论文标题
上三角矩阵代数上的多项式和警告类型问题的图像
The image of polynomials and Waring type problems on upper triangular matrix algebras
论文作者
论文摘要
令$ p $为非交易变量的多项式$ x_1，x_2，\ ldots，x_n $，在代数封闭的字段$ k $上，恒定项为零。本文研究的对象是上三角矩阵代数上的多项式图像$ t_m（k）$。我们介绍了一个多项式家庭，称为多项式多项式$ P $ p $ p $。使用这个家庭，我们将表明，如果$ p $是$ t_t（k）$的多项式身份，而不是$ t_ {t+1}（k）$的多项式身份，则是$ p \ left（t_m（k）\ right）\ subseteq t_m（k）^m（k）^{（t-1）} $。在$ t = 1，〜m-1 $的情况下达到平等，并提供了一个示例，以表明平等总体上不能保持。我们进一步证明了$ d $的存在，使得$ t_m（k）^{（t-1）} $的每个元素可以写成$ d $ $ d $的总和$ p \ left（t_m（k）\ right）$。还显示，单词映射下的$ t_m（k）^\ times $的图像是zariski密度在$ t_m（k）^\ times $中。
Let $p$ be a polynomial in non-commutative variables $x_1,x_2,\ldots,x_n$ with constant term zero over an algebraically closed field $K$. The object of study in this paper is the image of this kind of polynomial over the algebra of upper triangular matrices $T_m(K)$. We introduce a family of polynomials called multi-index $p$-inductive polynomials for a given polynomial $p$. Using this family we will show that, if $p$ is a polynomial identity of $T_t(K)$ but not of $T_{t+1}(K)$, then $p \left(T_m(K)\right)\subseteq T_m(K)^{(t-1)}$. Equality is achieved in the case $t=1,~m-1$ and an example has been provided to show that equality does not hold in general. We further prove existence of $d$ such that each element of $T_m(K)^{(t-1)}$ can be written as sum of $d$ many elements of $p\left( T_m(K) \right)$. It has also been shown that the image of $T_m(K)^\times$ under a word map is Zariski dense in $T_m(K)^\times$.
