The Combinatorial Nullstellensatz, Chevalley-Warning Theorem and weak Finitesatz in skew polynomial rings

Abstract

We study zeros of polynomials in the multivariate skew polynomial ring $D[x_1,\ldots,x_n; \sigma]$, where $\sigma$ is an automorphism of a division ring $D$. We prove a generalization of Alon's celebrated Combinatorial Nullstellensatz for such polynomials. In the case where $D$ is a finite field, we prove skew analogues of the Chevalley--Warning theorem, Ax's Lemma, and the weak case of Terjanian's Finitesatz.