Mesyan猜想的重述与修正
The Mesyan Conjecture: a restatement and a correction
摘要：
著名的Lvov-Kaplansky猜想指出，在$n\times n$矩阵上求值的多线性多项式$f$的像是向量空间。这个猜想的一个较弱的版本，被称为梅西亚猜想，指出如果$m=deg( f)$和$n\geq m-1$，那么它的图像包含迹零矩阵的集合。这种猜想已被证明为多项式的次数$m \leq 4$。$m=4$的证明在其中一个引理中包含错误。本文修正了该引理的证明，并给出了一些证据，使我们能够陈述关于新界$n \geq \frac{m+1}{2}$的Mesyan猜想，该猜想无法改进。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
