My research interests are Computer Algebra, Programming
Languages and Semantics.
Computer Algebra is manipulation of “mathematical data” with computers. I am
a former member of the CALFOR group leaded
by Daniel Lazard who was my thesis
advisor.
I started studying symbolic integration as part of my masters thesis. These
results are quite old but still current
the
Lazard-Rioboo
enhancement to Trager's integration formula is the standard way to perform
integration of rational functions. Comprehensive versions of this algorithm
are described in many Text-Books such as
Algorithms for computer algebra by Geddes, Czapor, and Labahn or
Modern Computer Algebra by Von Zur Gathen.
My thesis also contains an enhancement to express an integral in real terms
which is described in late
M. Bronstein's
Symbolic Integration book.
Both formulas and algorithms are used by major Computer Algebra Systems (aka
CAS) such as the free softwares
Axiom and
Maxima or the commercial softwares
Macsyma,
Maple and
Mathematica
My thesis was about
Cylindrical Algebraic
Decomposition (aka CAD) which is a technique to decompose the n-dimensional
real space into connected regions where the signs of a given set of
n-variables polynomials remain constants. I implemented a
Regular Axiom package
performing CAD.
Most of the improvements for CAD
were due to a better management of real algebraic numbers. These real
algebraic numbers manipulation algorithms were published in
ISSAC 92 and Imacs
93. I still maintain the
sources for
the RECLOS package since NAG Ltd included it in
Axiom and it is now part of the free software
Axiom distribution. This
implementation is still the only one which solves non trivial problems using a
general approach.
The technique of tower amnipulations I used for real
algebraic numbers was adapted from D5's
Dynamic
Evaluation. Together with Marc
Moreno Maza we proposed at
AAECC-11 another adaptation
of these techniques for general polynomial system solving which led to
working implementations of triangular resolution.
I gave extensions combining real arithmetics and generalized resultant
algorithms at ISSAC 2002 and in the
Journal of
Symbolic Computation. These will be the basis for other versions of the
RECLOS package.
Implementing these algorithms requires a lot of software machinery and use
dedicated programming languages such as Axiom or its successor
Aldor. As a computer scientist I started
wondering why Computer Algebra Systems use specially designed programming
languages. One of the main reasons is probably that inside CAS mathematics is
the data whereas inside general programs mathematics is part of the model.
Powerfull CAS such as Axiom or
Magma use a notion of domain
which is close to that of algebraic structure in mathematics. In order of
having a better understanding of the relations between those domains and
programming languages I started discussing with
Thérèse Hardin and we decided to
start the FoC project in 1997 as a joint effort of
the Computer Algebra and
Semantics groups of
LIP6. Our purpose was to build a programming
language able to include formal properties of algebraic structures as part of
their definition and to provide their proofs when implementing these
structures.
The FoCaL project is now a collaboration between
researchers of LIP6,
INRIA and Cedric and
has become one the major research subject of the
SPI group of
LIP6. Since I am dedicating most of my time to the
project I decided to join the group in 2004.
I have defended my Habilitation on December 18 2002.
I was one of the organizers of the Calculemus 2003
conference in Rome. I am also involved in the
Calculemus Interest Group and part of the
board of trustees since 2002 and reelected in 2006.
I was member of the Calculemus program commitee in 2002, 2004 and will be part
of the 2006 edition.
