as a system of polynomial equations.

Corollary 1. Let h = maxi=1,...,r

hi. If f1, . . . , fN ∈ C{u1, . . . , ur}, then system F has a solution in

CZ if and only if there exist tuples u1, . . . ,ur ∈ CB(d,D)+h−1, where ui := (ui,1, . . . , ui,B(d,D)+h),such that

fi(u1,j, . . . , ur,j) = 0 for all 1 6 i 6 N, 1 6 j 6 B(d,D) + h− 1.

Remark 2. We do not prove an effective strong Nullstellensatz generalizing Corollary 1, be-cause such a statement is false as shown in Section 5.

3.2. Effective elimination. We will introduce the notation that will be used in Theorem 2.Let x = (x1, . . . , xm) and u = (u1, . . . , ur) be two sets of unknowns. Consider a systemF = {f1 = 0, . . . , fN = 0} of difference equations, where f1, . . . , fM ∈ k{x,u}. We set

hi := maxj=1,...,N

ordui fj and H = h1 + . . .+ hr + r.

Let E be the field of fractions of k{x} and X denote the affine variety defined by f1 = . . . =fN = 0 over E. We denote the dimension and the sum of the degrees of the components of Xby d(F ) and D(F ), respectively.

Theorem 2. For all integers d > 0 and D > 1 and systems F in x and u with d(Σ) = dand D(Σ) = D, there exists a non-zero difference equation f(x) = 0 that is a consequence ofsystem F if and only if the ideal generated by the 0-th,. . . , (B(d,D) − 1)-th transforms of Fcontains a nonzero polynomial depending only on x and their transforms.

3.3. Consequences for computation. Theorem 1 and Corollary 1 reduce consistency ques-tions for systems of difference equations to consistency questions (in algebraically closed fields)of polynomial systems in finitely many variables and Theorem 2 reduces the question of ex-istence/finding a consequence in the x variables of a system of difference equations in thevariables x and u to a question about a polynomial ideal in a polynomial ring in finitely manyvariables. These algebraic problems are classical and have been computationally solved using,for example, Grobner bases, triangular sets, numerical algebraic geometry, etc. For all of thesemethods, implementations exist in many computer algebra systems and independent softwarepackages (see, for example, [8, 3, 32]).

6 ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON

4. Numerical values and practical examples

In the following table, we compute B(d,D)− 1 for small d and D.

d \D 1 2 3 4 50 1 2 3 4 51 2 5 11 21 36

Remark 3. Almost all examples of modeling phenomena in the sciences using polynomialdifference equations that we have seen in the literature can be written as systems with the samenumber of equations as unknowns in such a way that none of the equations is a consequenceof the others. The above table is applicable to elimination problems for such systems with nequations if the problem is to eliminate dn/2e unknowns or less, as such problems typicallyresult in varieties X (see the notation of Section 3.2) of dimension 0 or 1.

Remark 4. One can significantly speed up checking if an elimination is possible by

(1) applying the number of transforms that is in the bound(2) substituting random values into the variables that are not being eliminated.

Similarly to [17], for each number p, 0 < p < 1, using the Schwartz-Zippel lemma [36,Proposition 98], we can find the range for the random substitution so that the probability ofthe elimination being possible if and only if the “substituted” system has no solutions is greaterthan p. So, this would give an efficient probabilistic test for the possibility of elimination.

Remark 5. Although there could be special tricks and methods for each of the examplesbelow, our approach provides a general and fully automated procedure.

Example 6. Consider the May-Leonard model for 2-plant annual competition, scaled downfrom [30]: {

xn+1 = (1−b)xnxn+α1yn

+ bxn,

yn+1 = (1−b)ynα2xn+yn

+ byn,

which can be rewritten as

(1)

{(x+ α1y)σ(x) = (1− b)x+ bx(x+ α1y),

(α2x+ y)σ(y) = (1− b)y + by(α2x+ y),

where F = Q(α1, α2, b), with σ acting as the identity on F . To verify whether y can beeliminated from (1), we then consider the affine variety X defined by (1) over the fieldQ(α1, α2, b, x, σ(x)) with coordinates y, σ(y). A computation shows that d = 0 and D = 1,and so B(d,D) − 1 = 2 − 1 = 1. A computation shows that it is not only sufficient but alsonecessary to apply this single transform to perform the elimination. So, our main result givesa sharp upper bound for this example.

Example 7. Consider the May-Leonard model for 3-plant annual competition [30]:xn+1 = (1−b)xn

xn+α1yn+β1zn+ bxn,

yn+1 = (1−b)ynα2xn+yn+β2zn

+ byn,

zn+1 = (1−b)znα3xn+β3yn+zn

+ bzn,

which can be rewritten as

(2)

(x+ α1y + β1z)σ(x) = (1− b)x+ bx(x+ α1y + β1z),

(α2x+ y + β2z)σ(y) = (1− b)y + by(α2x+ y + β2z),

(α3x+ β3y + z)σ(z) = (1− b)z + bz(α3x+ β3y + z),

EFFECTIVE DIFFERENCE ELIMINATION AND NULLSTELLENSATZ 7

where F = Q(α1, α2, α3, β1, β3, β3, b), with σ acting as the identity on F . To verify whether yand z can be eliminated from (2), we consider the affine variety X defined by (2) over the fieldQ (α1, α2, α3, β1, β3, β3, b, x, σ(x)) with coordinates y, σ(y), z, σ(z). A computation shows thatd = 1 and D = 3, and so B(d,D)− 1 = 11. A computation shows that

• two prolongations are necessary and sufficient• carrying out a computation with 11 transforms as described in Remark 4 to check if an

elimination is possible does not take significantly more time than doing this with twotransforms.

Example 8. Consider the stage structured Leslie-Gower model [9, eq. (5)]:Jn+1 = b1

11+d1An

An

An+1 = s11

1+Jn+c1jnJn

jn+1 = b21

1+d2anan

an+1 = s21

1+c2Jn+jnjn,

which can be rewritten as

(3)

(1 + d1A)σ(J) = b1A

(1 + J + c1j)σ(A) = s1J

(1 + d2a)σ(j) = b2a

(1 + c2J + j)σ(a) = s2j,

where F = Q(b1, b2, c1, c2, d1, d2, s1, s2) with σ acting as the identity on F . To verify whether Jand j can be eliminated from (3), we consider the affine variety X defined by (3) over the fieldQ(b1, b2, c1, c2, d1, d2, s1, s2, a, A, σ(a), σ(A)) with coordinates j, σ(j), J, σ(J). A computationshows that d = 0; D = 1 as the equations are linear in j, σ(j), J, σ(J). Then B(d,D) − 1 =2− 1 = 1. A computation shows that it is not only sufficient but also necessary to apply thissingle transform to perform the elimination. So, our main result gives a sharp upper boundfor this example.

Example 9. A discrete multi-population SI model from [2], similarly to the previous examples,can be rewritten as

(4)

σ(S) = S(

1− a·∆tN1I − b·∆t

N1i)

σ(s) = s(

1− c·∆tN2I − d·∆t

N2i)

σ(I) = I + S(a·∆tN1I + b·∆t

N1i)

σ(i) = i+ s(c·∆tN2I + d·∆t

N2i),

where F = Q(a, b, c, d,∆t, N1, N2) with σ acting as the identity on F .

• To verify whether I, i can be eliminated from (4), we consider the affine variety definedby (4) over Q(a, b, c, d,∆t, N1, N2, s, σ(s), S, σ(S)), and so d = 0, D = 1, thus B(d,D)−1 = 2− 1 = 1.• To verify whether I, i, s can be eliminated from (4), we consider the affine variety

defined by (4) over Q(a, b, c, d,∆t, N1, N2, S, σ(S)), and so d = 2, D = 2. One can showthat B(2, 2)−1 = 70−1 = 69, which turned out to be computationally feasible as well.

8 ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON

Example 10. Let Fn be the n-th Fibonacci number. It turns out [11, p. 856] that thesequence An := F2n satisfies a nonlinear difference equation. Such an equation can be foundusing difference elimination as follows. We introduce Bn := F2n+1. Then standard identitiesF2k = Fk(2Fk+1 − Fk) and F2k+1 = F 2

k+1 + F 2k for the Fibonacci numbers imply the following

system of difference equations

(5)

{An+1 = An(2Bn − An),

Bn+1 = A2n +B2

n.

