Central polynomials of minimal degree for matrices

TL;DR

This paper investigates the minimal degree of central polynomials for matrix algebras, confirming known conjectures for small sizes and establishing new lower bounds for 4x4 matrices, enhancing understanding of polynomial identities.

Contribution

It proves that 4x4 matrix algebra lacks central polynomials of degree ≤12 in two variables, advancing the knowledge of polynomial identities and central polynomials.

Findings

No central polynomials of degree ≤12 in two variables for 4x4 matrices.

The algebra of 4x4 matrices has no polynomial identities of degree ≤12 in two variables.

Supports Formanek's conjecture for small matrix sizes.

Abstract

Formanek made the conjecture that the minimal degree of the central polynomials for the $n\times n$ matrix algebra over a field of characteristic 0 is $(n^2+3n-2)/2$ and this is true for $n\leq 3$. For $n=4$ there are examples of central polynomials of degree $13=(4^2+3\cdot 4-2)/2$ and we do not know whether there are central polynomials of lower degree. In this paper we discuss methods for searching for central polynomials of low degree and prove that the algebra of $4\times 4$ matrices does not have central polynomials in two variables of degree $\leq 12$. As a byproduct of our computations we obtain that this algebra does not have also polynomial identities in two variables of degree $\leq 12$.