Evaluation of Polynomials over Finite Rings via Additive Combinatorics

TL;DR

This paper improves bounds on solving polynomial equations over finite nilpotent rings by leveraging additive combinatorics, providing insights into polynomial value sets and their complexity.

Contribution

It introduces a tighter polynomial bound on equation solvability over finite nilpotent rings using additive combinatorics techniques.

Findings

Enhanced bounds on polynomial equation solvability

Application of additive combinatorics to finite rings

Potential for broader implications in algebraic complexity

Abstract

We give an improved polynomial bound on the complexity of the equation solvability problem, or more generally, of finding the value sets of polynomials over finite nilpotent rings. Our proof depends on a result in additive combinatorics, which may be of independent interest.