## Section: Research Program

### Fundamental Algorithms and Structured Systems

Participants : Jérémy Berthomieu, Jean-Charles Faugère, Guénaël Renault, Mohab Safey El Din, Elias Tsigaridas, Dongming Wang, Matías Bender, Thi Xuan Vu.

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.