Considered as a system of polynomial equations in Bn and Bn+1, (5) defines an affine varietyof dimension zero and degree two over Q(An, An+1). Theorem 2 implies that it is sufficient toconsider system (5) and two of its transforms to eliminate B. Performing this elimination, wefind the difference equation

5F 42nF2n+1 − 2F 2

2nF2n+2 + F 32n+1 = 0,

giving an alternative to the difference equation stated in [11, p. 856].

Example 11. The following example shows that our bound is sharp in the case d = 0 (this isthe case in Examples 6, 8, and 10). We fix a positive integer D and consider the system

(6)

{x(x− 1) · . . . · (x−D + 1) = 0,

σ(x)− x− 1 = 0.

System (6) does not have a solution in CZ, because the elements of the solution can only takevalues from 0, 1, . . . , D − 1 and strictly increase. On the other hand, the system consisting ofthe 0-th,. . . , D−1 = (B(0, D)−2)-th transforms of (6) has a solution σi(x) = i for 0 6 i 6 D.Hence, it is necessary to consider one more transform in order to express 1 (i.e. eliminate x).

Example 12. The following example is based on an example constructed by E. Amzallag andR. Gustavson. Consider the system

(7)

{f1 := xσ(x) = 0,

f2 := x− σ(x)− σ2(x) + 1 = 0.

Let X be the affine variety defined by (7) in the affine 3-space with coordinates x, σ(x), σ2(x).We have dimX = 1 and degX = 2. Therefore, B(d,D)− 1 = 5. A calculation shows that

1 ∈(f1, . . . , σ

5(f1), f2, . . . , σ5(f2)

)but 1 /∈

(f1, . . . , σ

4(f1), f2, . . . , σ4(f2)

).

Thus, this shows that our upper bound for d = 1 and D = 2 is sharp.

5. Counterexamples

5.1. Failure of the standard DEP method. Consider the system of difference equationsgiven by any set of generators of the polynomial ideal I := I1 ∩ I2 of the polynomial ringQ[x, σ(x), y, σ(y), z, w], where

I1 := (σ(y)z − 1, x, σ(x)− y), I2 := (σ(x), σ(y)− 1, (y − 1)z − 1, (x− 1)w − 1).

We do not present the actual generators of I to size of this set. A computation shows that

1 ∈(I, σ(I), σ2(I), σ3(I), σ4(I)

).

EFFECTIVE DIFFERENCE ELIMINATION AND NULLSTELLENSATZ 9

Therefore, by Proposition 1, the system has no solutions in any difference ring. One can alsoshow that

I = (I, σ(I)) ∩Q[x, σ(x), y, σ(y), z, w],(8)

σ(I) = (I, σ(I)) ∩Q[σ(x), σ2(x), σ(y), σ2(y), σ(z), σ(w)].(9)

Most of the existing effective bounds for systems of ordinary differential and difference equa-tions [4, 10, 19, 20, 28] use geometric axioms [6, 29] as sufficient conditions for the existenceof a solution based on the system and its first prolongation (differential equations) or firsttransform (differencee quations), which are summarized under the DEP method mentioned inthe introduction. In our case, it is tempting to formulate an analogue of such conditions as:

Let Γ be the affine variety defined by the system and its first transform. Ifthe projections of Γ onto the varieties defined by the system and by its firsttransform alone, respectively, are dominant, then the system is consistent.

However, this is false in the above example as we have shown, where Γ is theaffine variety corresponding to the ideal (I, σ(I)) in the affine space with coordinatesx, σ(x), σ2(x), y, σ(y), σ2(y), z, σ(z), w, σ(w), and (the Zariski closures of) the projections aregiven by the intersections in (8) and (9).

5.2. Non-existence of coefficient-independent effective strong Nullstellensatz. A(non-effective) strong Nullstellensatz for systems of difference equations can be stated as fol-lows. Let f1 = . . . = fN = 0 be a system of difference equations. If a difference polynomialf vanishes at all solutions of the system in CZ, then there exists ` such that f belongs to theradical of the ideal generated by the 0-th,. . . , `-th transforms of f1, . . . , fN .

The following example shows that there is no uniform upper bound for this ` in terms of thedegree, order, and number of variables of f1, . . . , fN . For every positive integer M , consider

(10)

{f1 = σ(x)− x− 1

M= 0,

f2 = x (y(x− 1)− 1) = 0.

Let f = y(x − 1) − 1 and x = {xn}∞−∞ and y = {yn}∞−∞ any solution of (10) in CZ. Ifyk(xk − 1)− 1 6= 0 for some k, then xk = 0. Hence, xk+M = 1, and so

xk+M (yk+M(xk+M − 1)− 1) = −1.

Therefore, f vanishes at every solution of (10) in CZ. On the other hand, we claim that fdoes not belong to the radical of the ideal generated by the 0-th,. . . , (M − 1)-th transformsof f1 and f2. These transforms belong to the polynomial ring C[x, . . . , σM(x), y, . . . , σM−1(y)].Consider the substitution

σk(x) =k

Mfor every 0 6 k 6M, σk(y) =

M

k −Mfor every 1 6 k 6M − 1, y = 0.

A direct computation shows that the polynomials f1, . . . , σM−1(f1), f2, . . . , σ

M−1(f2) vanishafter this substitution, but f does not.

6. Proofs of the main results

6.1. Difference Nullstellensatz.

Definition 8. We say that a difference ring (A, σ) is inversive if σ : A→ A is an automorphism.

10 ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON

Remark 6. For any difference ring (A, σ), there is an inversive difference ring (Ainv, σ) anda map of difference ring (A, σ) → (Ainv, σ) that is universal for maps from (A, σ) to inversivedifference rings.

Definition 9. Given a difference ring (A, σ), the ring of inversive difference polynomials overA in the variables, A{x1, . . . , xn}∗, may be realized as the ordinary polynomial ring over A inthe formal variables σj(xi), for j ∈ Z and 1 ≤ i ≤ n, with σ extending the given endomorphismon A and

σ(σj(xi)) = σj+1(xi)

on the variables.

Remark 7. If (A, σ) is inversive, then so is A{x1, . . . , xn}∗.Definition 10. Let S ⊂ k{x1, . . . , xn} be a finite set of difference polynomials and h =max{ordP | P ∈ S}. The set of n tuples a1, . . . , an ∈ k`+h, where ai := (ai,0, . . . , ai,N+h−1), iscalled a partial solution of length ` if, for every P ∈ S and 0 6 s 6 ` − 1, polynomial σs(P )vanishes after the substitution

σi(xj) = aj,i for every 1 6 j 6 n, 0 6 i 6 `+ h− 1.

Let K be an inversive difference field. Then the difference ring of sequences KZ with respectto the shift automorphism can be endowed with a structure of a difference K-algebra via theembedding of difference rings iK : K → KZ defined by

iK(f) =(. . . , σ−1(f), f, σ(f), σ2(f), . . .

)for f ∈ K.

This can be similarly done for KN.

Proposition 1. For all uncountable algebraically closed fields K and finite sets S ⊆K{x1, . . . , xn}, the following statements are equivalent:

1. S has a solution in KZ.2. S has a solution in KN.3. S has finite partial solutions of length ` for all `� 0.4. The ideal [S] := ({σj(P ) | P ∈ S, j ∈ N}) ⊆ K{x1, . . . , xn} does not contain 1.5. The ideal [S]∗ := ({σj(P ) | P ∈ S, j ∈ Z}) ⊆ K{x1, . . . , xn}∗ does not contain 1.6. S has a solution in some difference K-algebra.

Proof. The implications 1 =⇒ 2, 2 =⇒ 3, and 6 =⇒ 4 are straightforward.We show that 3 =⇒ 4. Assume that there exists an arbitrary long partial solution, but

1 ∈ [S]. Then there is an expression of the form

(11) 1 =∑i=0

∑P∈S

ai,Pσi(P ),

where ai,P ∈ K{x1, . . . , xn}. Let h = max{ordP | P ∈ S}. Consider a partial solution of S ofthe length `+ h and plug it into the equality (11). Then the right-hand side will vanish, so wearrive at contradiction.

We show that 4 =⇒ 5. Assume that 1 ∈ [S]∗. We fix some representation of 1 as an elementof [S]∗. Let N be the maximum number such that σ−N(xi) occurs in the representation.Applying σN to the both sides of the representation, we obtain a representation of 1 as anelement of [S].

