Central polynomials for matrices over finite fields

TL;DR

This paper proves that multihomogeneous central polynomials for matrix algebras over infinite fields of positive characteristic can be adapted to finite fields of the same characteristic, using elementary combinatorial methods.

Contribution

It demonstrates the existence of multihomogeneous central polynomials over finite fields with coefficients in the prime field, extending known results from infinite fields.

Findings

Existence of multihomogeneous central polynomials over finite fields

Construction of such polynomials with coefficients in the prime field

Elementary combinatorial proof technique

Abstract

Let $c(x_1,...,x_d)$ be a multihomogeneous central polynomial for the $n\times n$ matrix algebra $M_n(K)$ over an infinite field $K$ of positive characteristic $p$. We show that there exists a multihomogeneous polynomial $c_0(x_1,...,x_d)$ of the same degree and with coefficients in the prime field $F_p$ which is central for the algebra $M_n(F)$ for any (possibly finite) field $F$ of characteristic $p$. The proof is elementary and uses standard combinatorial techniques only.