Stably hyperbolic polynomials

Abstract

A real polynomial in one real variable is hyperbolic if all its roots are real. Denote the set of monic hyperbolic polynomials of degree $n$ by $\prod_n$. Suppose that for a real polynomial $P(x)$ of degree $n$ there exists $k \in \mathbf{N}$ and a polynomial $Q(x)$ of $\mathrm{degree} \le k-1$ such that $x^k P+Q \in \prod_{n+k}$. Denote the set of such polynomials $P$ by $\prod_n (k)$. Call the set $\prod_n (\infty) = \overline{\cup_{k=0}^{\infty} \prod_n (k)}$ the domain of stably hyperbolic polynomials of degree $n$. In the present paper we explore the geometric properties of the set $\prod_4 (\infty)$.