We show that 5 =⇒ 6. Let π : K{x1, . . . , xn} → K{x1, . . . , xn}/[S]∗ be the canonicalsurjection. Then (π(x1), . . . , π(xn)) is a solution of S in K{x1, . . . , xn}/[S]∗.

EFFECTIVE DIFFERENCE ELIMINATION AND NULLSTELLENSATZ 11

We show 5 =⇒ 1. Let E be the inversive difference subfield of K generated by thecoefficients of elements of S over the prime subfield of K. Since 1 does not belong to [S]∗ ∩E{x1, . . . , xn}∗, there exists a maximal (not necessarily difference) ideal m ⊂ E{x1, . . . , xn}∗containing [S]∗∩E{x1, . . . , xn}∗. Then L := E{x1, . . . , xn}∗/m is a field, and the transcendencedegree of L over E is at most countable. Since K is algebraically closed and has an uncountabletranscendence degree, there exists an embedding ϕ : L → K over the common subfield E.Composing ϕ with the canonical surjection E{x1, . . . , xn}∗ → L, we obtain an E-algebrahomomorphism ψ : E{x1, . . . , xn} → K such that [S]∗ ⊂ Kerψ. For every 1 6 i 6 n, weconstruct a sequence ai := {ai,j}j∈Z ∈ KZ by the formula

ai,j = ψ(σj(xi)

).

A direct computation shows that (a1, . . . , an) is a solution of S in KZ. �

6.2. Variety and two projections. Let k be a difference field and

F = {f1 = 0, . . . , fN = 0}a system of difference equations, with f1, . . . , fN ∈ k{u1, . . . , ur}. We set

hi := maxj=1,...,N

ordui fj and H = h1 + . . .+ hr + r.

For the rest of Section 6, we fix K be an inversive algebraically closed difference field ofuncountable transcendence degree containing k. With the system F of difference equations,we associate the following geometric data:

• the variety X defined by the polynomials f1, . . . , fN in AH ;• two projections π1, π2 : AH → AH−r defined by

π1

(u1, . . . , σ

h1(u1), u2, . . . , σhr(ur)

):=(u1, . . . , σ

h1−1(u1), u2, . . . , σhr−1(ur)

),

π2

(u1, . . . , σ

h1(u1), u2, . . . , σhr(ur)

):=(σ(u1), . . . , σh1(u1), σ(u2), . . . , σhr(ur)

).

Let Z ⊂ AH be a variety defined by polynomials g1, . . . , gs ∈ K[V ]. Let σi(Z), where i ∈ Z,

denote the variety defined by the polynomials gσi

1 , . . . , gσi

s ∈ K[V ], where gσi

means the resultof applying σi to all coefficients of g. The coordinate-wise application of σi defines a bijectionbetween Z and σi(Z).

Definition 11. A sequence of points p1, . . . , p` ∈ AH(K) is said to be a partial solution of thetriple (X, π1, π2) if {

π1(pi+1) = π2(pi) for every 1 6 i < `,

pi ∈ σi−1(X)(K) for every 1 6 i 6 `.

A two-sided infinite sequence with such a property is called a solution of the triple (X, π1, π2).

Lemma 1. For every positive integer `, system F has a partial solution of length ` if and onlyif the triple (X, π1, π2) has a partial solution of length `.

System F has a solution in KZ if and only if the triple (X, π1, π2) has an infinite solution.

Proof. Let h = max16i6r

hi. Consider a partial solution u1, . . . ,ur ∈ K`+h of F , where ui =

(ui,1, . . . , ui,`+h) for every 1 6 i 6 r. We set

pj := (u1,j, . . . , u1,j+h1 , u2,j, . . . , ur,j+hr) for every 1 6 j 6 `.

By the construction

π2(pj) = (u1,j+1, . . . , u1,j+h1 , u2,j+1, . . . , ur,j+hr) = π1(pj+1),

12 ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON

so pj+1 ∈ π−11 (π2(pj)) for every 1 6 j 6 ` − 1. The definition of partial solution implies that

pj ∈ σj−1(X) for every 1 6 j 6 `. Hence, p1, . . . , p` is a partial solution of the triple (X, π1, π2).The above argument can be straightforwardly reversed to construct a partial solution of F froma partial solution of (X, π1, π2). The case of infinite solutions is completely analogous. �

In the introduced geometric language, we can formulate the following question equivalent toeffective difference Nullstellensatz

Question. Let X be an algebraic subvariety of AH and π1, π2 be surjective linear maps AH →AH−r. How long a partial solution of (X, π1, π2) is it sufficient to find in order to conclude thatthe triple (X, π1, π2) has an infinite solution?

Thus, in what follows, we fix a triple (X, π1, π2), where X is an algebraic variety of AH andπ1, π2 are surjective linear maps AH → AH−r defined over the σ-constants of K.

6.2.1. Trains. The goal of this section is to generalize the notion of a solution of the triple tonot necessarily zero-dimensional points.

Definition 12. For two irreducible subvarieties Y1, Y2 ⊂ V , we say that the generic point ofY1 maps to the generic point of Y2 (and denote it by Y1 7→ Y2) if π1(Y2) = π2(Y1).

Definition 13. For ` a positive integer or +∞, a sequence of irreducible subvarieties(Y1, . . . , Y`) in V is said to be a train of length ` in X if{

Yi 7→ Yi+1 for every 1 6 i < `,

Yi ⊂ σi−1(X) for every 1 6 i 6 `.

Remark 8. Every partial solution can be considered a train of zero-dimensional subvarieties.

Lemma 2. For every train (Y1, . . . , Y`) in X, there exists a partial solution p1, . . . , p` of(X, π1, π2) such that, for all i, 1 6 i 6 `, we have pi ∈ Yi.

Proof. We will prove the following statement by induction on `: there exists a nonempty opensubset U ⊂ Y` such that, for every point p` ∈ U , there exists a partial solution p1, . . . , p` of(X, π1, π2) such that, for every i, 1 6 i 6 `, we have pi ∈ Yi. In the case ` = 1, we can setU = Y1, because every single point in X is a partial solution of length one.

Assume that ` > 1. Applying the inductive hypothesis to the train (Y1, . . . , Y`−1), weobtain an open nonempty subset U0 ⊂ Y`−1. Since U0 is dense in Y`−1, π2(U0) is dense in

π2(Y`−1) = π1(Y`). Since π1(Y`) is a constructible dense subset in π1(Y`), π2(U0) ∩ π1(Y`) is

also a dense constructible in π1(Y`). Let U1 ⊂ π2(U0) ∩ π1(Y`) be an open dense subset of

π1(Y`). Then U2 := Y` ∩ π−11 (U1) is nonempty open in Y`. We claim that every point p` ∈ U2

can be extended to a partial solution p1, . . . , p` such that pi ∈ Yi. By the definition of U2,π1(p`) ∈ π2(U0), so there exists p`−1 ∈ U0 such that π2(p`−1) = π1(p`). By the inductivehypothesis, p`−1 can be further extended to a partial solution. �

Corollary 2. If there exists an infinite train in X, then there exists a solution for the triple(X, π1, π2).

Proof. Since there exists an infinite train, then there exists an arbitrarily long train. Due toLemma 2, there exists an arbitrarily long solution of (X, π1, π2). Lemma 1 implies that thereexists an arbitrarily long solution of the corresponding system F . Hence, due to Proposition 1,there exists a solution of F in KZ. Lemma 1 implies that there exists an infinite solution ofthe triple (X, π1, π2). �

EFFECTIVE DIFFERENCE ELIMINATION AND NULLSTELLENSATZ 13

Definition 14. For two trains Y and Y ′ of the same length, the inclusion Y ⊂ Y ′ is understoodas a component-wise containment.

For a train Y in X and i ∈ Z, σi(Y ) is the result of the component-wise application of σi toY , and, since π1 and π2 are defined over the constants, σi(Y ) is a train in σi(X).

Remark 9. Since the component-wise union of any chain of trains of the same length is againa train of this length, trains of fixed length satisfy Zorn’s lemma with respect to inclusion.Hence, maximal trains of a fixed length are well-defined.

