Algorithms for Computing Triangular Decompositions of Polynomial Systems

TL;DR

This paper introduces optimized algorithms for triangular decomposition of polynomial systems that significantly outperform existing solvers, leveraging a weakened GCD notion and shared computations.

Contribution

The paper presents novel incremental algorithms for polynomial system decomposition using a weakened GCD concept, improving efficiency and scalability.

Findings

Implementation outperforms similar solvers by orders of magnitude.

Algorithms effectively simplify and optimize sub-computations.

Experimental results demonstrate significant performance gains.

Abstract

We propose new algorithms for computing triangular decompositions of polynomial systems incrementally. With respect to previous works, our improvements are based on a {\em weakened} notion of a polynomial GCD modulo a regular chain, which permits to greatly simplify and optimize the sub-algorithms. Extracting common work from similar expensive computations is also a key feature of our algorithms. In our experimental results the implementation of our new algorithms, realized with the {\RegularChains} library in {\Maple}, outperforms solvers with similar specifications by several orders of magnitude on sufficiently difficult problems.