Effective difference elimination and Nullstellensatz
Abstract.
We prove effective Nullstellensatz and elimination theorems for difference equations in sequence rings. More precisely, we compute an explicit function of geometric quantities associated to a system of difference equations (and these geometric quantities may themselves be bounded by a function of the number of variables, the order of the equations, and the degrees of the equations) so that for any system of difference equations in variables and , if these equations have any nontrivial consequences in the variables, then such a consequence may be seen algebraically considering transforms up to the order of our bound. Specializing to the case of , we obtain an effective method to test whether a given system of difference equations is consistent.
1. Introduction
Let be an algebraically closed field of arbitrary characteristic. We say that a sequence from satisfies a difference equation with constant coefficients if there is a nonzero polynomial such that, for every natural number , the equation holds. This can also be defined for systems of difference equations in several variables. Such difference equations and the sequences that solve them are ubiquitous throughout mathematics and in its applications to the sciences, including such areas as combinatorics, number theory, and epidemiology, amongst many others (see Section 4 for some of the examples).
In this paper we resolve some fundamental problems about difference equations. The questions we answer include the following (for precise statements, including the way nonconstant coefficients can appear, see Section 3):

Under what conditions does a system of difference equations have a sequence solution?

Can these conditions be made sufficiently transparent to allow for efficient computation?

