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An improvement to Chevalley's theorem with restricted variables
Abstract
Recently Schauz and Brink independently extended Chevalley's theorem to polynomials with restricted variables. In this note we give an improvement to Schauz-Brink's theorem via the ground field method. The improvement is significant in the cases where the degree of the polynomial is large compared to the weight of the degree of the polynomial.
... U. Schauz and D. Brink considered solutions lying in rectangular subsets of F N [32,7]. In some special cases the degree can be replaced by the p-weight degree [8]. Schauz and Brink's results were used in [18,1] to solve equations over finite nilpotent rings, groups, and generalizations of these structures. ...
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.
... Katz generalized Chevalley's Theorem to a wider class of finite commutative rings [Sch74,Kat09]. D. Brink considered solutions lying in rectangular subsets of F N ( [Bri11], with a p-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. ...
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.
First, a generalization of Chevalley's classical theorem from 1936 on polynomial equations f(x1,…, xN) = 0 over a finite field K is given, where the variables xi are restricted to arbitrary subsets Ai ⊆ K. The proof uses Alon's Nullstellensatz. Next, a theorem on integer polynomial congruences f(x1,…, xN) ≡ 0 (mod pν) with restricted variables is proved, which generalizes a more recent result of Schanuel. Finally, an extension of Olson's theorem on zero-sum sequences in finite Abelian p-groups is derived as a corollary.
The main result of this paper is a coecien t formula that sharpens and general- izes Alon and Tarsi's Combinatorial Nullstellensatz. On its own, it is a result about polynomials, providing some information about the polynomial map PjX1 Xn when only incomplete information about the polynomial P (X1;:::;Xn) is given. In a very general working frame, the grid points x 2 X1 Xn which do not vanish under an algebraic solution { a certain describing polyno- mial P (X1;:::;Xn) { correspond to the explicit solutions of a problem. As a consequence of the coecien t formula, we prove that the existence of an algebraic solution is equivalent to the existence of a nontrivial solution to a problem. By a problem, we mean everything that \owns" both, a set S , which may be called the set of solutions; and a subset Striv S , the set of trivial solutions. We give several examples of how to nd algebraic solutions, and how to apply our coecien t formula. These examples are mainly from graph theory and combina- torial number theory, but we also prove several versions of Chevalley and Warning's Theorem, including a generalization of Olson's Theorem, as examples and useful corollaries. We obtain a permanent formula by applying our coecien t formula to the matrix polynomial, which is a generalization of the graph polynomial. This formula is an integrative generalization and sharpening of: 1. Ryser's permanent formula. 2. Alon's Permanent Lemma. 3. Alon and Tarsi's Theorem about orientations and colorings of graphs. Furthermore, in combination with the Vigneron-Ellingham-Goddyn property of pla- nar n-regular graphs, the formula contains as very special cases: 4. Scheim's formula for the number of edge n-colorings of such graphs. 5. Ellingham and Goddyn's partial answer to the list coloring conjecture.
In this work, we introduce the p-weight degree of a polynomial over a finite field with respect to a subset of the variables. Using this p-weight, we improve the results of Moreno and Moreno for polynomial equations and for exponential sums over finite fields. We prove that our results cannot be improved in general because a family of polynomials where our bounds are attained is provided. Combining our result with a result of Cao and Sun, we give a p-adic improvement to the p-divisibility of general diagonal equations.
Following the work of Chevalley and Warning, Ax obtained a bound on the p-divisibility of exponential sums involving multi-variable polynomials of fixed degree d over a finite field of characteristic p. This bound was subsequently improved by Katz. More recently, Moreno and Moreno, and Adolphson and Sperber, derived bounds that in many instances improved upon the Ax–Katz result. Here we derive a tight bound on the p-divisibility of the exponential sums. While exact computation of this bound requires the solution of a system of modular equations, approximations are provided which in several classes of examples, improve on the results of Chevalley and Warning, Ax and Katz, Adolphson and Sperber, and Moreno and Moreno. All of the above results readily translate into bounds on the p-divisibility of the number of zeros of multi-variable polynomials.
An important consequence of one of our main results is a method to find classes of examples for which bounds on divisibility of the number of solutions of a system of polynomial equations over finite fields are tight. In particular, we give classes of examples for which the Moreno–Moreno bound is tight.
It is important to note that we have also found applications of our results to coding theory (computation of the covering radius of certain codes) and to Waring's problem over finite fields. These will be described elsewhere. 2000 Mathematics Subject Classification 11L07 (primary), 11G25 (secondary).