Solutions to polynomial congruences in well shaped sets

TL;DR

This paper extends bounds on solutions to polynomial congruences within general sets, including convex sets, by leveraging advanced mean value theorems and Schmidt's ideas.

Contribution

It introduces a novel approach combining Vinogradov's mean value theorem and Schmidt's techniques to analyze polynomial congruences in broad geometric sets.

Findings

Established nontrivial bounds for solutions in convex and well-shaped sets

Generalized previous results to arbitrary polynomials

Applied advanced mean value theorems to polynomial congruences

Abstract

We use a generalization of Vinogradov's mean value theorem of S. Parsell, S. Prendiville and T. Wooley and ideas of W. Schmidt to give nontrivial bounds for the number of solutions to polynomial congruences, for arbitrary polynomials, when the solutions lie in a very general class of sets, including all convex sets.