6.2.2. The number of maximal trains.

Lemma 3. Let ϕX : X → Z and ϕY : Y → Z be dominant morphisms of affine varieties overan algebraically closed field. Assume that X and Y are irreducible. Consider the fibered productX ×Z Y of ϕX and ϕY , considered as a variety, and denote the natural morphisms to X andY by πX and πY , respectively. Then there exists an irreducible component V ⊂ X ×Z Y suchthat the restrictions of both πX and πY to V are dominant.

Proof. Denote the algebras of regular functions on X, Y , and Z by A, B, and C, respectively.Since X, Y , and Z are irreducible (Z is irreducible as an image of an irreducible variety undera dominant morphism), these algebras are domains. We denote the fields of fractions of A,B, and C by E, F , and L, respectively. The dominant maps ϕX and ϕY give rise to injectivehomomorphisms ϕ#

X : C → A and ϕ#Y : C → B. These homomorphisms equip A and B with a

C-algebra structure. Then, the algebra of regular functions on X×Z Y , as a scheme, is A⊗CB(see [1, Lemma 25.6.7]).

Let p be any prime ideal in E ⊗L F . Let D := (E ⊗L F )/p and π : E ⊗L F → D bethe canonical projection. Consider the natural homomorphism i : A ⊗C B → E ⊗L F . Since1 ∈ i(A ⊗C B), the composition π ◦ i is a nonzero homomorphism. Consider the naturalembeddings iA : A → A ⊗C B and iB : B → A ⊗C B. We will show that the compositionsπ ◦ i ◦ iA : A → D and π ◦ i ◦ iB : B → D are injective. Introducing the natural embeddingsiE : E → E ⊗L F and jA : A→ E, we can rewrite

π ◦ i ◦ iA = π ◦ iE ◦ jA.

The homomorphisms iE and jA are injective. The restriction of π to iE(E) is also injective,since E is a field. Hence, the whole composition π ◦ iE ◦ jA is injective. The argument forπ ◦ i ◦ iB is analogous.

Thus, we have an irreducible subvariety of X ×Z Y , and hence of the variety (X ×Z Y )red [1,Lemma 25.12.6], defined by the ideal Ker(π ◦ i) that projects dominantly on both X and Y .Hence, the component containing this subvariety also projects dominantly on X and Y . �

Definition 15. Let X = X1 ∪ X2 ∪ . . . ∪ Xs be the decomposition of X into irreduciblecomponents. A pair (Y, c), where Y = (Y1, . . . , Y`) is a train in X and c = (c1, . . . , c`) ∈[1, . . . , s]`, is called a marked train of length ` and signature c if Yi ⊂ σi−1(Xci) for every1 6 i 6 `.

For every train, one can assign a signature, so it becomes a marked train. However, thissignature might be not unique. Analogously to trains, we define a notion of a maximal trainof given length ` and signature c. Let c = (c1, . . . , c`) ∈ [1, . . . , s]` be a tuple. Consider

Xc := Xc1 × σ(Xc2)× . . .× σ`−1(Xc`) ⊂ (AH)`.

14 ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON

We denote the projections (AH)` → AH onto the components by ψ`,1, . . . , ψ`,`, respectively.We introduce

(12) Wc :={p ∈ Xc

∣∣ π2 (ψ`,i(p)) = π1 (ψ`,i+1(p)) for all i, 1 6 i < `}.

The restrictions of ψ`,1, . . . , ψ`,` to Wc will be denoted by the same symbols.

Lemma 4. For every irreducible subvariety Y ⊂ Wc,(ψ`,1(Y ), . . . , ψ`,`(Y )

)is a marked train of signature c.

Proof. For every i, 1 6 i 6 `,

Yi := ψ`,i(Y ) ⊂ ψ`,i(Wc) ⊂ σi−1(Xci).

Moreover, since Y is irreducible, ψ`,i(Y ) is also irreducible. Fix some i, 1 6 i < `. We will

show that π2(Yi) = π1(Yi+1). We can write π2(Yi) as π2(ψ`,i(Y )). By (12), the latter is equal

to π1(ψ`,i+1(Y )), which is the same as π1(Yi+1). �

Lemma 5. For every marked train (Y1, . . . , Y`) of signature c in X, there exists an irreducible

subvariety Y ⊂ Wc such that, for every i, 1 6 i 6 `, we have Yi = ψ`,i(Y ).

Proof. We will prove the lemma by induction on `. For ` = 1, c = (c1), Wc = Xc1 , and we canset Y = Y1.

Let ` > 1. We apply the inductive hypothesis to the train (Y1, . . . , Y`−1) of signature c′ :=(c1, . . . , c`−1) and obtain an irreducible subvariety Y ′ ⊂ Wc′ ⊂ V `−1. Then there is a naturalembedding of Y ′ × Y` into V `. Denote (Y ′ × Y`) ∩Wc by W . Since Y ′ is already contained inWc′ ,

(13) W = {p ∈ Y ′ × Y` | π2 (ψ`,`−1(p)) = π1 (ψ`,`(p))}.Denote the projection of V ` to the first H(`− 1) coordinates by ψ, and let

(14) Z := π2 (ψ`−1,`−1(Y ′)) = π1(Y`).

Then equality (13) implies (see [16, Ex. 2.26]) that W together with the morphisms ψ : W →Y ′ and ψ`,` : W → Y` is the fibered product of the morphisms π2 ◦ ψ`−1,`−1 : Y ′ → Z andπ1 : Y` → Z. Equality (14) implies that both of these morphisms are dominant.

Due to Lemma 3, there exists an irreducible subset Y ⊂ W such that ψ : Y → Y ′ andψ`,` : Y → Y` are dominant. For every i, 1 6 i < `, since ψ`,i = ψ`−1,i ◦ ψ, the restrictionψ`,i : Y → Yi is dominant as a composition of two dominant morphisms. �

Lemma 6. Let the degree of Xi be Di, and fix a tuple c = (c1, . . . , c`) ∈ [1, . . . , s]`. The numberof maximal trains of signature c in X does not exceed Dc1 ·Dc2 · . . . ·Dc`.

Proof. Since Wc is the intersection of Xc with a linear subspace,

(15) degWc 6 degXc = degXc1 · deg σ(Xc2) · . . . · deg σ`−1(Xc`).

Since application of σ to a variety does not change the degree, the product in (15) does notexceed Dc1 · . . . ·Dc` . Hence, the number of components of Wc does not exceed Dc1 · . . . ·Dc` .

Let (Y1, . . . , Y`) be a maximal train in X of signature c. Lemma 5 implies that there exists

an irreducible subvariety Y ⊂ Wc such that for every i, 1 6 i 6 `, we have Yi = ψ`,i(Y ). LetC be an irreducible component of Wc containing Y . Lemma 4 implies that(

ψ`,1(C), . . . , ψ`,`(C))

EFFECTIVE DIFFERENCE ELIMINATION AND NULLSTELLENSATZ 15

is also a train of signature c. Moreover, since C ⊃ Y , this train contains (Y1, . . . , Y`). Themaximality of the latter implies that these trains are equal. Hence, Y could be chosen to be anirreducible component of Wc. Thus, we obtain an injective map from the set of maximal trainsof signature c to the set of all irreducible component of Wc. Hence, the number of maximaltrains also does not exceed Dc1 · . . . ·Dc` . �

Corollary 3. Let D =s∑i=1

Di be the sum of the degrees of the components of X. Then the

number of maximal trains in X of length ` does not exceed D`.

Proof. Since every maximal train can be considered as a marked maximal train, the numberof maximal trains of length ` in X does not exceed the sum of products Dc1 · . . . ·Dc` over alltuples c of length `. This sum is equal to

s∑c1=1

s∑c2=1

. . .s∑

c`=1

∏i=1

Dci = (D1 + . . .+Ds)` = D`. �

6.2.3. A bound for trains.

Definition 16. For a train Y = (Y1, . . . , Y`) in X, we introduce the codimension of Y as

codimY := dimX − min16i6`

dimYi.

Definition 17. We define B′(d,D) to be the smallest natural number N such that, for everytriple (X, π1, π2) such that the sum of the degrees of the components of X does not exceed D,the existence of a train of length N and codimension at most d in X implies the existence ofan infinite train in X, or ∞ if such N does not exist.

The following statement implies that B′(d,D) is finite for all d ∈ Z>0 and D ∈ Z>0 andgives an upper bound for B′(d,D).

Proposition 2. For all D ∈ Z>0,

(1) B′(0, D) 6 D + 1 and(2) for every d ∈ Z>0, B′(d+ 1, D) 6 B′(d,D) +DB′(d,D).

