The Mesyan Conjecture: a restatement and a correction

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 $n\ge m-1$ then its image contains the set of trace zero matrices. Such conjecture has been proved for polynomials of degree $m\le 4$. 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 $n\ge \frac{m+1}{2}$, which cannot be improved.

arXiv commentResearch article dedicated to Professor Vesselin Drensky on the occasion of his 70th birthday

References in corpus (9)

• Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture
• The images of multilinear polynomials evaluated on $3\times 3$ matrices
• The images of Lie polynomials evaluated on matrices
• Polynomials of small degree evaluated on matrices
• Images of multilinear polynomials on $n\times n$ upper triangular matrices over infinite fields
• On Multilinear Polynomials In Four Variables Evaluated On Matrices
• Images of multilinear polynomials of degree up to four on upper triangular matrices
• Multilinear Polynomials Are Surjective on Algebras With Surjective Inner Derivations
• Images of Multilinear Polynomials in the Algebra of Finitary Matrices Contain Trace Zero Matrices

Cited by in corpus (2)

• A new approach to the Lvov-Kaplansky conjecture through gradings
• The L'vov-Kaplansky Conjecture for Polynomials of Degree Three