The Mesyan Conjecture: a restatement and a correction

TL;DR

This paper revisits the Mesyan conjecture related to polynomial images on matrices, corrects a proof error for degree 4, and extends the conjecture's validity to a new bound for matrix size.

Contribution

The paper corrects a proof error in the degree 4 case of the Mesyan conjecture and establishes its validity for a new matrix size bound n ≥ (m+1)/2.

Findings

Corrected the proof of the degree 4 case of the Mesyan conjecture.

Extended the conjecture's validity to matrices with size n ≥ (m+1)/2.

Provided evidence supporting the conjecture under the new bound.

Abstract

The well-known Lvov-Kaplansky conjecture states that the image of a multilinear polynomial $f$ evaluated on $n\times n$ matrices is a vector space. A weaker version of this conjecture, known as the Mesyan conjecture, states that if $m=deg( f)$ and $n\geq m-1$ then its image contains the set of trace zero matrices. Such conjecture has been proved for polynomials of degree $m \leq 4$. The proof of the case $m=4$ contains an error in one of the lemmas. In this paper, we correct the proof of such lemma and present some evidences which allow us to state the Mesyan conjecture for the new bound $n \geq \frac{m+1}{2}$, which cannot be improved.