Sparse FGLM algorithms

Sparse FGLM algorithms

Given a zero-dimensional ideal I ⊂ K[x1, . . . ,xn] of degree D, the transformation of the ordering of its Gröbner basis from DRL to LEX is a key step in polynomial system solving and turns out to be the bottleneck of the whole solving process. Thus it is of crucial importance to design efficient algorithms to perform the change of ordering. The main contributions of this paper are several effi...

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

Consider a zero-dimensional ideal I in K[X1, . . . ,Xn]. 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 para...

FGLM and coprime polynomial pairs

To obtain coprime polynomial pairs, it is often used an approach that consists of taking two random polynomials of the required degrees and testing whether they are relatively prime. If they are not, they are discarded and another pair is tried. The test can obviously be carried out very efficiently. Although in many cases the probability of success for the test is close to 1, this test approac...

Probabilistic Algorithms Sparse Polynomials

Group Sparse RLS Algorithms

Group sparsity is one of the important signal priors for regularization of inverse problems. Sparsity with group structure is encountered in numerous applications. However, despite the abundance of sparsity based adaptive algorithms, attempts at group sparse adaptive methods are very scarce. In this paper we introduce novel Recursive Least Squares (RLS) adaptive algorithms regularized via penal...