Block-Krylov techniques in the context of sparse-FGLM algorithms

TL;DR

This paper presents a new method using Block-Krylov techniques to efficiently compute the zero set of zero-dimensional ideals in polynomial rings, improving upon existing algorithms by reducing assumptions and enhancing parallelization.

Contribution

It introduces a novel approach combining Krylov sequences and generating series to compute entire solutions of zero-dimensional ideals without relying on generic linear forms.

Findings

Efficient computation of zero set descriptions with small overhead.

Implementation demonstrates practical viability using C++ libraries.

Method generalizes previous algorithms to broader classes of ideals.

Abstract

Consider a zero-dimensional ideal $I$ in $\mathbb{K}[X_1,\dots,X_n]$. Inspired by Faug\`ere and Mou's Sparse FGLM algorithm, we use Krylov sequences based on multiplication matrices of $I$ in order to compute a description of its zero set by means of univariate polynomials. Steel recently showed how to use Coppersmith's block-Wiedemann algorithm in this context; he describes an algorithm that can be easily parallelized, but only computes parts of the output in this manner. Using generating series expressions going back to work of Bostan, Salvy, and Schost, we show how to compute the entire output for a small overhead, without making any assumption on the ideal $I$ other than it having dimension zero. We then propose a refinement of this idea that partially avoids the introduction of a generic linear form. We comment on experimental results obtained by an implementation based on the C++ libraries Eigen, LinBox and NTL.