Warning’s Second Theorem with restricted variables

Abstract

We present a restricted variable generalization of Warning's Second Theorem (a result giving a lower bound on the number of solutions of a low degree polynomial system over a finite field, assuming one solution exists). This is analogous to Schauz-Brink's restricted variable generalization of Chevalley's Theorem (a result giving conditions for a low degree polynomial system not to have exactly one solution). Just as Warning's Second Theorem implies Chevalley's Theorem, our result implies Schauz-Brink's Theorem. We include several combinatorial applications, enough to show that we have a general tool for obtaining quantitative refinements of combinatorial existence theorems.

Warning’s second theorem with relaxed outputs

“…Apart from [21] there had been little further exploration of Theorem 1.1c) until [12], which established the following result. Theorem 1.2 (Restricted variable Warning's second theorem [12]) Let K be a number field with ring of integers Z K , let p be a nonzero prime ideal of Z K , and let q = #Z K /p, so Z K /p ∼ = F q . Let A 1 , .…”

“…The generalization from polynomial congruences to polynomial congruences with relaxed outputs enables a wide range of applications. As in [12], whenever one has a combinatorial existence theorem proved via the Schauz-Wilson-Brink Theorem (or an argument that can be viewed as a special case thereof), one can instead apply Theorem 1.2 to get a lower bound on the number of solutions. Moreover, most applications of Schauz-Wilson-Brink include a homogeneity condition ensuring the existence of a trivial solution.…”

“…All of these applications can be generalized by allowing relaxed outputs. In [12], we gave three combinatorial applications of Theorem 1.2: to hypergraphs, to generalizations of the Erdős-Ginzburg-Ziv Theorem, and to weighted Davenport constants. In the former two cases, we can (and shall) immediately apply the Main Theorem to get stronger results.…”

The Combinatorial Nullstellensätze Revisited

“…• In §3 we expose the close relation between [CFS14], so Theorem 17 is in some sense the last piece of the "Chevalley-to-Alon" conversion process.…”

On Zeros of a Polynomial in a Finite Grid

“…In this prefilled context, the greedy distribution y G is defined by starting with the bins prefilled with b 1 ,...,b n balls and then distributing the remaining balls from left to right, filling each bin completely before moving on to the next bin. One sees -for example by adapting the argument of [14 In general we do not know a simple description of m(a 1 ,...,a n ; b 1 ,...,b n ; N). In practice, it can be computed using dynamic programming.…”