Polynomials over structured grids

TL;DR

This paper explores multivariate polynomials over structured grids, introducing the nullity parameter to define structure and extending key theorems to relax degree constraints based on grid structure.

Contribution

It proposes a new interpretation of structured sets via nullity and extends classical polynomial theorems to these grids, allowing for relaxed degree constraints.

Findings

Extended the Combinatorial Nullstellensatz to structured grids.

Generalized the Coefficient Theorem for polynomials over structured sets.

Showed that grid structure permits relaxed polynomial degree bounds.

Abstract

We study multivariate polynomials over `structured' grids. We begin by proposing an interpretation as to what it means for a finite subset of a field to be structured; we do so by means of a numerical parameter, the nullity. We then extend several results--notably, the Combinatorial Nullstellensatz and the Coefficient Theorem--to polynomials over structured grids. The main point is that the structure of a grid allows the degree constraints on polynomials to be relaxed.