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Waring Problem for Matrices over Finite Fields

Krishna Kishore, Adrian Vasiu, Sailun Zhan

Abstract

We prove that for all integers $k \geq 1$, $q\ge (k-1)^4+ 6k$, and $m \geq 1$, every matrix in $ M_m(\mathbb F_q)$ is a sum of two kth powers: $M_m(\mathbb F_q)=\{A^k+B^k|A,B\in M_m(\mathbb F_q)\}$. We further generalize and refine this result in the cases when both $B$ and $C$ can be chosen to be invertible, cyclic, or split semisimple, when $k$ is coprime to $p$, or when $m$ is sufficiently large. We also give a criterion for the Waring problem in terms of stabilizers.

Waring Problem for Matrices over Finite Fields

Abstract

We prove that for all integers , , and , every matrix in is a sum of two kth powers: . We further generalize and refine this result in the cases when both and can be chosen to be invertible, cyclic, or split semisimple, when is coprime to , or when is sufficiently large. We also give a criterion for the Waring problem in terms of stabilizers.
Paper Structure (8 sections, 49 theorems, 51 equations)

This paper contains 8 sections, 49 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. On powers
  3. Stabilizers and Double Cosets
  4. Diagonal entries in conjugacy classes
  5. Proofs of the main theorems
  6. The case of F2
  7. The case of F3
  8. Open Problems

Key Result

Theorem 1.1

We assume that $(n,q)$ is neither $(2,2)$ nor $(2,3)$. Let $g\in S_{n,q}\setminus Z_{n,q}$. Then the following properties are true: (a) The equality $P=M_{n,q}$ holds if and only if $g\in L_P$. (b) If $p\nmid k$ and $-1\in\mathbb P_{d,q}$, then $Q=M_{n,q}$ if and only if $g\in L_Q$. (c) If $p\nmid k

Theorems & Definitions (106)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 96 more