Abstract:
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.
Key words:$T$-ideal, standard polynomial, Capelli polynomial.