Proof. Throughout the proof, we will use the observation that the existence of an infinite trainin σi(X) for some i ∈ Z implies (via component-wise application of σ−i) the existence of aninfinite train of the same codimension in X.

We will also use the following construction for obtaining infinite trains. Consider a train(Y1, . . . , Y`) of length ` > 1. If Y` = σ`−1(Y1), then we can construct an infinite train as follows:

(16)(Y1, Y2, . . . , Y`−1, σ

`−1(Y1), σ`−1(Y2), . . . , σ`−1(Y`−1), σ2`−1(Y1), . . .).

We prove the first statement of the proposition. Consider a train (Y1, . . . , YD+1) of codimen-sion zero and length D + 1. Since, for every i, 1 6 i 6 D + 1, we have dimYi = dimX, thenevery σ−i+1(Yi) is an irreducible component of X. The number of components of X does notexceed D, so some of the σ−i+1(Yi)’s coincide. If σ−i+1(Yi) = σ−j+1(Yj) for some i < j, thenYj = σj−i(Yi), so we can construct an infinite train of codimension zero, as in (16).

We prove the second statement of the proposition. Consider a train (Y1, . . . , YB) of codi-mension at most d + 1 and length B := B′(d,D) + DB′(d,D). We introduce N := DB′(d,D) + 1

16 ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON

trains Z(1), . . . , Z(N) of length ` := B′(d,D) in X, σ(X), . . . , σN−1(X), respectively, such that,for all i, 1 6 i 6 N , we have

Z(i) =(Z

(i)1 , . . . , Z

(i)`

):= (Yi, . . . , Yi+`−1).

For every i, 1 6 i 6 N , consider a maximal train Z(i) =(Z

(i)1 , . . . , Z

(i)`

)of length ` in σi−1(X)

containing Z(i). Then σ−i+1(Z(i)) is a maximal train of length ` in X. If there exists i such that

codim Z(i) 6 d, then there is an infinite train of codimension at most d due to the definition

of B′(d,D). Otherwise, codim Z(i) = d+ 1 for every 1 6 i 6 N .Corollary 3 implies that there are at most D` = N − 1 maximal trains of length ` in X.

Hence, there are a and b, 1 6 a < b 6 N , such that

σ−a+1(Z(a)) = σ−b+1(Z(b)).

Since codim Z(a) = codim Z(b) = d+ 1, there exists j, 1 6 j 6 `, such that

dim Z(a)j = dim Z

(b)j = dimX − (d+ 1).

Hence, since both Z(a)j and Z

(a)j are irreducible, dimZ

(a)j > dimX − (d + 1) and Z

(a)j ⊂ Z

(a)j ,

they are equal. Analogously, Z(b)j = Z

(b)j . Therefore,

σ−a+1(Ya+j−1) = σ−a+1(Z(a)j ) = σ−a+1(Z

(a)j ) = σ−b+1(Z

(b)j ) = σ−b+1(Z

(b)j ) = σ−b+1(Yb+j−1).

Hence,

Yb+j−1 = σb−a(Ya+j−1),

so, as in (16), there exists an infinite train of codimension at most d+ 1. �

The proof of Proposition 2 implies.

Corollary 4. If (X, π1, π2) has an arbitrarily long partial solution, that there exists an infiniteskew-cyclic train in X, i.e. a train of the form(

. . . , Y1, Y2, . . . , Y`, σ`(Y1), σ`(Y2), . . .

),

where Yi ∈ σi−1(X) for 1 6 i 6 `.

Proposition 3. B′(1, D) 6 2 +D2 + D(D−1)(D−2)6

for every D > 1.

Proof. Assume that there is no infinite train of codimension at most one in X. Let

X = X1 ∪X2 ∪ . . . ∪Xs

be the irreducible decomposition of X and Di := degXi. We construct a directed graph (withloops and multiple edges) G with vertices numbered from 1 to s as follows. For every maximaltrain among the marked trains of signature (i, j) in X, we draw an edge from i to j (thenumber of such trains is finite by Lemma 6). The codimension of an edge is defined to be thecodimension of the corresponding train.

Assume that there is a directed cycle (c1, . . . , c`) consisting of edges of codimension zero.Then there is a train

(Xc1 , σ(Xc2), . . . , σ`−1(Xc`), σ

`(Xc1)),

so, as in (16), there exists an infinite train of codimension zero in X. Hence, in what follows,we will assume that there is no such a directed cycle in G. Therefore, we can reenumerate thecomponents in such a way that i < j for every codimension zero edge (i, j).

EFFECTIVE DIFFERENCE ELIMINATION AND NULLSTELLENSATZ 17

Consider a train Y = (Y1, . . . , Y`) of codimension one in X. The train Y can be consideredas a marked train with respect to a signature c = (c1, . . . , c`). For every 1 6 i < `, we assignan edge ei in G corresponding to any maximal marked train of signature (ci, ci+1) containing(σ−i+1(Yi), σ

−i+1(Yi+1)). Hence, there exists a path

(e1, . . . , e`−1)

in G that corresponds to the whole train Y . Assume that some edge e corresponding to amaximal train (Z1, Z2) of codimension one occurs twice in this path, so e = ei = ej for some1 6 i < j < `. Without loss of generality, we may assume that

dimZ1 = dimX − 1.

Since dimYi and dimYj are both at least dimX− 1 and (Z1, Z2) is maximal, we conclude that

Z1 = σ−i+1(Yi) = σ−j+1(Yj).

Hence, as in (16), there exists an infinite train of codimension one in X. Thus, in what follows,we will assume that every edge of codimension one occurs in the path (e1, . . . , e`−1) at mostonce.

For an edge e = (i, j), we introduce the weight w(i, j) := i− j. Let

N+ := |{i | w(ei) > 0}| and N− := |{i | w(ei) < 0}|and

W+ :=`−1∑i=1

max{0, w(ei)} and W− :=`−1∑i=1

min{0, w(ei)}.

By the above reenumeration, all edges with positive weight are of codimension at least one.Therefore, N+ does not exceed the number of maximal marked trains with signatures of theform (i, j) with i > j. Hence, due to Lemma 6, we obtain

N+ 6∑i>j

DiDj.

Since the sum of weights along any path between vertices a and b is equal to a− b,W+ +W− > −s+ 1.

Combining this inequality with the fact that N− 6 −W−, we obtain

N− 6 W+ + s− 1.

Due to Lemma 6,

W+ 6∑i>j

(i− j)DiDj.

Thus,

(17) `−1 = N+ +N− 6∑i>j

DiDj +∑i>j

(i− j)DiDj +s−1 6 D2 +∑i>j

(i− j−1)DiDj +s−1.

For all integers q, z1, . . . , zq > 1, let

fq(z1, . . . , zq) :=∑

16j<i6q

(i− j − 1)zizj + q − 1.

We claim that, if the sum

z1 + . . .+ zq = D := D1 + . . .+Ds 6 D

18 ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON

is fixed, then the maximal value of f is achieved at

q = D ⇐⇒ z1 = z2 = . . . = zq = 1.

To prove the claim, consider any other integer p > 1 and a tuple of positive integers(w1, . . . , wp). Let r 6 p be the largest integer such that wr 6= 1. We have

fp+1(w1, . . . , wr − 1, . . . , wp, 1)− fp(w1, . . . , wp) =

= 1 +∑i 6=r

wi(p− i− (|r − i| − 1)) + (wr − 1)(p− r) =

= 1 +∑i<r

wi(p− i− r + i+ 1) +∑i>r

(p− i+ r − i+ 1) + (wr − 1)(p− r) =

= 1 +∑i<r

wi(p− r + 1) + (p+ r + 1)(p− r)−∑i>r

2i+ (wr − 1)(p− r) =

= 1 +∑i<r

wi(p− r + 1) + (wr − 1)(p− r) > 0.

Hence, the claim is proved. Combining the claim with (17), we obtain

` 6 1 + D2 +∑

16j<i6D

(i− j − 1) = 1 + D2 +D−2∑i=1

i(D − 1− i) = 1 + D2 + D(D−1)(D−2)6

. �

Propositions 2 and 3 imply

Corollary 5. For all d ∈ Z>0 and D ∈ Z>0, B′(d,D) 6 B(d,D).

