Geodesic
To Chang theorem. II
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 2, pp. 127-137 
S. Yu. Antonov; A. V. Antonova. To Chang theorem. II. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 17 (2017) no. 2, pp. 127-137. http://geodesic.mathdoc.fr/item/ISU_2017_17_2_a0/
@article{ISU_2017_17_2_a0,
     author = {S. Yu. Antonov and A. V. Antonova},
     title = {To {Chang} theorem. {II}},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {127--137},
     year = {2017},
     volume = {17},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2017_17_2_a0/}
}
TY  - JOUR
AU  - S. Yu. Antonov
AU  - A. V. Antonova
TI  - To Chang theorem. II
JO  - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
PY  - 2017
SP  - 127
EP  - 137
VL  - 17
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/ISU_2017_17_2_a0/
LA  - ru
ID  - ISU_2017_17_2_a0
ER  - 
%0 Journal Article
%A S. Yu. Antonov
%A A. V. Antonova
%T To Chang theorem. II
%J Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
%D 2017
%P 127-137
%V 17
%N 2
%U http://geodesic.mathdoc.fr/item/ISU_2017_17_2_a0/
%G ru
%F ISU_2017_17_2_a0

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Chang Q., “Some consequences of the standard polynomial”, Proc. Amer. Math. Soc., 104:3 (1988), 707–710 | DOI | MR | Zbl

[2] Antonov S. Yu., Antonova A. V., “To Chang theorem”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 15:3 (2015), 247–251 (in Russian) | DOI | Zbl

[3] Gateva T. V., “The complexity of a bundle of varieties of associative algebras”, Russian Math. Surveys, 36:1 (1981), 233 | DOI | MR | Zbl

[4] Kemer A. R., “Remark on the standard identity”, Math. Notes, 23:5 (1978), 414–416 | DOI | MR | Zbl

[5] Benanti F., Drensky V., “On the consequences of the standard polynomial”, Comm. Algebra, 26 (1998), 4243–4275 | DOI | MR | Zbl

[6] Leron U., “Multilinear identities of the matrix ring”, Trans. Amer. Math. Soc., 183 (1973), 175–202 | DOI | MR | Zbl

[7] Amitsur S. A., Levitzki J., “Minimal identities for algebras”, Proc. Amer. Math. Soc., 1:4 (1950), 449–463 | DOI | MR | Zbl

[8] Owens F. W., “Applications of graph theory to matrix theory”, Amer. Math. Soc., 51:1 (1975), 242–249 | DOI | MR | Zbl

[9] Rosset S., “A new proof of the Amitsur–Levitzki identity”, Israel J. Math., 23 (1976), 187–188 | DOI | MR | Zbl

[10] Szigeti J., Tuza Z., Revesz G., “Eulerian polynomial identities on matrix rings”, J. of Algebra, 161:1 (1993), 90–101 | DOI | MR | Zbl

[11] Lee A., Revesz G., Szigeti J., Tuza Z., “Capelli polynomials, almost-permutation matrices and sparse Eulerian graphs”, Discrete Math., 230:1–3 (2001), 49–61 | DOI | MR | Zbl

[12] Antonov S. Yu., “The least degree identities subspace $M_1^{(m, k)}(F)$ of matrix superalgebra $ M ^ {(m, k)} (F)$”, Russian Math. (Iz. VUZ), 56:11 (2012), 1–16 | DOI | MR | Zbl