## Section: Scientific Foundations

### Low level implementation and Dedicated Algebraic Computation and Linear Algebra.

Participants : Jean-Charles Faugère, Christian Eder, Elias Tsigaridas, F. Martani.

Here, the primary objective is to focus on *dedicated* algorithms
and software for the linear algebra steps in Gröbner bases
computations and for problems arising in Number Theory. As explained
above, linear algebra is a key step in the process of computing
efficiently Gröbner bases. It is then natural to develop specific
linear algebra algorithms and implementations to further strengthen
the existing software. Conversely, Gröbner bases computation is
often a key ingredient in higher level algorithms from Algebraic
Number Theory. In theses cases, the algebraic problems are very
particular and specific. Hence dedicated Gröbner bases algorithms
and implementations would provide a better efficiency.

**Dedicated linear algebra tools.** FGb is
an efficient library for Gröbner bases computations which can be used,
for instance, via Maple . However, the library is sequential. A
goal of the project is to extend its efficiency to new trend parallel
architectures such as clusters of multi-processor systems in order to
tackle a broader class of problems for several applications.
Consequently, our first aim is to provide a durable, long term
software solution, which will be the successor of the existing FGb library. To achieve this goal, we will first develop a high
performance linear algebra package (under the LGPL license). This
could be organized in the form of a collaborative project between the
members of the team. The objective is not to develop a general
library similar to the Linbox project but to propose a dedicated
linear algebra package taking into account the specific properties of
the matrices generated by the Gröbner bases algorithms. Indeed these
matrices are sparse (the actual sparsity depends strongly on the
application), almost block triangular and not necessarily of full
rank. Moreover, most of the pivots are known at the beginning of the
computation. In practice, such matrices are huge (more than ${10}^{6}$ columns) but taking into account their shape may allow us to speed
up the computations by one or several orders of magnitude. A variant of a Gaussian elimination algorithm together
with a corresponding C implementation has been presented. The main
peculiarity is the order in which the operations are performed. This
will be the kernel of the new linear library that will be developed.

Fast linear algebra packages would also benefit to the transformation of a Gröbner basis of a zero–dimensional ideal with respect to a given monomial ordering into a Gröbner basis with respect to another ordering. In the generic case at least, the change of ordering is equivalent to the computation of the minimal polynomial of a so-called multiplication matrix. By taking into account the sparsity of this matrix, the computation of the Gröbner basis can be done more efficiently using variant of the Wiedemann algorithm. Hence, our goal is also to obtain a dedicated high performance library for transforming (i.e. change ordering) Gröbner bases.

**Dedicated algebraic tools for Algebraic Number
Theory.** Recent results in Algebraic Number Theory tend to show that
the computation of Gröbner bases is a key step toward the resolution
of difficult problems in this
domain ( P. Gaudry, *Index calculus for abelian
varieties of small dimension and the elliptic curve discrete logarithm
problem*, Journal of Symbolic Computation 44,12 (2009)
pp. 1690-1702). Using existing resolution methods is simply not enough
to solve relevant problems. The main algorithmic lock to overcome is
to adapt the Gröbner basis computation step to the specific
problems. Typically, problems coming from Algebraic Number Theory
usually have a lot of symmetries or the input systems are very
structured. This is the case in particular for problems coming from
the algorithmic theory of Abelian varieties over finite
fields ( e.g. point counting, discrete logarithm, isogeny.)
where the objects are represented by polynomial system and are endowed
with intrinsic group actions. The main goal here is to provide
dedicated algebraic resolution algorithms and implementations for
solving such problems. We do not restrict our focus on problems in positive
characteristic. For instance, tower of algebraic fields can be
viewed as triangular sets; more generally, related problems (e.g. effective Galois
theory) which can be represented by polynomial systems will receive
our attention. This is motivated by the fact that, for example,
computing small integer solutions of Diophantine polynomial systems in
connection with Coppersmith's method would also gain in efficiency by
using a dedicated Gröbner bases computations step.