Journ ees GECKO / TERA2008 En l'honneur du 60 e anniversaire de Marc GIUSTI - 24-28 novembre 2008 Ecole polytechnique Amphith eˆatre Pierre ...
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Journées GECKO / TERA2008
En l’honneur du 60e anniversaire de
Marc GIUSTI
24–28 novembre 2008
École polytechnique
Amphithéâtre Pierre Faurre
Programme
1
1 Programme
LUNDI 24 NOVEMBRE
10h30 Accueil. Café
11h15 Luis Miguel PARDO Quelques réflexions incomplètes sur la résolution non-universelle des équations
polynomiales
11h45 Mike STILLMAN Computing in intersection theory and intersection rings of flag bundles and
Grassmannians
12h45-14h30 Déjeuner
14h30 Joris VAN DER HOEVEN On the art of guessing
15h30 Frédéric CHYZAK Products of Ordinary Differential Operators by Evaluation and Interpolation
16h–16h15 Pause café Coffee break
16h15 Laurent BUSÉ On the singularities of plane rational curves
16h45 Antonio CAFURE Bit complexity for polynomial solving over the integers
17h15 Guillermo MATERA Lower bounds for robust interpolation algorithms.
17h45 Pablo SOLERNÓ Sur l’indice et l’ordre de systèmes d’équations algebro-différentielles ordinaires
18h30 Cocktail WELCOME RECEPTION
MARDI 25 NOVEMBRE
10h Joos HEINTZ Natural geometric objects and intrinsic complexity
11h Pause Café Coffee break
11h15 Teo MORA An FGLM-like algorithm for computing the radical of a 0-dim. ideal
11h45 Dave BAYER Graph colorings and toric algebra
12h45-14h30 Déjeuner
14h30 Bernd BANK Wavelet construction via algorithmic real algebraic geometry
15h30 Ian MORRISON New Gröbner Approaches to Hilbert Stability
16h–16h15 Pause café Coffee break
16h15 Bernard MOURRAIN Border bases, perturbations and walk on the Hilbert scheme
16h45 Teresa KRICK On the effective Nullstellensatz
17h15 Carlos BELTRÀN Integration in the space of singular maps : Where Geometry seems to link Real
and Complex Analysis
17h45 Xavier DAHAN Lexicographic Gröbner bases and equiprojectable decompositions
MERCREDI 26 NOVEMBRE
10h André GALLIGO Polynomials, Factorization and Randomness
11h Pause café Coffee break
11h15 Éric SCHOST Evaluation properties of invariant polynomials
11h45 Norbert A’CAMPO Computation of Monodromy
12h45 Déjeuner
JEUDI 27 NOVEMBRE ATTENTION, le jeudi matin c’est en amphi Cauchy.
10h Grégoire LECERF Recent theoretical and practical advances in mutivariate polynomial factorization
11h Pause café Coffee break
11h15 Michel DEMAZURE Réécriture algébrique sans soustraction
11h45 Bruno SALVY Combinatorial Newton Iteration and Efficient Random Generation
212h15 Alin BOSTAN The full counting function of Gessel walks is algebraic
12h45-14h30 Déjeuner
14h30 Joris VAN DER HOEVEN Mathemagix I : General introduction
15h Grégoire LECERF Mathemagix II : C++ architecture, fast algebraic operations
15h30–15h45 Pause café Coffee break
15h45 Bernard MOURRAIN Mathemagix III : polynomial systems, geometry
16h Joris VAN DER HOEVEN Mathemagix IV : analysis, singularities
16h15 Daouda Niang DIATTA Mathemagix V : topology of curves and surfaces
16h45–17h Pause café Coffee break
17h GECKO BUSINESS MEETING
20h30 DÎNER Restaurant “Bel Canto”, 72, quai de l’Hôtel de Ville, 75004 Paris.
VENDREDI 28 NOVEMBRE
10h Mike SHUB Homotopy Methods for solving systems of polynomial equations
11h Pause café Coffee break
11h15 Jean-Claude YAKOUBSOHN Calcul certifié du pgcd approché
11h45 Paola BOITO and Jean-Pierre DEDIEU Condition Geodesics in Matrix Spaces
12h45-14h30 Déjeuner
THE END
32 Résumés
Computation of Monodromy
Norbert A’CAMPO, Université de Bâle
26/XI 11h45
The TQFT-monodromy for singularities of plane curves is a strong invariant. It is an interesting problem
to compute this monodromy.
