The idealizer of the set of quasi-stable polynomials

TL;DR

This paper investigates the structure of the set of stable polynomials under Hadamard product, proposing a conjecture for its idealizer and proving it for degrees up to five, with partial results supporting the conjecture.

Contribution

It formulates a conjecture characterizing the idealizer of stable polynomials and proves it for degrees up to five, advancing understanding of polynomial stability structures.

Findings

Conjecture characterizes the idealizer of stable polynomials.

Proved the conjecture for degrees n ≤ 5.

Established necessary and sufficient conditions supporting the conjecture.

Abstract

It follows from the Garloff-Wagner Theorem that the set of stable polynomials of degree $n$, denoted by $\mathcal{H}_n$, i.e., those whose zeros all lie in the open left complex half-plane, with the Hadamard product $*$, forms an abelian semigroup contained in the abelian group $\mathbb{R}_n^+$ of polynomials of degree $n$ with positive real coefficients. By the idealizer of the set $\mathcal{H}_n$, we refer to the largest subsemigroup of $\mathbb{R}_n^+$ in which $\mathcal{H}_n$ is an ideal. In this paper, we formulate a conjecture characterizing the idealizer of $\mathcal{H}_n$ and prove it for $n \leqslant 5$. In addition, we show that the proposed condition is necessary for any polynomial to belong to the idealizer and establish, in a distinguished special case, a sufficient condition of a similar nature that supports the conjecture.