Chevalley Warning type results on abelian groups
Abstract.
We develop a notion of degree for functions between two abelian groups that allows us to generalize the Chevalley Warning Theorems from fields to noncommutative rings or abelian groups of prime power order.
Key words and phrases:
Chevalley Warning Theorem, functional degree, abelian groups2010 Mathematics Subject Classification:
20K01 (13F20, 20C05)1. Introduction
A classical result by C. Chevalley [Che35] states that if a system of polynomial equations over a finite field has exactly one solution in , then the sum of the total degrees of the ’s is at least . E. Warning [War35] improved this result by showing that under the hypothesis that is strictly larger than the sum of the total degrees of the ’s, the number of solutions, which cannot be by Chevalley’s result, is divisible by the characteristic of the field (Warning’s First Theorem), and the system has either no or at least solutions (Warning’s Second Theorem). Proofs of these results can be found, e.g., in [Asg18]. All three results have been considerably strengthened: for Warning’s First Theorem, [Ax64, Kat71] provide lower bounds for such that divides the number of solutions. O. Moreno and C.J. Moreno showed that in these bounds, the total degree of a polynomial can be replaced with the weight degree [MM95]. S.H. Schanuel and D.J. Katz generalized Chevalley’s Theorem to a wider class of finite commutative rings [Sch74, Kat09]. D. Brink considered solutions lying in rectangular subsets of ([Bri11], with a weight degree version given in [CGM19]). Brink’s result was used in [KS18, Aic19] to solve equations over finite nilpotent rings, groups, and generalizations of these structures.
In this article, we generalize the Chevalley Warning Theorems into a different direction: instead of polynomial functions on finite fields, we consider arbitrary functions on abelian groups. Unlike for polynomial functions on fields, there is no generally agreed concept of the degree of such functions. In the first half of the present article, we develop a notion of degree for functions between two abelian groups based on [May12, VL83]. Since this degree does not depend on a term representation of the function, we suggest the name functional degree for this concept. Using this functional degree, we obtain variants of Chevalley’s and Warning’s First Theorem for finite abelian groups (Theorems 10.2 and 11.3). From these results, one easily derives Moreno and Moreno’s weight improvement of Warning’s First Theorem (Theorem 12.4). We also see that Warning’s First Theorem remains true if we replace “finite field” with “not necessarily commutative ring of prime power order” (Theorem 11.5). The proof of the last result takes advantage of the fact that every function, be it polynomial or not, has a functional degree. For polynomial functions over finite fields, the functional degree specializes to the weight degree from [MM95]. This allows us to generalize Asgarli’s proof of Warning’s Second Theorem [Asg18] to derive its weight degree improvement [MM95, Theorem 2] (Theorem 14.1). A similar improvement of Brink’s Theorem [Bri11, Theorem 1] can be obtained in the special case that the domain is restricted to a subgroup of (Section 13) and we also obtain that the number of solutions in the subgroup is divisible by the field characteristic (Theorem 13.2). Warning’s First Theorem can be strenghtened if we know that the functions are not surjective: such “restricted range” versions are given in Theorem 13.5 and Corollary 13.6.
The functional degree defined in this note has its origins in [BAE00, May12, VL83]. In [May12], P. Mayr defines the degree of every finitary operation on an algebra with a Mal’cev term [May12, (3.9)]. Our definition applies to functions from one abelian group into another abelian group , and it involves the augmentation ideal of the group ring that acts on a function by shifting its arguments. This follows an idea from [VL83], where such group rings were successfully applied in the structure theory of nilpotent algebras in congruence modular varieties (cf. [FM87] and [May12, Corollary 3.10]). In those situations where both definitions apply, Mayr’s degree and the functional degree coincide. A pivotal result is that the functional degree of the composed function is at most the product of the functional degrees of and (Theorems 4.2 and 5.3). For arbitrary finite abelian groups, there may be functions of infinite degree, but if domain and codomain are finite abelian groups (for the same ), then the degree of every function is finite (Section 9).
2. Definition of the functional degree
In this section, we will introduce the functional degree of a function between two abelian groups. We write for the set of positive integers, , and for , the set is abbreviated by . In general, we will write groups additively, and we sometimes simply write for the abelian group . By , we denote its group ring over the integers [Pas77]. The elements of this ring are integer tuples indexed by with finite. We will write such a tuple in the form , where for all , instead of the more common . The multiplication of then satisfies for all ; thus is a commutative ring with unity . The augmentation ideal of is the ideal generated by , and it will be denoted by . For every ideal of , the power is defined as , and is the ideal generated by . For , is generated by the set . Let be an abelian group, and let . The ring operates on the group by
and hence
We will use this module operation also for a function from an abelian group into a ring, or even a field, . The multiplication on is then immaterial for the module operation. For and an ideal of , .
Definition 2.1 (Functional degree).
Let and be abelian groups, and let . Let be the augmentation ideal of . The functional degree of is defined by
with if there is no with .
For all with , we have , and thus if , also . Hence for each function and for each , we have
We also see that , where is for all infinite subsets of . Hence for each , we have
Another description is given in the following Lemma.
Lemma 2.2.
Let and be abelian groups such that is generated by . Let , and let . Then the following are equivalent:

.

