Time: Friday, March 25, 2022, 1:00pm-2:00pm.
Location: DH 245
Title: A Chevalley Warning Theorem Modulo Differing Prime Powers
Speaker: David J. Grynkiewicz, U. Memphis
Abstract: Polynomial methods are a powerful algebraic tool that can be used to prove many interesting and otherwise difficult combinatorial results. One such tool is the Chevalley-Warning Theorem. It states that if $p$ is a prime, $f_1,\ldots,f_s\in \mathbb Z[X_1,\ldots, X_n]$ are polynomials and $$V=\{\textbf x\in [0,p-1]^n:\; f_1(\textbf x)\equiv 0,\;\ldots,\;f_s(\textbf{x})\equiv 0\mod p\}$$ is the set of all common roots of the polynomials $f_1,\ldots,f_s$ (modulo $p$), then $$|V|\equiv 0\mod p$$ provided $n> \sum_{i=1}^{s}\mathsf{deg} f_i$. In particular, if the polynomials have no constant term and the number of variables is larger than the sum of degrees, then there is guaranteed to be a nontrivial common root to all the polynomials.
There are many generalizations of this result, including those quantifying just how large a prime power $p^m$ divides the cardinality $|V|$. We will discuss these results and illustrate them by giving some examples of simple proofs of otherwise very difficult theorems. We will then talk about a seemingly un-noticed generalization of the Chevalley-Warning Theorem. By a rather straightforward modification of a proof of R. Wilson, one can obtain a version of the Chevalley-Warning Theorem applicable when the equations $f_1(\textbf x)\equiv 0\mod p^{m_1},\;\ldots,\;f_n(\textbf x)\equiv 0\mod p^{m_s}$ are considered modulo \emph{differing prime powers} of the common prime $p$. Using Hensel’s Lemma, which we will discuss in detail for those unfamiliar, we will explain how this result means any typical Chevalley-Warning argument modulo $p$ can effectively be implemented modulo varying primes powers $p^{m_1},\ldots,p^{m_s}$ of a common prime, with tight bounds for how large a prime power $p^m$ divides the cardinality of the corresponding set $V$. As one simple example, this gives a new proof of the Davenport Constant of a finite abelian $p$-group without need of using group algebras.