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Finite fields whose members are the sum of a potent and a 4-potent

Stephen D. Cohen, Peter V. Danchev, Tomás Oliveira e Silva

TL;DR

This work classifies finite fields $ abla{F}_q$ for which every element can be written as the sum of an $n$-potent and a $4$-potent, under the natural condition $(n-1)\mid(q-1)$ and $n>1$. Using a case analysis on the parity of $q$ and the specific forms $n=(q+3)/4$, $(q+1)/2$, and $(q+2)/3$, the authors derive contradictions via cardinality arguments and character-sum bounds (quadratic and cubic) together with Weil-type estimates and Jacobi sums. They reduce the problem to finitely many large-$q$ obstructions and then confirm, via explicit computations up to $q\le 1000$, that exactly ten non-trivial pairs $(q,n)$ satisfy the Restricted Problem, listing them explicitly. The paper also outlines a natural extension to $m=5$, provides computational evidence, and discusses potential applications to ring and matrix theory. Overall, it extends prior tripotent results and tightens the landscape of when potent-sum representations cover finite fields.

Abstract

We classify those finite fields $\mathbb{F}_q$, for $q$ a power of some fixed prime number, whose members are the sum of an $n$-potent element with $n>1$ and a 4-potent element. It is shown that there are precisely ten non-trivial pairs $(q,n)$ for which this is the case. This continues a recent publication by Cohen-Danchev et al. in Turk. J. Math. (2024) in which the tripotent version was examined in-depth as well as it extends recent results of this branch established by Abyzov-Tapkin in Sib. Math. J. (2024).

Finite fields whose members are the sum of a potent and a 4-potent

TL;DR

This work classifies finite fields for which every element can be written as the sum of an -potent and a -potent, under the natural condition and . Using a case analysis on the parity of and the specific forms , , and , the authors derive contradictions via cardinality arguments and character-sum bounds (quadratic and cubic) together with Weil-type estimates and Jacobi sums. They reduce the problem to finitely many large- obstructions and then confirm, via explicit computations up to , that exactly ten non-trivial pairs satisfy the Restricted Problem, listing them explicitly. The paper also outlines a natural extension to , provides computational evidence, and discusses potential applications to ring and matrix theory. Overall, it extends prior tripotent results and tightens the landscape of when potent-sum representations cover finite fields.

Abstract

We classify those finite fields , for a power of some fixed prime number, whose members are the sum of an -potent element with and a 4-potent element. It is shown that there are precisely ten non-trivial pairs for which this is the case. This continues a recent publication by Cohen-Danchev et al. in Turk. J. Math. (2024) in which the tripotent version was examined in-depth as well as it extends recent results of this branch established by Abyzov-Tapkin in Sib. Math. J. (2024).

Paper Structure

This paper contains 8 sections, 6 theorems, 36 equations.

Table of Contents

  1. Introduction and Motivation
  2. Principal Results
  3. $q$ odd and $n=\frac{q+3 }{4}$
  4. $q$ odd and $n=\frac{q+ 1}{2}$
  5. $q$ odd or even and $n=\frac{q+ 2}{3}$
  6. Completion of proof of Theorem \ref{['main']}
  7. A conjecture and a further problem
  8. Pari-gp code

Key Result

Theorem 1.1

Suppose $q$ is a prime power and $n$ is a positive integer such that $n<q$ and $n-1$ divides $q-1$. Then, every element of $\mathbb{F}_q$ is the sum of a tripotent and an $n$-potent if, and only if, $n=\frac{q-1}{2}$ and $q \in \{3,5,7,9\}$.

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • Conjecture 3.1