Warning's Second Theorem with Resricted Variables

TL;DR

This paper extends Warning's Second Theorem to systems with restricted variables, providing a new tool for combinatorial existence theorems and their quantitative refinements over finite fields.

Contribution

It introduces a restricted variable generalization of Warning's Second Theorem, linking it to Brink's theorem and enabling new combinatorial applications.

Findings

Provides a lower bound on solutions for restricted polynomial systems

Establishes a connection between Warning's and Brink's theorems

Offers new combinatorial tools for quantitative analysis

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 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 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.