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Full-Text Articles in Physical Sciences and Mathematics
On The Mesyan Conjecture, Pedro Souza Fagundes, Thiago Castilho De Mello, Pedro Henrique Da Silva Dos Santos
On The Mesyan Conjecture, Pedro Souza Fagundes, Thiago Castilho De Mello, Pedro Henrique Da Silva Dos Santos
Turkish Journal of Mathematics
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 …
On Central Polynomials And Codimension Growth, Fabrizio Martino
On Central Polynomials And Codimension Growth, Fabrizio Martino
Turkish Journal of Mathematics
Let $A$ be an associative algebra over a field of characteristic zero. A central polynomial is a polynomial of the free associative algebra that takes central values of $A.$ In this survey, we present some recent results about the exponential growth of the central codimension sequence and the proper central codimension sequence in the setting of algebras with involution and algebras graded by a finite group.