For all , we have
Proof.
Let . We first observe that is generated, as an ideal of , by . To show this, let be the ideal generated by . Obviously, . For the other inclusion, we note that the set is a subgroup of because and hence, if and are both elements of , then so is . Furthermore, if , then , and thus . Thus is indeed a subgroup of . Now by the definition of , we have , and since is generated by , . Thus , which implies . We will now prove the equivalence of (1) and (2).
3. Elementary properties of the functional degree
In this section, we list some properties of the functional degree that follow quite immediately from its definition.
Lemma 3.1.
Proof.
Let . For (1), we observe that is an invertible element in the ring . The ring is commutative. Therefore, for each ideal of , we have we have if and only if . Hence and have the same functional degree. This completes the proof of (1).
For proving (2), we first observe that the statment is obvious if . Hence we assume and set . We show that . To this end, we observe that for all , . Since , we have . From the definition of the functional degree, we see that implies that , and therefore .
For proving , we first show that for every with there exists such that . For this purpose, let be such that . Then there are such that : If this product is for all , then , contradicting . Take . Hence , and therefore . Hence there exists a with the required property.
Therefore, if , then taking , we obtain .
In the case , we obtain a sequence from such that . Hence .
This completes the proof of the inequality of (3).
(4) For the “if”direction, we see that for all , if is a constant function. Therefore and thus . For the “only if”direction, we assume that . Now let . Then , hence . Thus is constant.
(5) We assume that is a group homomorphism, and we let . Then , and therefore is a constant function. By item (4), . Now by (3), .
∎
Lemma 3.2 (Addition).
Let and be abelian groups, and let . Then we have:

.

If , then .
Lemma 3.3 (Restriction of a function).
Let and be abelian groups, let , let be a subgroup of , and let be the restriction of to . Then .
Proof: Suppose that , and let . Let . Seeing as a subring of , we observe that . Hence . For every , we have . Thus . Therefore, . ∎
Lemma 3.4 (Combination of functions).
Let , and be abelian groups, let , and let . We define a function by for all . Then .
Proof.
For every , we have . Hence for every , if and only if both and , which implies the result. ∎
4. The degree of composed functions
The aim of this section is to prove that the functional degree of a composition is at most the product of the functional degrees of and . For this purpose, we characterize the functional degree of a function by certain “linearity” properties. Similar linearity properties have been used in [May12] for defining the degree of finitary operations in an algebra with a Mal’cev term. For a set and , we write for the power set of and for the set . We write for ( and ).
Lemma 4.1 (Characterization of the degree).
Let and be abelian groups, let , and let . Then the following are equivalent:

.

For every , we have
(4.1) 
For every , there exists a family of integers such that for all , we have

There exist functions such that for all , we have
(4.2) and for each , the function does not depend on its th argument.
Proof.
(2)(3): We proceed by induction on . If , then we set and for all subsets of that are not equal to . Now assume . For all , we have
We will now expand each using the induction hypothesis. Since every proper subset of has less than elements, the induction hypothesis yields for every a family of integers such that
for all . Note that we may take for those with . Hence
for all , and therefore with satisfies the required property, which completes the induction step.
(3)(4): By (3), we have a family of integers such that for all , , which is equal to
Hence
does not depend on its th argument, and the ’s satisfy (4.2).
(4)(1): We prove that for each and for each , the existence of such implies . We proceed by induction on . For , we observe that the identity with not depending on implies that is constant, and therefore of functional degree by Lemma 3.1(4). For the induction step, let and assume that , where does not depend on its th argument. The induction hypothesis is that every with and not depending on its th argument satisfies . We want to prove . If , there is nothing to prove, hence we may assume that . We will use Lemma 3.1(3) to compute . To this end, we let and estimate . For each , we have
Now for and , we define . Since does not depend on its th argument, we have
for all . The function does not depend on its th argument. Therefore, the induction hypothesis yields . Now by Lemma 3.1(3), . ∎
The interplay of the degree with functional composition will be central in our further development.
Theorem 4.2 (Composition).
Let be abelian groups, let and with and . Then .
Proof.
Let , . We show that satisfies condition (4) of Lemma 4.1. To this end, let . Then from Lemma 4.1(3), applied first to and then to , we obtain two families and such that
Now for each , the corresponding summand can only depend on those with , and hence on at most arguments. Therefore we can write as a sum of functions each of which depends on at most arguments. Collecting these functions into summands, we obtain the functions that satisfy condition (4) of Lemma 4.1, and hence . ∎
5. Partial degree
We will now define the partial degree of a function in each of its variables. Intuitively, is the maximal degree of those functions from that we obtain by setting all arguments except for the th one to constants. More formally, we proceed as follows: For and , we define the function by if , and for all . Hence .
Definition 5.1 (Partial degree).
Let , let be a family of abelian groups, and let be an abelian group. Let , and let . Then the partial degree of in its th argument, , is defined by
Theorem 5.2.
Let , let be a family of abelian groups, let be an abelian group, and let . Let . Then .
Proof.
For proving the first inequality , we fix and estimate the degree of the function from to . Since the degree of a constant function is and the degree of the identity mapping on is at most (it is if and if ), Lemma 3.4 yields that the function is of degree at most . Hence by Theorem 4.2, , which completes the proof of .
We will now prove
(5.1) 
We first consider the case . To this end, let , , and . Let and . We assume and . Let , , . Let be the ring homomorphism defined by for all , and let be defined by . We observe that
for all and . Therefore for all elements and , we have
(5.2) 
for all and .
Since , we have , and similarly for all