The disconnectedness of certain sets defined after uni-variate polynomials

TL;DR

This paper investigates the topological structure of specific sets of univariate polynomials with fixed coefficient signs and root counts, revealing disconnectedness in certain cases and providing comprehensive existence criteria.

Contribution

It proves the disconnectedness of a particular polynomial set for degree 6 with specified root configuration and characterizes the existence of polynomials with given root counts for various signs and degrees.

Findings

The set of degree 6 polynomials with specified roots is not connected.

Complete characterization of when such polynomials exist for given root counts and coefficient signs.

Provides an exhaustive answer for cases with total root count up to 3.

Abstract

We consider the set of monic real univariate polynomials of a given degree $d$ with non-vanishing coefficients, with given signs of the coefficients and with given quantities $pos$ of their positive and $neg$ of their negative roots (all roots are distinct). For $d=6$ and for signs of the coefficients $(+,-,+,+,+,-,+)$, we prove that the set of such polynomials having two positive, two negative and two complex conjugate roots, is not connected. For $pos+neg\leq 3$ and for any $d$, we give the exhaustive answer to the question for which signs of the coefficients there exist polynomials with such values of $pos$ and $neg$.