Given a system of difference equations on tuples of sequences, how does one eliminate some of the variables so as to deduce the consequences of these equations on the first variables?
Our solution to the first question is a conceptual difference Nullstellensatz, to the second, an effective difference Nullstellensatz, and to the third, an effective difference elimination algorithm. Even though the abstract Nullstellensatz is intellectually satisfying in that conditions of different kinds are shown to be equivalent, namely the existential condition that there is a sequence solution to a system of difference equations and the universal condition that the difference ideal generated by the equations is proper, the difficult work and applications, both theoretical and practical, comes with our main effective theorems.
Effective elimination theorems and methods have a long history and play central roles in computational algebra. Row reduction, or Gaussian elimination, is a fundamental technique in linear algebra. Elimination for polynomial equations is substantially more complicated and has been the subject of intensive and sophisticated work [Brownawell, Kollar, Jelonek]. In recent work of the first two authors joined by Vo [OPV2017], effective elimination theorems were obtained for algebraic differential equations through a reduction to the polynomial case through the decompositioneliminationprolongation method.
While these questions are important and difference equations have been studied intensively both for their applications and theory, to our knowledge, none of these questions has received a satisfactory answer in the literature. We explain below how some known results, both positive and negative, may help explain the existence of this lacuna. In particular, in some essential ways, the effective Nullstellensatz and elimination problems for difference equations are substantially more difficult than the corresponding problems for differential equations and the methods of [OPV2017] do not routinely transpose to this context.
The foundational work on difference algebra, that is, the study of the theory of difference rings and of difference equations as encoded through the algebraic properties of rings of difference polynomials, was initiated by Cohn in [Cohn], following the tradition of Ritt and Kolchin in differential algebra. Deep results have been obtained in this subject, but their relevance to the problems at hand is hampered by their restrictions, for the Nullstellensatz and elimination theorems, to the case in which solutions are sought in difference fields, and thus have little bearing on the structures used in practice, namely difference rings presented as rings of sequences, such as given with the shift operator . 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 [ChHr99]. There, they established a recursive axiomatization for the theory of existentially closed difference fields and proved a quantifier simplification theorem. From this it follows that in principle there are effective procedures to check the consistency of difference equations in difference fields and to perform difference elimination in difference fields. More recent work of Tomašić [Tomasic, Tomasic2018] geometrizes the quantifier simplification theorem and brings the complexity 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 practical implementation of this work is infeasible. In symbolic computation, steps have been taken towards extending the characteristic set method from differential algebra to the study of difference and differencedifferential equations in works of Gao, van der Hoeven, Li, Yuan, Zhang [GLY2009, Gao2009, LYG2015, LL15]. These methods are more efficient than those coming from logic, but as they are restricted to the study of inversive prime difference ideals, they, too, are fundamentally results about solutions to difference equations in difference fields and the constructions of difference resultants depend on restrictive hypotheses.
The situation for difference equations in sequence rings differs starkly. Simple examples show that consistency checking in difference fields is not the same problem as consistency checking for sequences. For example, the system of difference equations has no solution in a difference field, but the sequence is a solution in .
More seriously, theorems of Hrushovski and Point [HrPo] show that the logical methods used for difference fields fail dramatically for sequence rings. In particular, they show that the firstorder theory of regarded as a difference ring is undecidable. Thus, we cannot derive a consistency checking method from a recursive axiomatization of this theory nor can we produce an elimination algorithm from an effective quantifier elimination theorem; no such axiomatization or quantifier elimination procedure exists. That we succeed in solving the effective consistency checking and effective elimination problems for difference equations in sequence 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 our arguments. We have two main theorems: Theorem 1 an effective Nullstellensatz and Theorem 2 an effective difference elimination theorem. Strictly speaking, the effective Nullstellensatz is a special case of an effective elimination theorem, but we prove elimination by bootstrapping through the Nullstellensatz.
The key to our work is a new proof technique based on the spirit of the decompositioneliminationprolongation (DEP) method. As is completely standard, a system of difference equations may be regarded as a system of algebraic equations in more variables together with specifications that certain coordinates should be obtained from others by the application of the distinguished endomorphism and the usual DEP methods allow for one to cleverly reduce questions about the original system of difference equations to questions entirely about algebraic equations. A version of the DEP method for difference equations in difference fields is employed in [Hr01] for the purpose of computing explicit bounds in Diophantine geometric problems. This DEP method cannot work for the problems at hand as explained in Section 5. We overcome this obstacle by taking a different approach to reducing the question about the original system to the question about algebraic equations. The core of this reduction is for us to show that every system of difference equations that has a solution actually has what we call a skewperiodic solution with the components being (not necessarily closed!) points of the affine variety corresponding to the original system, and the length of the period can be bounded in terms of the geometric data of the original system (see Section 6.2.3).
