标题： 结合代数的三重运算恒等式 显示英文标题

标题： Operator identities of multiplicity 3 for associative algebras

摘要： 我们考虑结合代数上线性算子的代数恒等式，其中每个项的次数为2（变量的数量）且乘数为3（算子出现的次数）。 我们应用作者与Elgendy先前工作中使用的方法，该方法对次数为2、乘数为1和2的算子恒等式进行了分类。 我们从乘数为3的一般算子恒等式开始，该恒等式有10个项和不确定的系数。 我们使用操作概念的部分组合来生成该恒等式在次数为3、乘数为4时的所有结果。 这些结果的系数矩阵大小为$105 \times 20$，且其元素为不确定的。 我们计算该矩阵的部分史密斯形式，并利用行列式理想上的格罗布纳基来发现哪些不确定量的取值会使得矩阵的秩为次最大。 可能的次最大秩值只有16和19：有6个秩为16的新恒等式，以及8个秩为19的新恒等式。 显示英文摘要

摘要： We consider algebraic identities for linear operators on associative algebras in which each term has degree 2 (the number of variables) and multiplicity 3 (the number of occurrences of the operator). We apply the methods of earlier work by the author and Elgendy which classified operator identities of degree 2, multiplicities 1 and 2. We begin with the general operator identity of multiplicity 3 which has 10 terms and indeterminate coefficients. We use the operadic concept of partial composition to generate all consequences of this identity in degree 3, multiplicity 4. The coefficient matrix of these consequences has size $105 \times 20$ and indeterminate entries. We compute the partial Smith form of this matrix and use Gr\"obner bases for determinantal ideals to discover which values of the indeterminates produce a matrix of submaximal rank. The only possible submaximal values of the rank are 16 and 19: there are 6 new identities of rank 16, and 8 new identities of rank 19.