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

TL;DR

This paper extends classical polynomial theorems to multivariate skew polynomial rings over division rings, providing new algebraic tools for analyzing zeros in non-commutative settings.

Contribution

It generalizes the Combinatorial Nullstellensatz and Chevalley-Warning theorem to skew polynomial rings, broadening their applicability to non-commutative algebra.

Findings

Generalized Nullstellensatz for skew polynomials

Proved Chevalley-Warning type theorems in skew polynomial rings over finite fields

Established analogues of Ax's Lemma and Terjanian's Finitesatz in this context

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.