Section: Research Program
Fundamental Algorithms and Structured Systems
Participants : Jérémy Berthomieu, Jean-Charles Faugère, Mohab Safey El Din, Elias Tsigaridas, Dongming Wang, Matías Bender, Thi Xuan Vu.
Efficient algorithms
(i) developing dedicated linear algebra routines performing the Gaussian elimination steps: this is precisely the objective 2 described below;
(ii) generating smaller or simpler matrices to which we will apply Gaussian elimination.
We describe here our goals for the latter problem. First, we focus on algorithms for computing a Gröbner basis of general polynomial systems. Next, we present our goals on the development of dedicated algorithms for computing Gröbner bases of structured polynomial systems which arise in various applications.
Algorithms for general systems. Several
degrees of freedom are available to the designer of a Gröbner
basis algorithm to generate the matrices occurring during the
computation. For instance, it would be desirable to obtain matrices
which would be almost triangular or very sparse. Such a goal can be
achieved by considering various interpretations of the
Algorithms dedicated to structured polynomial systems. A complementary approach is to exploit the structure of the input polynomials to design specific algorithms. Very often, problems coming from applications are not random but are highly structured. The specific nature of these systems may vary a lot: some polynomial systems can be sparse (when the number of terms in each equation is low), overdetermined (the number of the equations is larger than the number of variables), invariants by the action of some finite groups, multi-linear (each equation is linear w.r.t. to one block of variables) or more generally multihomogeneous. In each case, the ultimate goal is to identify large classes of problems whose theoretical/practical complexity drops and to propose in each case dedicated algorithms.