Bounds on trilinear and quadrilinear exponential sums

TL;DR

This paper derives explicit upper bounds for trilinear and quadrilinear exponential sums in prime finite fields, improving previous results by leveraging recent estimates on solutions to specific equations.

Contribution

It provides new explicit bounds on exponential sums, enhancing the understanding of their behavior in finite fields and building on recent advances in solution estimates.

Findings

Improved upper bounds for trilinear exponential sums.

Explicit bounds for quadrilinear exponential sums.

Enhanced understanding of exponential sums in prime finite fields.

Abstract

We use an estimate of Aksoy Yazici, Murphy, Rudnev and Shkredov (2016) on the number of solutions of certain equations involving products and differences of sets in prime finite fields to give an explicit upper bound on trilinear exponential sums which improves the previous bound of Bourgain and Garaev (2009). We also obtain explicit bounds for quadrilinear exponential sums.