A Condition for Multiplicity Structure of Univariate Polynomials

TL;DR

This paper introduces a new, more efficient condition for determining the multiplicity structure of univariate polynomials with a fixed number of distinct roots, improving upon previous gcd-based methods.

Contribution

It presents a novel condition that does not rely on repeated gcd's, with fewer and lower-degree polynomials, and proves its optimality.

Findings

New condition avoids repeated gcd's

Fewer and lower-degree polynomials in the condition

Proven optimality of the number of polynomials

Abstract

We consider the problem of finding a condition for a univariate polynomial having a given multiplicity structure when the number of distinct roots is given. It is well known that such conditions can be written as conjunctions of several polynomial equations and one inequation in the coefficients, by using repeated parametric gcd's. In this paper, we give a novel condition which is not based on repeated gcd's. Furthermore, it is shown that the number of polynomials in the condition is optimal and the degree of polynomials is smaller than that in the previous condition based on repeated gcd's.