Parametric "Non-nested" Discriminants for Multiplicities of Univariate Polynomials

TL;DR

This paper introduces a new class of non-nested discriminants for univariate polynomials that simplify root multiplicity classification by reducing complexity and degree compared to traditional nested determinants.

Contribution

It proposes a novel approach to discriminants that are non-nested determinants, offering a simpler and lower-degree alternative to existing methods based on repeated gcds.

Findings

New non-nested discriminants are simpler to compute.

Discriminants have smaller maximum degrees.

Method improves root multiplicity classification efficiency.

Abstract

We consider the problem of complex root classification, i.e., finding the conditions on the coefficients of a univariate polynomial for all possible multiplicity structures on its complex roots. It is well known that such conditions can be written as conjunctions of several polynomial equations and one inequation in the coefficients. Those polynomials in the coefficients are called discriminants for multiplicities. It is well known that discriminants can be obtained by using repeated parametric gcd's. The resulting discriminants are usually nested determinants, that is, determinants of matrices whose entries are determinants, and so son. In this paper, we give a new type of discriminants which are not based on repeated gcd's. The new discriminants are simpler in that they are non-nested determinants and have smaller maximum degrees.