6.3. Proof of effective Nullstellensatz.

Proof of Theorem 1. We will use the notation introduced in Section 6.2. The fact that thesystem consisting of 0-th, . . ., B(d,D)−1-th transforms of F considered as a polynomial systemis consistent implies that F has a partial solution of length B(d,D) > B′(d,D). Lemma 1implies that there is a partial solution of the triple (X, π1, π2) of length B′(d,D). This partialsolution is a train in X of codimension dimX = d and length B′(d,D). Definition of B′(d,D)implies that there exists an infinite train in X. Then Lemma 1 and Corollary 2 imply thatsystem F has a solution in some difference ring extending k. �

Proof of Corollary 1. We will use the notation introduced in Section 6.2. If k = C, then Kcan be chosen to be C, too. The statement of the corollary implies that system F has a partialsolution of length B(d,D). Analogously to the proof of Theorem 1, we have that system F isconsistent. Then Proposition 1 implies that F has a solution in CZ. �

6.4. Proof of effective elimination.

Proof of Theorem 2. Let E0 ⊃ E be any difference field extension such that E0 is algebraicallyclosed and has an uncountable transcendence degree over the prime subfield. Since the differ-ence ideal generated by Σ in k{x,y} contains a nonzero polynomial depending only on x andtheir transforms, the difference ideal generated by Σ in E0{y} contains 1. So, the system doesnot have a solution in EZ

0 . Theorem 1 implies that system Σ does not have a partial solution

EFFECTIVE DIFFERENCE ELIMINATION AND NULLSTELLENSATZ 19

in E0 of length B(d,D). Hence, the ideal generated by B(d,D) transforms of Σ contains 1.Since the ideal is defined over E, there is an expression of 1 over E of the form

1 =

B(d,D)−1∑i=0

(N∑j=1

ci,jσi(gj) +

m∑j=1

c′i,jσi(σ(yj)− ykj

)),

where ci,j, c′i,j ∈ E. Multiplying both sides of the above equality by the product of the denom-

inators of ci,j’s and c′i,j’s, we obtain an expression of a nonzero polynomial from k{x} as ank{x,y}-linear combination of B(d,D) transforms of Σ. �

7. Difference Nullstellensatz over small fields

With our Proposition 1 we had required that the field from which we construct the sequencering in which solutions to differences are sought be uncountable. In practice, this is a harmlessassumption as one might take that field to be C. However, this result may be conceptually un-satisfying and one might wish to find solutions to difference equations in sequences taken froma small field, such as the field of algebraic numbers. With the next proposition we show how toweaken the uncountability hypothesis by appealing to a more refined equivalent condition tothe consistency of a system of difference equations coming from our work towards the effectiveNullstellensatz and a theorem of Hrushovski on the limit theory of the Frobenius automor-phisms [18]. Our invocation of Hrushovski’s theorem is essentially contained in Fakhruddin’sproof of the density of periodic points for polarized algebraic dynamical systems in [13]. Forour purposes a slightly weaker result due to Varshavsky [35] suffices.

Proposition 4. For all algebraically closed fields K ( without any restriction on the cardinal-ity) and finite sets S ⊆ K{x1, . . . , xn}, the following statements are equivalent:

1. S has a solution in KZ.2. S has a solution in KN.3. S has finite partial solutions in KN for all N � 0.4. The ideal [S] := ({σj(P ) | P ∈ S, j ∈ N}) ⊆ K{x1, . . . , xn} does not contain 1.5. The ideal [S]∗ := ({σj(P ) | P ∈ S, j ∈ Z}) ⊆ K{x1, . . . , xn}∗ does not contain 1.6. S has a solution in some difference K-algebra.

In order to prove Proposition 4, we will extract two consequences of [35]. In Lemma 7 and 8,φs denotes the s-th power of the Frobenius automorphism.

Lemma 7. Let R be a finitely generated difference subring of a difference field K. Then thereexist a prime p, a positive integer s, and a difference homomorphism ψ : R → F, where F isthe algebraic closure of Fp considered as a difference ring with respect to φs.

Proof. Let R be generated by a1, . . . , a`. Since R is a subring of a field, the ideal

(18) I := {f ∈ Z{X1, . . . , X`} | f(a1, . . . , a`) = 0}is a prime difference ideal. As such, because every finitely generated difference ring is a Rittdifference ring [7, Chapter 3, Theorems II, and V], I is finitely generated as a perfect differenceideal. Let S be a finite set of such generators for I and h the largest order of the elements ofS. Let b1, . . . , bN be the elements of

{σj(ai) | 1 ≤ i ≤ `, 0 ≤ j < h}written in some order, so N = `h. Then R is also generated by b1, . . . , bN , and the correspond-ing vanishing ideal in the difference polynomial ring Z{Y1, . . . , YN} is generated as a perfect

20 ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON

difference ideal by difference polynomials of order one. Replacing a1, . . . , a` by b1, . . . , bN , wemay assume that I is already generated as a perfect difference ideal by order one differencepolynomials.

Let p be the ideal of all polynomials in x1, . . . , x` vanishing on (a1, . . . , a`) and let q ⊆Z[x1, . . . , x`, y1, . . . , y`] be the ideal of all polynomials vanishing on (a1, . . . , a`, σ(a1), . . . , σ(a`)).If K has positive characteristic, let p be its characteristic. Otherwise, let p be any sufficientlylarge prime so that the ideal (p) + q is proper. Let Q be a minimal prime extending (p) + qand P := Q ∩ Z[x1, . . . , x`]. Then

X := Spec(Z[x1, . . . , x`]/P) and C := Spec(Z[x1, . . . , x`, y1, . . . , y`]/Q)

are irreducible schemes of finite type over Fp. Then, by [35, Theorem 0.1], there exists apositive integer s such that the intersection of C with the graph of φs is nonempty. Let(a∗1, . . . , a

∗` , φs(a

∗1), . . . , φs(a

∗`)) be a point in the intersection. Since the substitution σj(xi) =

φjs(a∗i ) annihilates q, it also annihilates every polynomial in its perfect closure I (see (18)). Let

F be the algebraic closure of Fp. Then the map ψ : R → F defined by ψ (σj(ai)) = φjs(a∗i ) is a

desired homomorphism of difference rings (R, σ) and (F, φs). �

Lemma 8. For every

• prime number p,• positive integer s,• scheme X of finite type defined over F, the algebraic closure of Fp,• irreducible subvariety Γ ⊂ X × φs(X) such that the projections to X and φs(X) are

dominant,

there exist positive integers ` and t and an infinite sequence (ai)∞i=−∞ such that (ai, ai+1) ∈

φsi(Γ) for every i ∈ Z.

Proof. Since X and Γ are defined over some finite subfield of F, there is a positive integer `with φs`(X) = X and φs`(Γ) = Γ. Lemma 3 implies that there exists an irreducible componentΞ of the fiber product

Γ×φs(X) φs(Γ)×φ2s(X) · · · ×φ(`−1)s(X) φ(`−1)s(Γ).

such that the projections of Ξ onto Γ, φs(Γ), . . . , φ(`−1)s(Γ) are dominant. We denote theprojection Ξ→ φsi(Γ) by ρi for every 0 6 i 6 `− 1.

Let τ1 : Γ → X and τ2 : Γ → φs(X) denote the projections. We define projections πi : Ξ →φsi(X) for 0 6 i 6 ` as follows:

πi =

τ1 ◦ ρ0, for i = 0,

τ1 ◦ ρi = τ2 ◦ ρi−1, for 0 < i < `,

τ2 ◦ ρ`−1, for i = `.

Consider the fiber product of Ξ ×X Ξ where the first Ξ → X is π` and the second mapΞ → X is π0. Lemma 3 implies that there exists an irreducible component Υ of this productsuch that the projections of Υ onto both Ξ’s are dominant. Take r so that Ξ and Υ are bothdefined over Fpr and

s` | r.By [35, Theorem 0.1], there is a power φt of φr and a point a ∈ Ξ(F) with (a, φt(a)) ∈ Υ(F).Note that φt leaves invariant Γ, Ξ, and Υ. Since coefficients of πi’s and ρj’s are invariant under

EFFECTIVE DIFFERENCE ELIMINATION AND NULLSTELLENSATZ 21

φ, we will denote the conjugation of any of these maps by any power of φ by the same letter.For 0 ≤ i < ` and j ∈ Z, define

ai+j` := πi(φtj(a)).

