An Algorithm for Discriminating the Complete Multiplicities of a Parametric Univariate Polynomial
Simin Qin, Bican Xia, Jing Yang
TL;DR
This work tackles the parametric complete multiplicity problem for univariate polynomials by replacing the classical repeated gcd approach with incremental gcds built from high-order derivatives and non-nested subresultants. The authors establish an icgcd framework showing that incremental gcds $G_i$ correspond to specific subresultants $R_{(\bar{\mu}_1,\ldots,\bar{\mu}_i)}$ of $(P^{(0)},P^{(1)},\ldots)$, enabling a compact expression of multiplicity structure via root data. They formulate an algorithm ParametricCompleteMultiplicity that combines icgcd with non-nested discriminants and discriminant sequences to produce necessary and sufficient conditions for every possible complete multiplicity $\boldsymbol{\mu_c}$, and they demonstrate substantial reductions in both the number and degree of polynomials in the output relative to the classical YHZ method. Empirical results show that the new approach scales better and is faster on larger problems, with notable improvements over YHZ on degrees where the old method struggles. The work also highlights underlying connections between incremental gcds and pseudo-remainders, pointing to potential non-nested determinant techniques and deeper structural insights for future efficiency gains.
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.