We construct inside of the Milnor fiber of an isolated hypersurface singularity a stratified subset. For
the case of functions of three complex variables this subset is build with strata of real dimension two. The
visualization of this subset would give new understanding of monodromy.
Wavelet construction via algorithmic real algebraic geometry
Bernd BANK & Lutz LEHMANN, Humboldt Universität zu Berlin
Joint work with Marc Giusti, Joos Heintz, Luis Miguel Pardo
25/XI 14h30
As a result of the TERA–project a new type, highly efficient probabilistic algorithm for the solution of
systems of polynomial equations was developed and implemented for the complex case. The geometry of
polar varieties allows to extend this algorithm to a method that finds real solutions of systems of polynomial
equations.
In order to test this method special emphasis was placed on the fact that example problems are of real-life
and practical importance.
In the talk we report on the application of the TERA–method to real polynomial equation systems solving
basic for the design of fast wavelet transforms. The wavelet transforms we have in mind should reflect the
practical important properties of symmetry and orthogonality. These requirements are expressible by a finite
number of real parameters satisfying a finite system of polynomial equations.
If these equations have a real solution at all, the solution set can be finite or a variety of positive dimension.
Examples with real solution sets of positive dimension have the advantage that one can search for optimal
solutions in the sense that the wavelets have additionally desired analytic properties.
It turns out that the TERA–algorithm performes very well with this task and is able to solve larger systems
than the best known commercial polynomial solvers.
Graph colorings and toric algebra
Dave BAYER, Barnard College, Columbia University
25/XI 11h45
The close link between integer programming and toric algebra relates maximal lattice-free polytopes to
the semigroup structure of an associated toric ring. Monomial modules allow us to think of this toric ring
as an infinite-periodic monomial ideal, whose injective hull determines the maximal lattice-free polytopes.
Given a graph, there is a natural toric ring whose injective hull determines the colorability of the graph.
4Integration in the space of singular maps : Where Geometry
seems to link Real and Complex Analysis
Carlos BELTRÀN, Dept. of Mathematics, University of Toronto
25/XI 17h15
Complex varieties in the affine space have very special properties. Some of them are :
1) they are minimal manifolds,
2) when intersecting the variety with a ball of increasing radius, the Hausdorff measure grows at least as in
the linear case,
3) the volume growth of a tube around the variety is controlled by the Hausdorff measure of the variety.
Indeed, these three facts are intimately related, as 1 ⇒ 2 ⇒ 3. In this talk, I will show that property 3 is
also satisfied by a very particular class of real varieties : the set of rank-deficient matrices, and I will discuss
the geometric resemblance of the real and complex situations, which naturally poses the following question :
Are these real varieties minimal manifolds ? One motivation for this study is related to Numerical Analysis :
The volume growth of the tube around these varieties describes the probability that a rank-deficient matrix
is “easy to solve” in floating point computations, in the sense that the condition number of the problem is
small. I will show very precise bounds for this numerical problem.
The full counting function of Gessel walks is algebraic
Alin BOSTAN, INRIA Paris-Rocquencourt
27/XI 12h15
The aim of the talk is to show how a difficult combinatorial problem has been recently solved using an
experimental-mathematics approach combined with (rather involved) computer algebra techniques. More
precisely, let f (n, i, j) denote the number of lattice walks in the quarter plane which start at the origin, end
at the point (i, j), and consist of n unit steps going either west, south-west, east, or north-east. In the early
nineties, Ira Gessel conjectured that the sequence of excursions f (n, 0, 0) is holonomic. We will present the
computer-driven discovery and proof of the following generalization,
X obtained in August 2008 together with
Manuel Kauers : the full generating series F (t, x, y) = f (n, i, j)xi y j tn is an algebraic function.
i,j,n≥0
On the singularities of plane rational curves
Laurent BUSÉ, INRIA Sophia Antipolis
24/XI 16h15
Given a rational plane algebraic curve C, we will show that some informations on the singularities of
C can be recovered from a (birational) parameterization of C. In particular, we will describe some explicit
adjoint linear systems on C and will interpret them as equations of a certain Rees algebra. We will end by
giving an extension of Abhyankar’s Taylor-resultant for an arbitrary rational plane curve.
5Bit complexity for polynomial solving over the integers
Antonio CAFURE, Universidad Nacional de General Sarmiento
24/XI 16h45
When solving polynomial systems over the integers, to avoid the so-called intermediate expression swell,
the input system is reduced modulo a suitable prime, and from a resolution modulo this prime, by p-adic
lifting, a resolution of the input system is computed.