Let us show that the sequence {ai}∞i=−∞ satisfies the requirement of the lemma. Considerj ∈ Z and 0 6 i < `. Then ai+j` = τ1 (ρi(φtj(a))). We also have

ai+j`+1 = τ1 (ρi+1(φtj(a))) = τ2 (ρi(φtj(a))) for i < `− 1,

ai+j`+1 = τ1

(ρ0(φt(j+1)(a))

)= τ2 (ρ`−1(φtj(a))) for i = `− 1

because Ξ and Υ are components of the corresponding fiber products. In both cases,

(ai+j`, ai+j`+1) ∈ ρi (φtj(Ξ)) = ρi(Ξ) ⊂ φsi(Γ) = φs(i+j`)(Γ). �

In Lemmas 9 and 10, for a valued field (K, v), we write

O = {x ∈ K : v(x) ≥ 0}

for the valuation ring,

m = {x ∈ K : v(x) > 0}for the maximal ideal of O, and k = O/m for the residue field. We denote the reduction mapr : O → k by r and will abuse notation writing r for the reduction map on associated objects.

Lemma 9. Let (K, v) be a Henselian field, n 6 m positive integers, f1, . . . , fn ∈ O[x1, . . . , xm]and a = (a1, . . . , am) ∈ km. We assume that, for each i, we have r(fi)(a) = 0 and that the

matrix(∂r(fi)∂xj

(a))

16i6n, 16j6mhas rank n. Then there is c = (c1, . . . , cm) ∈ Om such that

f1(c) = . . . = fn(c) = 0 and r(c) = a.

Proof. By hypothesis, there is some J ⊆ {x1, . . . , xm} with |J | = n and invertible matrix(∂r(fi)∂xj

(a))

16i6n, j∈J. Relabeling the variables, we may assume that J = {1, . . . , n}. Define

fi := xi for n < i ≤ m. Then the square matrix(∂r(fi)∂xj

(a))

16i6n, 16j6nis invertible. There

exists some b ∈ Om such that r(b) = a. Then, by to [24, Section 4, Multidimensional Hensel’sLemma], there is some c ∈ On with f1(c) = · · · = fn(c) = 0 and r(c) = r(a). �

Lemma 10. Let (K, v) be a Henselian field and f : X → Y a smooth map of schemes of finitetype over O. Suppose that a ∈ X(k) and b ∈ Y (O) satisfy f(a) = r(b). Then there is a pointc ∈ X(O) with f(c) = b and r(c) = a.

Proof. [1, Lemma 28.32.11] implies that there are affine open neighborhoods U ⊆ X and V ⊆ Yof a and r(b), respectively, for which fU : U → V is standard smooth. That is, there exist:

• positive integers m and n,• a finitely generated O-algebra S,• polynomials g1, . . . , gn ∈ S[x1, . . . , xm], and• isomorphisms of schemes ϕ1 : U → Spec(T ) and ϕ2 : V → Spec(S), where T =S[x1, . . . , xm]/(g1, . . . , gn),

such that

• some n× n minor of the Jacobian ( ∂gi∂xj

) is an invertible element of T and

• fU = ϕ−12 ◦ g ◦ ϕ1, where g : Spec(T ) → Spec(S) is the dual morphism of schemes to

the O-algebra homomorphism S → T .

22 ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON

Since O is a local ring and r(b) is a reduction of b modulo m ⊂ O, the point b belongs toany open neighborhood of r(b), in particular, b ∈ V (O). By pre-composing with the dual toϕ2, this corresponds to an O-algebra homomorphism b] : S → O. Let a] denote the O-algebrahomomorphism obtained by pre-composing the dual to ϕ1 with the O-algebra homomorphismO[U ]→ k corresponding to a ∈ U(k).

For each i, 1 6 i 6 n, consider the polynomials gb]

i ∈ O[x1, . . . , xm] and ga]

i ∈ k[x1, . . . , xm].

The fact r(b) = f(a) implies that r(gb

]

i

)= ga

]

i . Let aj := a](xj) for 1 6 j 6 m. Since a] is a

homomorphism, ga]

i (a1, . . . , am) = 0 for every 1 6 i 6 n and also the Jacobian matrix

(∂ga

]

i

∂xj

)has full rank at (a1, . . . , am).

Then, by Lemma 9, we may find (c1, . . . , cm) ∈ Om for which gb]

i (c1, . . . , cm) = 0 for 1 6 i 6 n

and r(cj) = aj for 1 6 j 6 m. Since gb]

i (c1, . . . , cm) = 0 for 1 6 i 6 n, the map c] : T → Odefined by c]|S = b] and by c](xi) = ci for 1 6 i 6 m is a well-definedO-algebra homomorphism.This gives us a point c ∈ U(O) such that f(c) = b. Moreover, r(c) = a since r(ci) = ai forevery 1 6 i 6 m. �

Corollary 6. Let (K, v) be a Henselian field and X a scheme of finite type over O such thatthe canonical morphism X → Spec(O) is smooth. Then, for every a ∈ X(k), there existsc ∈ X(O) such that r(c) = a.

Proof. The corollary follows from Lemma 10 applied to Y = Spec(O) and b being the identitymap Spec(O)→ Spec(O). �

With these statements in place, we finish the proof of Proposition 4.In what follows, for a positive integer m and a commutative ring R, Rm denotes the com-

mutative ring generated by the set {rm | r ∈ R}. For an affine scheme X over a perfect ring Rof characteristic p and q = pn, we define a scheme X(q) by X(q) := Spec(OqX). There is a mapFn : X → X(q) that is dual to the inclusion OqX ↪→ OX . This map is a special case of what iscalled the relative Frobenius morphism. See [1, Tag 0CC6] for more details. In coordinates,when X ⊂ Am, Fn can be written as (a1, . . . , am) 7→ (aq1, . . . , a

qm). Since R is perfect, Fn defines

a bijection between X(R) and X(q)(R). If p = 0, we will assume that q = 1 and Fn is theidentity map.

Lemma 11. If µ : Γ → Z is morphism of irreducible affine varieties over an algebraicallyclosed field K, then there exist

• an affine variety Υ,• morphisms ν : Γ→ Υ and τ : Υ→ Z,• a positive integer n and a morphism γ : Υ→ Γ(q), where q = pn,

such that µ = τ ◦ ν, γ ◦ ν = Fn, ν is finite, and τ is generically smooth.

Proof. If charK = 0, take Υ = Γ, ν = idΓ and τ = µ by [31, Theorem 2.27].Let charK = p > 0, t1, . . . , t` be a transcendence basis of K(Γ) over E := Quot(µ∗(OZ)),

and L be the relative separable closure of E(t1, . . . , t`) in K(Γ). Then, as K(Γ) is a finite purelyinseparable extension of L, for n� 0 we have K(Γ)p

n ⊆ L. Let q := pn and A = µ∗(OZ)[OqΓ],the ring generated by µ∗(OZ) and OqΓ. Set Υ := Spec(A) over K.

Dual to the homomorphisms of rings OZ → A and A→ OΓ, we have morphisms τ : Υ→ Zand ν : Γ → Υ with µ = τ ◦ ν. Since the field extensions E ↪→ K(Υ) is a subextension of thethe separable extension E ↪→ L, the morphism τ : Υ → Z is smooth at the generic point of

EFFECTIVE DIFFERENCE ELIMINATION AND NULLSTELLENSATZ 23

Υ due to [1, Lemma 10.138.9]. Form the inclusion OqΓ ↪→ A = OΥ, we obtain the morphismγ : Υ→ Γ(q) with Fn = γ ◦ ν.

Since OqΓ ⊂ OΓ is a finite integral extension and OqΓ ⊂ A ⊂ OΓ, the extension A ⊂ OΓ isalso a finite integral extension. Hence, the dual map ν : Γ→ Υ is a finite morphism. �

With these lemmas in place, we finish the proof of Proposition 4.

Proof of Proposition 4. The only implication whose proof in the original argument for Propo-sition 1 used uncountability is from 5. to 1. We observe that 5. implies 3., because 1 isnot contained in any ideal generated by finitely many transforms of the system, so Hilbert’sNullstellensatz implies that there exist arbitrarily long partial solutions of the system over K.This is exactly 3.

