2010/01/22 by Jean‐Charles Faugère, Jean-Charles Faugère, Faugère, Jean-Charles +4
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Commutative Algebra and Its Applications #FOS: Computer and information sciences #Polynomial and algebraic computation #Symbolic Computation (cs.SC) #Tensor decomposition and applications #cs.SC
paper · pdf · doi:10.48550/arxiv.1001.4004
31 pages
openalex publication_date 2010/01/22 · arxiv created 2010/02/24 · arxiv updated 2010/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Solving multihomogeneous systems, as a wide range of structured algebraic systems occurring frequently in practical problems, is of first importance. Experimentally, solving these systems with Gröbner bases algorithms seems to be easier than solving homogeneous systems of the same degree. Nevertheless, the reasons of this behaviour are not clear. In this paper, we focus on bilinear systems (i.e. bihomogeneous systems where all equations have bidegree (1,1)). Our goal is to provide a theoretical explanation of the aforementionned experimental behaviour and to propose new techniques to speed up the Gröbner basis computations by using the multihomogeneous structure of those systems. The contributions are theoretical and practical. First, we adapt the classical F5 criterion to avoid reductions to zero which occur when the input is a set of bilinear polynomials. We also prove an explicit form of the Hilbert series of bihomogeneous ideals generated by generic bilinear polynomials and give a new upper bound on the degree of regularity of generic affine bilinear systems. This leads to new complexity bounds for solving bilinear systems. We propose also a variant of the F5 Algorithm dedicated to multihomogeneous systems which exploits a structural property of the Macaulay matrix which occurs on such inputs. Experimental results show that this variant requires less time and memory than the classical homogeneous F5 Algorithm.