With our theorems we explicitly bound the number of prolongations required to solve the problems at hand, i.e. testing a system of difference equations for consistency or computing a nontrivial element of the elimination ideal. For the elimination problem, our bound is not sensitive to the number of variables that are not being eliminated, see Remark 3. The bounds are small enough in many cases to permit efficient computation, see Section 4.
We draw an interesting theoretical conclusion from our work towards the explicit bounds for the difference elimination problem in Section 7. Specifically, with Theorem 3 we show that for any algebraically closed difference field, whenever a finite system of difference equations over is consistent in the sense that it has a solution in some difference ring, then it already has a solution in the ring of sequences of elements of . We give a soft proof of such a difference Nullstellensatz under the hypothesis that is uncountable with Proposition 1. The proof of Theorem 3 is much more difficult than it may have been expected it to be. In extending this difference Nullstellensatz to general we use crucially our result that a system of difference equations is consistent if and only if it has a skewperiodic solution and then appeal to remarkable theorems of Hrushovski on the firstorder theory of the Frobenius automorphism and of Varshavsky on intersections of correspondences with the graph of the Frobenius.
The paper is organized as follows. We give the basic definitions in Section 2, and then introduce the notation and terminology specific to our paper. The main results, Theorem 1 for the effective Nullstellensatz and Theorem 2 for the effective elimination, are expressed in Section 3. In Section 4, we illustrate our results in several practical examples. With Section 5, we present counterexamples to an effective strong difference Nullstellensatz and to the application of the usual DEP method to these problems. The proofs of the main theorems are presented in Section 6. Finally, in Section 7, we strengthen the difference Nullstellensatz giving equivalent criteria for the existence of sequence solutions to systems difference equations over any algebraically closed field.
2. Preliminaries
A detailed introduction to difference rings can be found in [Cohn, LevinBook].
Definition 1.
A difference ring is a pair where is a commutative ring and is a ring endomorphism.
Example 1.
If is any commutative ring, then the sequence rings and are difference rings with defined by (, respectively).
Definition 2.
A map of difference rings is given by a map of rings such that that .
Remark 1.
We often abuse notation saying that is a difference ring when we mean the pair .
Definition 3.
If is a difference ring, then the free difference algebra in one generator over , , also called the ring of difference polynomials in over , may be realized as the ordinary polynomial ring in the indeterminates . Iterating this procedure, one obtains the difference polynomial ring in variables.
Definition 4.
For and , we define the order of with respect to , denoted to be the maximal for which appears in . If no appears, we set .
Example 2.
)=1.
Definition 5.
If is a difference ring and is a set of difference polynomials over , is an extension of difference rings, and is an tuple from , then we say that is a solution of the equations if, under the unique map of difference rings given by extending the given map and sending for , every element of is sent to .
Example 3.
Let and , where is the shift (to the left) operator. Then the tuple is a solution of the equations
Definition 6.
If is a difference ring and and is a nonnegative integer, the th transform of is the set . So, the th transform of is . The th transform of a system of difference equations is defined similarly.
Example 4.
The 2nd transform of the system
is the system
Definition 7.
A difference equation is said to be a consequence of a system of difference equations if there exists a nonnegative integer such that belongs to the polynomial ideal generated by the th,th transforms of , that is
Example 5.
Let be the system
The equation is a consequence of with , because
3. Main results
For all and we define
3.1. Effective difference Nullstellensatz
Theorem 1.
Let be a difference field and a system of difference equations, with . We set
Let and denote the dimension and the sum of the degrees of the components of the affine variety defined by over in the affine space, respectively. Then system has a solution in a difference ring containing if and only if the system consisting of the th,th transforms of is consistent as a system of polynomial equations.
Corollary 1.
Let . If , then system has a solution in if and only if there exist tuples , where , such that
3.2. Effective elimination
We will introduce the notation that will be used in Theorem 2. Let and be two sets of unknowns. Consider a system of difference equations, where . We set
Let be the field of fractions of and denote the affine variety defined by over . We denote the dimension and the sum of the degrees of the components of by and , respectively.
Theorem 2.
For all integers and and systems in and with and , there exists a nonzero difference equation that is a consequence of system if and only if the ideal generated by the th,th transforms of contains a nonzero polynomial depending only on and their transforms.
3.3. Consequences for computation
Theorem 1 and Corollary 1 reduce consistency questions 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 existence/finding a consequence in the variables of a system of difference equations in the variables and to a question about a polynomial ideal in a polynomial ring in finitely many variables. These algebraic problems are classical and have been computationally solved using, for example, Gröbner bases, triangular sets, numerical algebraic geometry, etc. For all of these methods, implementations exist in many computer algebra systems and independent software packages (see, for example, [CLO, Bertini, STV2008]).
4. Numerical values and practical examples
In the following table, we compute for small and .
Remark 4.
Almost all examples of modeling phenomena in the sciences using polynomial difference equations that we have seen in the literature can be written as systems with the same number of equations as unknowns in such a way that none of the equations is a consequence of the others. The above table is applicable to elimination problems for such systems with equations if the problem is to eliminate unknowns or less, as such problems typically result in varieties (see the notation of Section 3.2) of dimension or .
Remark 5.
One can significantly speed up checking if an elimination is possible by