Consider a triple (X, π1, π2) constructed in Section 6.2. Due to Lemma 1, 3. implies that(X, π1, π2) has an arbitrarily long partial solution. On the other hand, the existence of asolution to S in KZ is equivalent to the existence of a two-sided infinite solution to (X, π1, π2)over K (see Lemma 1). We thus reduce to finding a solution to (X, π1, π2) over K. ThenCorollary 4 implies that there exists an infinite skew-cyclic train

(19)(. . . , Y1, Y2, . . . , Y`, σ

`(Y1), . . .)

in X. Let Z be an irreducible subvariety of the fiber product as in Lemma 5 applied to thetrain (Y1, . . . , Y`). For 1 ≤ i ≤ `, let ρi : Z → Yi be the dominant projection to Yi (which isψ`,i|Z in the notation of Lemma 5). The projection σj(Z)→ σj(Yi) obtained by conjugation byσj of ρi will be denoted by σj(ρi) for every j ∈ Z. Recall that, since (19) is a train, π2 ◦ ρ` andπ1 ◦ σ`(ρ1) are dominant onto the same variety. Due to Lemma 3, there exists an irreduciblecomponent Γ, which we fix, of the fiber product of Z with σ`(Z) over π2 ◦ ρ` and π1 ◦ σ`(ρ1)such that µ1 : Γ→ Z and µ2 : Γ→ σ`(Z) are dominant.

Let us call a sequence (ai)∞i=−∞ with ai ∈ σi`(Z) and (ai, ai+1) ∈ σi`(Γ) for all i a weak

solution to (Z,Γ). Such a weak solution gives rise to the solution

(. . . , σ−`(ρ1)(a−1), . . . , σ−`(ρ`)(a−1), ρ1(a0), ρ2(a0), . . . , ρ`(a0), σ`(ρ1)(a1), . . . , σ`(ρ`)(a1), . . .)

of (X, π1, π2). Thus, it suffices for us to find a weak solution.

• Υi and Υ2 be affine varieties• n a positive integer,• νi : Γ→ Υi, τi : Υi → Z, γi : Υi → Γ(q), morphisms, where q = pn and i = 1, 2,

so that, for i = 1, 2,

• γi ◦ νi = Fn,• τi is generically smooth,• µi = τi ◦ νi,

which is possible to do by Lemma 11.We fix some equations defining Γ, Z, Υi, τi, νi, and µi for i = 1, 2. Denote the difference ring

generated by the coefficients of these equations by R. Let Γ, Z, Υ1, and Υ2 be the models of Γ,Z, Υ1, and Υ2 defined by these fixed equations over R. Thus, we have the following diagram:

24 ALEXEY OVCHINNIKOV, GLEB POGUDIN, AND THOMAS SCANLON

Γ

Fn

��

ν1zzν2 $$

µ1

��

µ2

��

Υ1

��

τ1��

γ1

##

Υ2

��

τ2 ""

γ2

{{Z

((

Γ(q)

π

��

σ`(Z)

uuSpec(R)

Let Υ1, Υ2, and Γ(q) be dense open subsets in Υ1, Υ2, and Γ(q), respectively, such that τ1, τ2,and π, respectively, are smooth on these subsets, which exist since smoothness of a morphismis an open condition (see the discussion just after [1, Definition 28.32.1]). Let

Γ = ν−11 (Υ1) ∩ ν−1

2 (Υ1) ∩ F−1n (Γ(q)),

which is dense open in Γ. Let Γ′ be a non-empty open subset of Γ is defined by a singleinequality f 6= 0, where f ∈ OΓ. The image of Γ′ under Fn is open dense in (Γ′)(q) ⊂ Γ(q),

defined by f q 6= 0. Let Υ′i = γ−1i

((Γ′)(q)

)∩ Υi, i = 1, 2. Then νi (Γ′) ⊂ Υ′i, and (Γ′)(q) ⊂ Γ(q).

We apply Lemma 7 to (R, σ`) and obtain ψ : (R, σ`) → (F, φs), where F is the algebraicclosure of Fp and, in the case charK = 0, p is some prime number provided by Lemma 7. LetXF denote the base change of a scheme X over R to F via ψ. Let (ai)

∞i=−∞ be a sequence such

that, for each i ∈ Z,(ai, ai+1) ∈ φsi(Γ′F)(F).

Such a sequence exists by Lemma 8. Fix an extension of ψ to a place ϑ on K (see [12,Theorem 3.1.1]). Let O be the valuation ring of ϑ and v be a valuation on K. Note thatR ⊂ O. Also note that we do not assert that ϑ respects σ on all of O nor even that O ispreserved by σ. Let E be the residue field of O. Since K is algebraically closed, E is alsoalgebraically closed [12, Theorem 3.2.11]. Since Fp ⊂ E, F is embedded into E.

[1, Lemma 28.32.5] implies that the morphisms of schemes (Υ′1)O → ZO, (Υ′2)O → σ`(Z)O,

and (Γ′)(q)O → Spec(O) are smooth as well as all their shifts/conjugations by σ`. We shall now

build a weak solution (bi)∞i=−∞ to (ZO(O), Γ′O(O)) so that

ϑ(bi) = ai for each i ∈ Z.Since K is algebraically closed, (K, v) is Henselian [24, Lemma 4.1]. For i = 0 and i = 1,

since (Γ′)(q)O → Spec(O) is smooth, every point in (Γ′)

(q)F (F) lifts to a point in (Γ′)

(q)O (O) due

to Corollary 6. Thus, we may choose some (b0, b1) ∈ (Γ′)(q)O (O) specializing to (Fn(a0), Fn(a1))

and set b0 = F−1n (b0) and b1 = F−1

n (b1).Assume that we have already constructed bi for some i > 0 so that ϑ(bi) = ai. Due to

Lemma 10 applied to the morphism of schemes σi`◦τ1◦σ−i` : σi` ((Υ′1)O)→ σi` (ZO) and points(σi` ◦ ν1 ◦σ−i`) ((ai, ai+1)) and bi, there exists P ∈ σi` ((Υ′1)O) such that (σi` ◦ τ1 ◦σ−i`)(P ) = biand ϑ(P ) = (φis ◦ ν1 ◦ φ−is ) ((ai, ai+1)). Consider

Q = F−1n

(σi` ◦ γ1 ◦ σ−i`(P )

)∈ σi` (Γ′O(O)) .

Since ν1 is a finite morphism, it is surjective on O-points due to [31, Theorem 1.12] togetherwith [12, Theorem 3.1.3]. Using this and the fact that Fn is bijective on O-points, σi` ◦ ν1 ◦

EFFECTIVE DIFFERENCE ELIMINATION AND NULLSTELLENSATZ 25

σ−i`(Q) = P . Hence, (σi` ◦ µ1 ◦ σ−i`)(Q) = (σi` ◦ τ1 ◦ σ−i`)(P ) = bi, so Q can be written as(bi, c). Since

F−1n ◦ φis ◦ γ1 ◦ ν1 ◦ φ−is = φis ◦ F−1

n ◦ γ1 ◦ ν1 ◦ φ−is = φis ◦ id ◦φ−is = id,

we haveϑ(Q) = F−1

n ◦ φis ◦ γ1 ◦ ν1 ◦ φ−is ((ai, ai+1)) = (ai, ai+1).

Thus, we can set bi+1 = c. In the same way, we produce the bi with i < 0 using the fact that(Υ′2)O → σ`(Z)O is smooth. �

Acknowledgments

This work has been partially supported by the NSF grants CCF-0952591, CCF-1563942,DMS-1413859, DMS-1363372, by the NSA grant #H98230-15-1-0245, by PSC-CUNY grant#60098-00 48, by Queens College Research Enhancement, and by the Austrian Science FundFWF grant Y464-N18. The authors are grateful to the CCiS at CUNY Queens College for thecomputational resources.

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EFFECTIVE DIFFERENCE ELIMINATION AND NULLSTELLENSATZ 27

CUNY Queens College, Department of Mathematics, 65-30 Kissena Blvd, Queens, NY 11367and CUNY Graduate Center, Ph.D. programs in Mathematics and Computer Science, 365 FifthAvenue, New York, NY 10016

E-mail address: aovchinnikov@qc.cuny.edu

New York University, Courant Institute of Mathematical Sciences, New York, NY 10012E-mail address: pogudin@cims.nyu.edu

University of California at Berkeley, Department of Mathematics, Berkeley, CA 94720E-mail address: scanlon@math.berkeley.edu