Applying different results obtained by the TERA group in the preceding years we give an upper bound
on the height of the primes that enable us to perform such a modular reduction. We use this upper bound
in order to obtain an upper bound on the bit complexity of solving polynomial systems over the integers.
Products of Ordinary Differential Operators by Evaluation and
Interpolation
Frédéric CHYZAK, INRIA Paris - Rocquencourt
24/XI 15h30
It is known that multiplication of linear differential operators over ground fields of characteristic zero can
be reduced to a constant number of matrix products. We give a new algorithm by evaluation and interpolation
which is faster than the previously-known one by a constant factor, and prove that in characteristic zero,
multiplication of differential operators and of matrices are computationally equivalent problems. In positive
characteristic, we show that differential operators can be multiplied in nearly optimal time. Theoretical
results are validated by intensive experiments.
Lexicographic Gröbner bases and equiprojectable decompositions
Xavier DAHAN, Universite de Kyushu, Fukuoka, Japon
25/XI 17h45
In 1985, D. Lazard gave a complete description of the structure of lexicographic Gröbner bases of zero-
dimensional and radical ideals of bivariate polynomials. He somehow linked this structure to the primary
decomposition of the ideal. We show that a more relevant decomposition is the so-called ”equiprojectable”
one, not the primary. Then, relying on this decomposition, the generalization of this ”structure theorem”, to
ideals of multivariate polynomials is not difficult. Two consequences can de deduced : size on the coefficients
of lexicographic Gröbner bases, and a geometric-numerical criterion for choosing a ”lucky prime” (term
borrowed to E. Arnold, JSC 2003), in the context of modular computation of Gröbner bases.
Condition Geodesics in Matrix Spaces
Paola BOITO and Jean-Pierre DEDIEU, Institut de Mathématiques,
Université Paul Sabatier, Toulouse
28/XI 11h45
6The condition metric for spaces of polynomial systems has been introduced and studied in a series of
papers by Beltrán, Dedieu, Malajovich and Shub. The interest of this metric comes from the fact that the
associated geodesics avoid ill-conditioned problems and are a useful tool to improve classical complexity
bounds for Bézout’s theorem. The linear case is examined here : Using nonsmooth nonconvex analysis
techniques, we study the behaviour of condition geodesics in the space of full rank, real or complex rectangular
matrices. The main results include an existence theorem for the boundary problem, a differential inclusion
for such geodesics based on Clarke’s generalized gradients, regularity properties and a detailed description
of a few particular cases (diagonal and unitary matrices). Moreover, we study condition geodesics from a
numerical viewpoint and develop an effective algorithm that allows to compute geodesics with given endpoints
and helps to illustrate theoretical results and formulate new conjectures.
Réécriture algébrique sans soustraction
Michel DEMAZURE, professeur des universités retraité
27/XI 11h15
On interprète un article de Fiore et Leinster en termes de réécriture.
Polynomials, Factorization and Randomness
André GALLIGO, université de Nice
26/XI 10h00
The development of algorithms for polynomial factorization leads me to approximate computations and
randomness. I will present 3 aspects : generic change of coordinates, early detection, distribution laws, and
their use for multivariate polynomial factorization,
Natural geometric objects and intrinsic complexity
Joos HEINTZ, Universidad de Buenos Aires and CONICET, Argentina and
Universidad de Cantabria, Santander, Spain
25/XI 10h00
My friendship and scientific collaboration with Marc Giusti started 1982 during the meeting Algorith-
mique et Calcul Formel in Limoges. This led in the following years to a intensive interchange and cross
fertilization of semantical concepts from algebraic geometry and syntactical views from theoretical computer
science, mainly complexity theory, but also data structures and types and programming paradigms. Toge-
ther with our coworkers from the international TERA (Turbo Evaluation and Rapid Algorithms) group we
succeded finally to establish an agenda in symbolic
and seminumeric elimination theory, which someone may like and someother not, but which is difficult
to ignore. An outcome of this synergy was the development of the Kronecker Software package by G. Lecerf
and collaborators and a couple of applications, one of them to image processing.
In my talk I shall try to give an account of the ideas and questions which guided us during 26 years, the
mathematical difficulties we met and the solutions which we finally found. In this context Marcs intuition that
7in effective elimination theory computations should evolve along “natural” geometric objets which determine
in their turn the size and complexity of the intermediate results, plays a crucial role. This view is reflected
in the quest for the intrinsic complexity of elimination problems.
