Symmetric *-polynomials on C^n - Vasylyshyn, T.V. (Carpathian Mathematical Publications, 2018)
*-Polynomials are natural generalizations of usual polynomials between complex vector spaces. A *-polynomial is a function between complex vector spaces X and Y, which is a sum of so-called (p,q)-polynomials. In turn, for nonnegative integers p and q, a (p,q)-polynomial is a function between X and Y, which is the restriction to the diagonal of some mapping, acting from the Cartesian power X^p+q to Y, which is linear with respect to every of its first p arguments, antilinear with respect to every of its last q arguments and invariant with respect to permutations of its first p arguments and last q arguments separately. In this work we construct formulas for recovering of (p,q)-polynomial components of *-polynomials, acting between complex vector spaces X and Y, by the values of *-polynomials. We use these formulas for investigations of *-polynomials, acting from the n-dimensional complex vector space C^n to C, which are symmetric, that is, invariant with respect to permutations of coordinates of its argument. We show that every symmetric *-polynomial, acting from C^n to C, can be represented as an algebraic combination of some "elementary" symmetric *-polynomials. Results of the paper can be used for investigations of algebras, generated by symmetric *-polynomials, acting from C^n to C.