## Section: Scientific Foundations

### Fundamental Algorithms and Structured Systems

Participants : Jean-Charles Faugère, Mohab Safey El Din, Elias Tsigaridas, Guénaël Renault, Dongming Wang, Jérémy Berthomieu, Pierre-Jean Spaenlehauer, Chenqi Mou, Jules Svartz, Louise Huot, Thibault Verron.

Efficient algorithms ${F}_{4}/{F}_{5}$ (J.-C. Faugère. *A
new efficient algorithm for computing Gröbner bases without reduction
to zero (F5).* In Proceedings of ISSAC '02, pages 75-83, New York, NY,
USA, 2002. ACM.) for computing the Gröbner basis of a
polynomial system rely heavily on a connection with linear
algebra. Indeed, these algorithms reduce the Gröbner basis
computation to a sequence of Gaussian eliminations on several
submatrices of the so-called Macaulay matrix in some degree. Thus, we expect to
improve the existing algorithms by

*(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 ${F}_{5}$
algorithm with respect to different monomial orderings. To address
this problem, the tight complexity results obtained for ${F}_{5}$ will be used to
help in the design of such a general algorithm. To illustrate this
point, consider the important problem of solving boolean polynomial
systems; it might be interesting to preserve the sparsity of the
original equations and, at the same time, using the fact that
overdetermined systems are much easier to solve.

**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.