Symmetric ∗∗-polynomials on Cn
Keywords:
(p,q)-polynomial, ∗-polynomial, symmetric ∗-polynomialAbstract
∗-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 Xp+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 Cn 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 Cn 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 Cn to C.