applying the number of transforms that is in the bound

substituting random values into the variables that are not being eliminated.
Similarly to [HOPY2017], for each number , , using the SchwartzZippel lemma [Zippel, Proposition 98], we can find the range for the random substitution so that the probability of the elimination being possible if and only if the “substituted” system has no solutions is greater than . So, this would give an efficient probabilistic test for the possibility of elimination.
Remark 6.
Although there could be special tricks and methods for each of the examples below, our approach provides a general and fully automated procedure.
Example 6.
Consider the MayLeonard model for 2plant annual competition, scaled down from [RA2004]:
which can be rewritten as
(1) 
where , with acting as the identity on . To verify whether can be eliminated from (1), we then consider the affine variety defined by (1) over the field with coordinates . A computation shows that and , and so . A computation shows that it is not only sufficient but also necessary to apply this single transform to perform the elimination. So, our main result gives a sharp upper bound for this example.
Example 7.
Consider the MayLeonard model for 3plant annual competition [RA2004]:
which can be rewritten as
(2) 
where , with acting as the identity on . To verify whether and can be eliminated from (2), we consider the affine variety defined by (2) over the field with coordinates . A computation shows that and , and so . A computation shows that

two prolongations are necessary and sufficient

carrying out a computation with transforms as described in Remark 5 to check if an elimination is possible does not take significantly more time than doing this with two transforms.
Example 8.
Consider the stage structured LeslieGower model [CHR07, eq. (5)]:
which can be rewritten as
(3) 
where with acting as the identity on . To verify whether and can be eliminated from (3), we consider the affine variety defined by (3) over the field with coordinates . A computation shows that ; as the equations are linear in . Then . A computation shows that it is not only sufficient but also necessary to apply this single transform to perform the elimination. So, our main result gives a sharp upper bound for this example.
Example 9.
A discrete multipopulation SI model from [Allen1994], similarly to the previous examples, can be rewritten as
(4) 
where with acting as the identity on .
Example 10.
Let be the th Fibonacci number. It turns out [Zeilberger2014, p. 856] that the sequence satisfies a nonlinear difference equation. Such an equation can be found using difference elimination as follows. We introduce . Then standard identities and for the Fibonacci numbers imply the following system of difference equations
(5) 
Considered as a system of polynomial equations in and , (5) defines an affine variety of dimension zero and degree two over . Theorem 2 implies that it is sufficient to consider system (5) and two of its transforms to eliminate . Performing this elimination, we find the difference equation
giving an alternative to the difference equation stated in [Zeilberger2014, p. 856].
Example 11.
The following example shows that our bound is sharp in the case (this is the case in Examples 6, 8, and 10). We fix a positive integer and consider the system
(6) 
System (6) does not have a solution in , because the elements of the solution can only take values from and strictly increase. On the other hand, the system consisting of the th,th transforms of (6) has a solution for . Hence, it is necessary to consider one more transform in order to express (i.e. eliminate ).
Example 12.
The following example is based on an example constructed by E. Amzallag and R. Gustavson. Consider the system
(7) 
Let be the affine variety defined by (7) in the affine 3space with coordinates . We have and . Therefore, . A calculation shows that
Thus, this shows that our upper bound for and is sharp.
5. Counterexamples
5.1. Failure of the standard DEP method
Consider the system of difference equations given by any set of generators of the polynomial ideal of the polynomial ring , where
We do not present the actual generators of to size of this set. A computation shows that
Therefore, by Proposition 1, the system has no solutions in any difference ring. One can also show that
(8)  
(9) 
Most of the existing effective bounds for systems of ordinary differential and difference equations [Gal, AJS, Hr01, HrushovskiPillaybounds, OPV2017] use geometric axioms [ChHr99, PP1998] as sufficient conditions for the existence of a solution based on the system and its first prolongation (differential equations) or first transform (differencee quations), which are summarized under the DEP method mentioned in the introduction. In our case, it is tempting to formulate an analogue of such conditions as:
Let be the affine variety defined by the system and its first transform. If the projections of onto the varieties defined by the system and by its first transform alone, respectively, are dominant, then the system is consistent.
However, this is false in the above example as we have shown, where is the affine variety corresponding to the ideal in the affine space with coordinates , and (the Zariski closures of) the projections are given by the intersections in (8) and (9).
5.2. Nonexistence of coefficientindependent effective strong Nullstellensatz
A (noneffective) strong Nullstellensatz for systems of difference equations can be stated as follows. Let be a system of difference equations. If a difference polynomial vanishes at all solutions of the system in , then there exists such that belongs to the radical of the ideal generated by the th,th transforms of .
The following example shows that there is no uniform upper bound for this in terms of the degree, order, and number of variables of . For every positive integer , consider
(10) 
Let and and any solution of (10) in . If for some , then . Hence, , and so
Therefore, vanishes at every solution of (10) in . However, does not belong to the radical of the ideal generated by the th,th transforms of and . These transforms belong to the polynomial ring . Consider the substitution
A direct computation shows that the polynomials vanish after this substitution, but does not.
6. Proofs of the main results
6.1. Difference Nullstellensatz
Definition 8.
We say that a difference ring is inversive if is an automorphism.
Remark 7.
For any difference ring , there is an inversive difference ring and a map of difference ring that is universal for maps from to inversive difference rings.
Definition 9.
Given a difference ring , the ring of inversive difference polynomials over in the variables, , is realized as the ordinary polynomial ring over in the formal variables , for and , with extending the given endomorphism on and
on the variables.
Remark 8.
If is inversive, then so is .
Definition 10.
Let be a finite set of difference polynomials and . The set of tuples , where , is called a partial solution of length if, for every and , polynomial vanishes after the substitution
Let be an inversive difference field. Then the difference ring of sequences with respect to the shift automorphism can be endowed with a structure of a difference algebra via the embedding of difference rings defined by
This can be similarly done for .
Proposition 1.
For all uncountable algebraically closed fields and finite sets , the following statements are equivalent:

has a solution in .

has a solution in .

has finite partial solutions of length for all .

The ideal does not contain .

The ideal does not contain .

has a solution in some difference algebra.
Proof.
The implications , , and are straightforward.
. Assume that there exists an arbitrary long partial solution, but . Then there is an expression of the form
(11) 
where . Let . Consider a partial solution of of the length and plug it into the equality (11). Then the righthand side will vanish, so we arrive at contradiction.
. Assume that . We fix some representation of as an element of . Let be the maximum number such that occurs in the representation. Applying to the both sides of the representation, we obtain a representation of as an element of .
. Let be the canonical surjection. Then is a solution of in .
. Let be the inversive difference subfield of generated by the coefficients of elements of over the prime subfield of . Since does not belong to , there exists a maximal (not necessarily difference) ideal containing . Then is a field, and the transcendence degree of over is at most countable. Since is algebraically closed and has an uncountable transcendence degree, there exists an embedding over the common subfield . Composing with the canonical surjection , we obtain an algebra homomorphism such that . For every , we construct a sequence by the formula
A direct computation shows that is a solution of in . ∎
6.2. Variety and two projections
Let be a difference field and
a system of difference equations, with . We set
For the rest of Section 6, we fix be an inversive algebraically closed difference field of uncountable transcendence degree containing . With the system of difference equations, we associate the following geometric data:

the variety defined by the polynomials in ;

two projections defined by
Let be a variety defined by polynomials . Let , where , denote the variety defined by the polynomials , where means the result of applying to all coefficients of . The coordinatewise application of defines a bijection between and .
Definition 11.
A sequence is a partial solution of the triple if
A twosided infinite sequence with such a property is called a solution of the triple .
Lemma 1.
For every positive integer , system has a partial solution of length if and only if the triple has a partial solution of length .
System has a solution in if and only if the triple has an infinite solution.
Proof.
Let . Consider a partial solution of , where for every . We set
By the construction
so for every . The definition of partial solution implies that for every . Hence, is a partial solution of the triple . The above argument can be straightforwardly reversed to construct a partial solution of from a partial solution of . The case of infinite solutions is completely analogous. ∎
In the introduced geometric language, we can formulate the following question equivalent to effective difference Nullstellensatz
Question.
Let be an algebraic subvariety of and be surjective linear maps . How long a partial solution of is it sufficient to find in order to conclude that the triple has an infinite solution?
Thus, in what follows, we fix a triple , where is an algebraic variety of and are surjective linear maps defined over the constants of .
6.2.1. Trains
The goal of this section is to generalize the notion of a solution of the triple to not necessarily zerodimensional points.
Definition 12.
For two irreducible subvarieties , we say that the generic point of maps to the generic point of (and denote it by ) if .
Definition 13.
For a positive integer or , a sequence of irreducible subvarieties in is said to be a train of length in if