https://hal-unilim.archives-ouvertes.fr/hal-01661690v2Hyun, Seung GyuSeung GyuHyunDavid R. Cheriton School of Computer Science - University of Waterloo [Waterloo]Neiger, VincentVincentNeigerXLIM-MATHIS - Mathématiques & Sécurité de l'information - XLIM - XLIM - UNILIM - Université de Limoges - CNRS - Centre National de la Recherche ScientifiqueRahkooy, HamidHamidRahkooyDavid R. Cheriton School of Computer Science - University of Waterloo [Waterloo]Schost, ÉricÉricSchostDavid R. Cheriton School of Computer Science - University of Waterloo [Waterloo]Block-Krylov techniques in the context of sparse-FGLM algorithmsHAL CCSD2019Polynomial systemsBlock-Krylov algorithmsSparse FGLM[INFO.INFO-SC] Computer Science [cs]/Symbolic Computation [cs.SC]Neiger, Vincent2019-01-15 13:23:452022-06-26 02:36:392019-01-23 16:38:16enPreprints, Working Papers, ...https://hal-unilim.archives-ouvertes.fr/hal-01661690v1application/pdf2Consider a zero-dimensional ideal $I$ in $\mathbb{K}[X_1,\dots,X_n]$. Inspired by Faugère 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 LinBox, Eigen and NTL.