Symmetric $*$-polynomials on $\mathbb C^n$

$*$-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 $\mathbb{C}^n$ to $\mathbb{C},$ which are symmetric, that is, invariant with respect to permutations of coordinates of its argument. We show that every symmetric $*$-polynomial, acting from $\mathbb{C}^n$ to $\mathbb{C},$ 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 $\mathbb{C}^n$ to $\mathbb{C}.$