On the Waring problem with Dickson polynomials modulo a prime

TL;DR

This paper advances the understanding of the Waring problem involving Dickson polynomials over prime finite fields by leveraging recent bounds on exponential sums, leading to improved results.

Contribution

It provides new bounds and results for the Waring problem with Dickson polynomials in prime finite fields, improving upon previous work.

Findings

Enhanced bounds on the Waring problem with Dickson polynomials

Application of recent Kloosterman and Gauss sum bounds

Improved results over previous studies

Abstract

We improve recent results of D. Gomez and A. Winterhof (2010) and of A. Ostafe and I. E. Shparlinski (2011) on the Waring problem with Dickson polynomials in the case of prime finite fields. Our approach is based on recent bounds of Kloosterman and Gauss sums due to A. Ostafe, I. E. Shparlinski and J. F. Voloch (2021).