Warning's Second Theorem with Resricted 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 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.
Cite
@article{arxiv.1404.7793,
title = {Warning's Second Theorem with Resricted Variables},
author = {Pete L. Clark and Aden Forrow and John R. Schmitt},
journal= {arXiv preprint arXiv:1404.7793},
year = {2014}
}
Comments
15 pages; some expository changes, including the correction of an inaccuracy in the exposition of Section 2.3