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On the Waring problem with Dickson polynomials modulo a prime

Igor E. Shparlinski, José Felipe Voloch

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).

On the Waring problem with Dickson polynomials modulo a prime

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).

Paper Structure

This paper contains 13 sections, 11 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Previous results
  3. Main results
  4. Preliminaries
  5. Notation
  6. Value sets of Dickson Polynomials
  7. Bounds of some exponential sums
  8. Bounds on the number of solutions to some equations
  9. Proof of Theorem \ref{['thm:medium e']}
  10. Small $\gcd(e,p-1)$
  11. Small $\gcd(e,p+1)$
  12. Proof of Theorem \ref{['thm:small e']}
  13. Proof of Theorem \ref{['thm:monom med k']}

Key Result

Theorem 1.1

Let $p$ be prime. There is an absolute constant $C > 0$ such that for any fixed even integer $s \geqslant 4$, uniformly over $a\in \mathbb{F}_p^*$ the inequality $g_a(e,p) \leqslant s$ holds provided that and, if $a$ is a quadratic residue modulo $p$, also provided that

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 6 more