Turkish Journal of Mathematics
DOI
10.55730/1300-0098.3232
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 evidence which allows us to state the Mesyan conjecture for the new bound $n \geq \frac{m+1}{2}$, which cannot be improved.
Keywords
Images of polynomials, Lvov-Kaplansky conjecture, Mesyan conjecture, polynomial identities, central polynomials
First Page
1794
Last Page
1808
Recommended Citation
FAGUNDES, PEDRO SOUZA; MELLO, THIAGO CASTILHO DE; and SANTOS, PEDRO HENRIQUE DA SILVA DOS
(2022)
"On the Mesyan conjecture,"
Turkish Journal of Mathematics: Vol. 46:
No.
5, Article 11.
https://doi.org/10.55730/1300-0098.3232
Available at:
https://journals.tubitak.gov.tr/math/vol46/iss5/11
