Root Configurations of Real Univariate Cubics and Quartics

TL;DR

This paper derives simple polynomial conditions from Sturm sequences to determine the root multiplicities and orderings of real and complex roots in general monic cubic and quartic polynomials with real coefficients.

Contribution

It provides a straightforward method to identify root configurations of real univariate cubics and quartics using Sturm sequences.

Findings

Polynomial conditions for root multiplicities are explicitly derived.

Conditions determine the order of real roots and their multiplicities.

Method simplifies root configuration analysis for cubics and quartics.

Abstract

For the general monic cubic and quartic with real coefficients, polynomial conditions on the coefficients are derived as directly and as simply as possible from the Sturm sequence that will determine the real and complex root multiplicities together with the order of the real roots with respect to multiplicity.