An Algorithm for Discriminating the Complete Multiplicities of a Parametric Univariate Polynomial

TL;DR

This paper presents a new algorithm for determining the complete multiplicities of roots in a parametric univariate polynomial, improving efficiency and simplicity over classical gcd-based methods.

Contribution

The authors introduce a novel incremental gcd approach using non-nested subresultants, reducing complexity and computational time compared to traditional methods.

Findings

Algorithm is faster for large problems

Conditions produced are simpler than classical methods

Effective in handling parametric polynomial multiplicities

Abstract

In this paper, we tackle the parametric complete multiplicity problem for a univariate polynomial. Our approach to the parametric complete multiplicity problem has a significant difference from the classical method, which relies on repeated gcd computation. Instead, we introduce a novel technique that uses incremental gcds of the given polynomial and its high-order derivatives. This approach, formulated as non-nested subresultants, sidesteps the exponential expansion of polynomial degrees in the generated condition. We also uncover the hidden structure between the incremental gcds and pseudo-remainders. Our analysis reveals that the conditions produced by our new algorithm are simpler than those generated by the classical approach in most cases. Experiments show that our algorithm is faster than the one based on repeated gcd computation for problems with relatively big size.