#### For citation:

Antonov S. Y., Antonova A. V. To Chang Theorem. II. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2017, vol. 17, iss. 2, pp. 127-137. DOI: 10.18500/1816-9791-2017-17-2-127-137

# To Chang Theorem. II

Multilinear polynomials H +(¯x, ¯y| ¯ w), H −(¯x, ¯y| ¯ w) ∈ F{X ∪ Y }, the sum of which is a polynomial H (¯x, ¯y| ¯ w) Chang (where F{X∪Y } is a free associative algebra over an arbitrary field F of characteristic not equal two, generated by a countable set X ∪ Y ) have been introduced in this paper. It has been proved that each of them is a consequence of the standard polynomial S−(¯x). In particular it has been shown that the Capelli quasi-polynomials b2m−1(¯xm, ¯y) and h2m−1(¯xm, ¯y) are also consequences of the polynomial S−m (¯x). The minimal degree of the polynomials b2m−1(¯xm, ¯y), h2m−1(¯xm, ¯y) in which they are a polynomial identity of matrix algebra Mn(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.

- Chang Q. Some consequences of the standard polynomial. Proc. Amer. Math. Soc., 1988, vol. 104, no. 3, pp. 707–710. DOI: https://doi.org/10.2307/2046778.
- Antonov S. Yu., Antonova A. V. To Chang theorem. Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2015, vol. 15, iss. 3, pp. 247–251 (in Russian). DOI: https://doi.org/10.18500/1816-9791-2015-15-3-247-251.
- Gateva T. V. The complexity of a bundle of varieties of associative algebras. Russian Math. Surveys, 1981, vol. 36, no. 1, pp. 233. DOI: https://doi.org/10.1070/RM1981v036n01ABEH002548.
- Kemer A. R. Remark on the standard identity. Math. Notes, 1978, vol. 23, no. 5, pp. 414–416. DOI: https://doi.org/10.1007/BF01789011.
- Benanti F., Drensky V. On the consequences of the standard polynomial. Comm. Algebra, 1998, vol. 26, pp. 4243–4275.
- Leron U. Multilinear identities of the matrix ring. Trans. Amer. Math. Soc., 1973. vol. 183, pp. 175–202.
- Amitsur S. A., Levitzki J. Minimal identities for algebras. Proc. Amer. Math. Soc., 1950, vol. 1, no. 4, pp. 449–463.
- Owens F. W. Applications of graph theory to matrix theory. Amer. Math. Soc., 1975, vol. 51, no. 1, pp. 242–249.
- Rosset S. A new proof of the Amitsur–Levitzki identity. Israel J. Math., 1976, vol. 23, pp. 187–188.
- Szigeti J., Tuza Z., Revesz G. Eulerian polynomial identities on matrix rings. J. of Algebra, 1993, vol. 161, iss. 1, pp. 90–101. DOI: https://doi.org/10.1006/jabr.1993.1207.
- Lee A., Revesz G., Szigeti J., Tuza Z. Capelli polynomials, almost-permutation matrices and sparse Eulerian graphs. Descrete Math., 2001, vol. 230, no. 1–3, pp. 49–61.
- Antonov S. Yu. The least degree identities subspace M1(m,k)(F) of matrix superalgebra M(m,k) (F). Russian Math. (Iz. VUZ), 2012, vol. 56, no. 11, pp. 1–16. DOI: https://doi.org/10.3103/S1066369X12110011.

- 312 reads