Fattening up Warning's Second Theorem

TL;DR

This paper generalizes Warning's Second Theorem to polynomial systems over finite local principal rings, providing new proofs and broad applications in additive group theory, graph theory, and polynomial interpolation.

Contribution

It extends Warning's Second Theorem to a wider class of rings and offers new proofs, enabling diverse applications in mathematics.

Findings

Generalization of Warning's Second Theorem to finite local principal rings

New proof of the generalized theorem

Applications demonstrated in additive group theory, graph theory, and polynomial interpolation

Abstract

We present a generalization of Warning's Second Theorem to polynomial systems over a finite local principal ring with suitably restricted input and output variables. This generalizes a recent result with Forrow and Schmitt (and gives a new proof of that result). Applications to additive group theory, graph theory and polynomial interpolation are pursued in detail.