Multiplicity Preserving Triangular Set Decomposition of Two Polynomials

TL;DR

This paper introduces a novel algorithm for decomposing systems of two polynomials into triangular sets that preserve multiplicities, with a focus on bivariate cases, supported by complexity analysis and experimental validation.

Contribution

It presents a new multiplicity preserving triangular set decomposition algorithm specifically for two polynomials, including a complete method for bivariate systems with positive multiplicities.

Findings

Effective decomposition into triangular sets demonstrated

Algorithm preserves multiplicities accurately

Complexity analysis confirms efficiency in bivariate case

Abstract

In this paper, a multiplicity preserving triangular set decomposition algorithm is proposed for a system of two polynomials. The algorithm decomposes the variety defined by the polynomial system into unmixed components represented by triangular sets, which may have negative multiplicities. In the bivariate case, we give a complete algorithm to decompose the system into multiplicity preserving triangular sets with positive multiplicities. We also analyze the complexity of the algorithm in the bivariate case. We implement our algorithm and show the effectiveness of the method with extensive experiments.