Degree 5 polynomials and Descartes' rule of signs

TL;DR

This paper investigates the realizability of root configurations of degree 5 polynomials with specific sign patterns, explaining non-existence cases using geometric representations of discriminant sets.

Contribution

It provides a detailed geometric analysis of the discriminant set for degree 5 polynomials, clarifying which root and sign configurations are possible or impossible.

Findings

Certain root-sign configurations are impossible for degree 5 polynomials.

Geometric visualization of the discriminant set explains non-existence cases.

All other configurations compatible with Descartes' rule are realizable.

Abstract

For a univariate real polynomial without zero coefficients, Descartes' rule of signs (completed by an observation of Fourier) says that its numbers $pos$ of positive and $neg$ of negative roots (counted with multiplicity) are majorized respectively by the numbers $c$ and $p$ of sign changes and sign preservartions in the sequence of its coefficients, and that the differences $c-pos$ and $p-neg$ are even numbers. For degree 5 polynomials, it has been proved by A.~Albouy and Y.~Fu that there exist no such polynomials having three distinct positive and no negative roots and whose signs of the coefficients are $(+,+,-,+,-,-)$ (or having three distinct negative and no positive roots and whose signs of the coefficients are $(+,-,-,-,-,+)$). For degree 5 and when the leading coefficient is positive, these are all cases of numbers of positive and negative roots (all distinct) and signs of the coefficients which are compatible with Descartes' rule of signs, but for which there exist no such polynomials. We explain this non-existence and the existence in all other cases with $d=5$ by means of pictures showing the discriminant set of the family of polynomials $x^5+x^4+ax^3+bx^2+cx+d$ together with the coordinate axes.