论文标题
$ 3 \ times 3 $类似矩阵的无效矩阵的零件理想
Null ideals of sets of $3 \times 3$ similar matrices with irreducible characteristic polynomial
论文作者
论文摘要
让$ f $为一个字段，$ m_n（f）$ $ n \ times n $矩阵的戒指超过$ f $。给定一个$ m_n（f）$的子集$ s $，$ s $的null理想是所有多项式$ f $的集合，其系数为$ m_n（f）$，因此$ f（a）= 0 $ for s $ in s $中的所有$ a \。我们说，如果$ s $的无效理想是多项式环$ m_n（f）[x] $的双面理想，那么$ s $是核心。我们研究了足够的条件，在$ S $由$ 3 \ times 3 $矩阵组成的情况下，$ s $是核心，所有这些条件共享相同的不可约性多项式。特别是，我们表明，如果$ f $是有限的，$ q $元素和$ | s | \ geqslant q^3-q^2+1 $，然后$ s $是核心。作为我们工作的副产品，我们可以在vandermonde矩阵，可逆矩阵换向器以及通过可逆差异关系定义的图形上获得一些结果。
Let $F$ be a field and $M_n(F)$ the ring of $n \times n$ matrices over $F$. Given a subset $S$ of $M_n(F)$, the null ideal of $S$ is the set of all polynomials $f$ with coefficients from $M_n(F)$ such that $f(A) = 0$ for all $A \in S$. We say that $S$ is core if the null ideal of $S$ is a two-sided ideal of the polynomial ring $M_n(F)[x]$. We study sufficient conditions under which $S$ is core in the case where $S$ consists of $3 \times 3$ matrices, all of which share the same irreducible characteristic polynomial. In particular, we show that if $F$ is finite with $q$ elements and $|S| \geqslant q^3-q^2+1$, then $S$ is core. As a byproduct of our work, we obtain some results on block Vandermonde matrices, invertible matrix commutators, and graphs defined via an invertible difference relation.
