矩阵环的欧拉恒等式与标准多项式恒等式

摘要

Szigeti-Tuza和Revesz使用Swan图论定理构造了 n× n矩阵环M n （C）的欧拉恒等式[1]。本文中证明这些恒等式可由标准多项式生成，即：若欧拉图Γp ，q从某顶点t到u（t ，u可为同一点）至少有 n条边，则该欧拉图对应的欧拉多项式 fΓp ，q （X）可由标准多项式 Sn （X）生成。该结果不仅推广了Chang[2]和Giambruno-Sehal[3]的结果，而且找到由欧拉恒等式生成的 T-理想的一个有限生成集。%Using Swan's graph theoretic theorem ,Szigeti-Tuza and Revesz constructed Eulerian identities of the n × n matrix ring Mn (C) [1] .In this paper ,we showed that those identities could be generated by standard polynomical identities ,i .e . If the Eulerian graph Γp ,q had at least n edges starting at some vertex t and ending at some vertex u(t= u was allowed) ,then Eulerian polynomial fΓp ,q (X) can be generated by Sn (X) .By this result ,the generalization of the results of chang [2] and Giambruno-Sehgal[3] were obtained ,and also a finite generating set of the T-ideal generated by the Eulerian identities was found .