The main result of this short note is a generic version of Paz's conjectureon the lengths of generating sets in matrix algebras. Consider a genericg-tuple A=(A_1,..., A_g) of nxn matrices over a field. We show that whenever$g^{2d}\geq n^2$, the set of all words of degree 2d in A spans the full nxnmatrix algebra. Our proofs use generic matrices, are combinatorial and dependon the construction of a special kind of directed multigraphs with fewedge-disjoint walks.
展开▼
机译：此简短笔记的主要结果是Paz猜想的通用形式，即矩阵代数中生成集的长度。考虑一个字段上nxn个矩阵的通用元组A =（A_1，...，A_g）。我们证明，每当$ g ^ {2d} \ geq n ^ 2 $时，A中2d阶所有单词的集合就会跨越整个nxnmatrix代数。我们的证明使用通用矩阵，是组合的，并且依赖于特殊的有向多重图的构造，这种有向图具有很少的边不相交的走动。
展开▼
