Geodesic
Quasi-polynomials of Capelli
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 4, pp. 371-382.

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This paper deals with the class of Capelli polynomials in free associative algebra $F\{Z\}$ where $F$ is an arbitrary field and $Z$ is a countable set. The interest to these objects is initiated by assumption that the polynomials (Capelli quasi-polynomials) of some odd degree introduced will be contained in the basis ideal $Z_2$-graded identities of $Z_2$-graded matrix algebra $M^{(m,k)}(F)$ when $\mathrm{char}\,F=0$. In connection with this assumption the fundamental properties of Capelli quasi-polynomials have been given in the paper. In particularly, the decomposition of Capelli type polynomials have been given by the polynomials of the same type and some betweeness of their $T$-ideals have been shown. Besides, taking into account some properties of Capelli quasi-polynomials obtained and also the Chang theorem we show that all Capelli quasi-polynomials of even degree $2n$ $(n>1)$ are consequence of standard polynomial $S_n^-$ in case when the characteristic of field $F$ is not equal to two. At last we find the least $n \in N$ at which any of Capelli quasi-polynomials of even degree $2n$ belongs to ideal of matrix algebra $M_m(F)$ identities. 
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S. Yu. Antonov; A. V. Antonova. Quasi-polynomials of Capelli. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 4, pp. 371-382. http://geodesic.mathdoc.fr/item/ISU_2015_15_4_a1/

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