To Chang theorem. II

S. Yu. Antonov ; A. V. Antonova

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 2, pp. 127-137

Résumé

Multilinear polynomials $\mathcal{ H}^+(\bar x, \bar y \vert \bar w)$, $\mathcal{ H}^-(\bar x, \bar y \vert \bar w)\in F\{X\cup Y\}$, the sum of which is a polynomial $\mathcal{ H}(\bar x, \bar y \vert \bar w)$ Chang (where $F\{X\cup Y\}$ is a free associative algebra over an arbitrary field $F$ of characteristic not equal two, generated by a countable set $X\cup Y$) have been introduced in this paper. It has been proved that each of them is a consequence of the standard polynomial $S^-(\bar x)$. In particular it has been shown that the Capelli quasi-polynomials $b_{2m-1}(\bar x_m, \bar y)$ and $h_{2m-1}(\bar x_m, \bar y)$ are also consequences of the polynomial $S^-_m(\bar x)$. The minimal degree of the polynomials $b_{2m-1}(\bar x_m, \bar y)$, $h_{2m-1}(\bar x_m, \bar y)$ in which they are a polynomial identity of matrix algebra $M_n(F)$ has been also found in the paper. The obtained results are the translation of Chang results to some Capelli quasi-polynomials of odd degree.