Commutative monoid of self-dual symmetric polynomials

Abstract: We consider a set ${\mathfrak R}{\mathfrak S}\left(\lambda,S_n\right)$ of self- and skew-reciprocal polynomials in $\lambda$, of degree $mn$, where $m\in{ \mathbb Z}_{\geq}$, $n\in{\mathbb Z}_>$, based on polynomial invariants $I_{n, r}({\bf x}^n)$ of symmetric group $S_n$, acting on the Euclidean space ${\mathbb E}^n$ over the field of real numbers ${\mathbb R}$, where ${\bf x^n}=\{x_1, \ldots,x_n\}\in{\mathbb E}^n$. We prove that ${\mathfrak R}{\mathfrak S}\left( \lambda,S_n\right)$ exhibits a commutative monoid under multiplication. Real solutions $\lambda\left({\bf x^n}\right)$ of skew-reciprocal equations have many remarkable properties: a homogeneity of the 1st order, a duality under inversion of variables $x_i\to x_i^{-1}$ and function $\lambda\to\lambda^{-1}$, a monotony of $\lambda\left({\bf x^n}\right)$ with respect to every $x_i$ and others. We find the bounds of $\lambda\left({\bf x^n}\right)$ which are given by arithmetic and harmonic means of the set $\{x_1,\ldots,x_n\}$.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)