非交换多项式在$2\times 2$矩阵上的象
The images of non-commutative polynomials evaluated on $2\times 2$ matrices
摘要：
设$p$是具有任意特征的二次闭域$K$中系数的多个非交换变量的多线性多项式.已经证明，对于任何$n$，$p$在$n$的集合$M_n(K)$上由$n$矩阵求值的像或者是零，或者是标量矩阵的集合，或者是迹0的矩阵的集合$sl_n(K)$，或者是$M_n(K)$的全部。证明了对$n=2$的猜想.
英文摘要：
Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field $K$ of any characteristic. It has been conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set $sl_n(K)$ of matrices of trace 0, or all of $M_n(K)$. We prove the conjecture for $n=2$.
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。
