论文标题
基质代数的分级多项式图像
Images of graded polynomials on matrix algebras
论文作者
论文摘要
本文的目的是开始研究完整矩阵代数上的分级多项式图像。我们使用矩阵代数$ m_n（k）$在字段上$ k $赋予其规范$ \ mathbb {z} _n $ -grading（Vasilovsky的分级）。我们明确地确定了在字段上多项式多项式图像的线性跨度的可能性，并在$ \ mathbb Q $中的理性数字Q $ Q $，并陈述了L'Vov-Kaplansky猜想的类似物，涉及$ n \ n $ ns $ n是$ n是$ n is a A $ ns a a a $ ns p prime prime prime prime prime prime prime。当$ k $是零或大于$ n $的特征零或大于$ n $的特征性封闭的$ m_n（k）$的多项式多项式的这种猜想，而对第2级矩阵的任意程度的多项式。我们还确定了对$ m_2（k）$ m_2（k）的半均匀渐变层次多项式的所有可能图像。
The aim of this paper is to start the study of images of graded polynomials on full matrix algebras. We work with the matrix algebra $M_n(K)$ over a field $K$ endowed with its canonical $\mathbb{Z}_n$-grading (Vasilovsky's grading). We explicitly determine the possibilities for the linear span of the image of a multilinear graded polynomial over the field $\mathbb Q$ of rational numbers and state an analogue of the L'vov-Kaplansky conjecture about images of multilinear graded polynomials on $n\times n$ matrices, where $n$ is a prime number. We confirm such conjecture for polynomials of degree 2 over $M_n(K)$ when $K$ is a quadratically closed field of characteristic zero or greater than $n$ and for polynomials of arbitrary degree over matrices of order 2. We also determine all the possible images of semi-homogeneous graded polynomials evaluated on $M_2(K)$.
