Exponential sum estimate for systems including linear polynomials

TL;DR

This paper extends exponential sum estimates to systems that include linear polynomials, enabling new applications of the Hardy-Littlewood circle method in such contexts.

Contribution

It provides an analogous exponential sum estimate for systems with linear polynomials, complementing Schmidt's previous work on non-linear systems.

Findings

Established an exponential sum estimate for systems including linear polynomials

Enabled applications of the Hardy-Littlewood circle method to these systems

Extended the theoretical framework for analyzing polynomial systems

Abstract

In his paper, W. M. Schmidt obtained an exponential sum estimate for systems of polynomials not including linear polynomials, which was then used to apply the Hardy-Littlewood circle method. We prove an analogous estimate for systems including linear polynomials.