Computations modulo regular chains

TL;DR

This paper introduces new algorithms for polynomial GCDs and regularity tests modulo regular chains, leveraging modular methods and fast arithmetic, resulting in significantly faster computations compared to existing software.

Contribution

The paper presents novel algorithms for core operations in triangular decompositions, improving efficiency through modular methods and optimized polynomial arithmetic.

Findings

New algorithms outperform existing implementations by several orders of magnitude.

Extensive experiments demonstrate significant speedups in polynomial computations.

Code comparisons show superior performance over Maple and Magma functions.

Abstract

The computation of triangular decompositions are based on two fundamental operations: polynomial GCDs modulo regular chains and regularity test modulo saturated ideals. We propose new algorithms for these core operations relying on modular methods and fast polynomial arithmetic. Our strategies take also advantage of the context in which these operations are performed. We report on extensive experimentation, comparing our code to pre-existing Maple implementations, as well as more optimized Magma functions. In most cases, our new code outperforms the other packages by several orders of magnitude.