On the Waring Problem for Matrices over Finite Fields

TL;DR

The paper addresses the Waring problem for matrices over finite fields by proving two main results: (1) over $\mathbb{F}_q$ with $q\neq 2$, if $q^n>(k-1)^4$ then every $n\times n$ matrix is a sum of three $k$-th powers; (2) if $n\ge 7$ and $k<q$, every $n\times n$ matrix is a sum of two $k$-th powers. The approach blends additive decompositions via companion and Frobenius normal forms, construction of $k$-power matrices through irreducible polynomials, and trace-control techniques to realize $k$-th power summands with prescribed spectra. Key contributions include a detailed framework for decomposing non-scalar matrices into components with desired characteristic polynomials, the existence of $k$-power irreducible polynomials of degree $n$ with arbitrary traces for large $n$, and a unified route to obtain two- and three-term Waring decompositions under explicit field-size and dimension conditions. Overall, the work advances a Larsen-type conjecture for matrix rings over finite fields and provides concrete bounds and constructions for practical Waring decompositions in this noncommutative setting.

Abstract

We prove that if $k$ is a positive integer then for every finite field $\mathbb{F}$ of cardinality $q\neq 2$ and for every positive integer $n$ such that $q^n>(k-1)^4$, every $n\times n$ matrix over $\mathbb{F}$ can be expressed as a sum of three $k$-th powers. Moreover, if $n\geq 7$ and $k<q$, every $n\times n$ matrix over $\mathbb{F}$ can be written as a sum of two $k$-th powers.

Theorems & Definitions (34)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Remark 2.4
• Theorem 3.1
• Lemma 3.2
• proof