On the effective Nullstellensatz
Teresa KRICK, Université de Buenos Aires
25/XI 16h45
The subject of the talk will be the Nullstellensatz, a cornerstone in Algebraic Geometry, in its effective
aspects.
I will mention a method, developed by Marc Giusti and others in the 90’s, which produces the coefficients
g1 , . . . , gs in an expression 1 = g1 f1 + · · · + gs fs (when possible) and allowed to obtain the best effective
arithmetic results up to now. I also intend to introduce a new recent proof by Z. Jelonek which makes use
only of elementary and natural tools of classical algebraic geometry and which will hopefully allow to improve
those arithmetic results.
Recent theoretical and practical advances in multivariate
polynomial factorization
Grégoire LECERF, université de Versailles
27/XI 10h00
This talk will survey the recent advances within the factorization of multivariate polynomials. We will
show algorithms that are essentially subquadratic in time for almost all the tasks, with a focus on the
methods developped during the period covered by the ANR GECKO.
Lower bounds for robust interpolation algorithms
Guillermo MATERA, Universidad Nacional de General Sarmiento
24/XI 17h15
In this talk we discuss lower bounds on the complexity of robust algorithms for solving families of
interpolation problems. Our notion of robustness models the behavior of all known universal methods for
solving families of interpolation problems avoiding unnecessary branchings and allowing the solution of
certain limit problems. We first show that a robust algorithm solving a family of Lagrange interpolation
problems with N nodes encoded by a Zariski open subset of the space CN of nodes has a cost which is
at least linear in N , showing thus that standard interpolation methods are essentially optimal. Then we
consider families of interpolation problems with singularities. In
particular, we consider the family of problems which consists of interpolating a polynomial given by a
straight-line program of length L from its value in a correct-test sequence. We show that any robust algorithm
solving such a family of problems requires a number of arithmetic operations which is exponential in L.
Joint work with Nardo Giménez, Joos Heintz and Pablo Solernó.
8An FGLM-like algorithm for computing the radical of a
0-dim. ideal
Teo MORA, Universita’ di Genova
25/XI 11h15
Mainly motivated by Auzinger–Stetter Algorithm and Cardinal’s Conjecture, in MEGA-92 Traverso at
al. proposed the notion of Gröbner representation of a 0-dimensional ideal J ⊂ Q = K[Z1 , . . . , Zr ] which is
the assignement of
– a K-linearly independent = {q1 , . . . , qs } ⊂ Q,
n set q 2
o
(h)
– the set M = M(q) := alj ∈ Ks , 1 ≤ h ≤ r of r square matrices
(l)
– s3 values γij∈K
which satisfy
1. Q/J = ∼ Span (q),
K
P (h)
2. Zh ql ≡ j alj qj (mod J) for each l, j, h, 1 ≤ l, j ≤ s, 1 ≤ h ≤ r,
P (l)
3. qi qj ≡ l γij ql (mod J) for each l, j, h, 1 ≤ i, j, l ≤ s.
In connection with Rouillier’s Rational Univartiate Representation (actually with Kronecker Parame-
trization), at the ICPSS conference in honour of Daniel Lazard, Arai and Muritzugu posed the following
Problem. — Given a 0-dimensional ideal J ⊂ Q = K[Z1 , . . . , Zr ] via a Gröbner representation, compute
√ Pr
a Gröbner representation of J and a separating linear form Y = i=1 ai Zi for its roots.
An algorithm (based on repeated application of Buchberger Algorithm) for computing both the radical
and a separating linear form for its roots of a given zero-dimensional ideal was proposed in 1987 by Gianni.
Inspired by old results by Giusti–Heintz and Alonso–Raimondo on decomposition, I reformulate here an
FGLM-like linear algebra adaptation of Gianni’s Algorithm thus solving the given problem.
New Gröbner Approaches to Hilbert Stability
Ian MORRISON, Fordham University
25/XI 15h30
Recently, Gieseker’s GIT construction of M g as a quotient of the ν-canonical Hilbert scheme for ν ≥ 5 has
found new applications in Hassett’s log minimal model program for M g . These models arise as quotients for
ν < 5 and are moduli spaces whose boundary curves satisfy variants of Deligne-Mumford stability. Pinning
down these variants, and constructing the models involve understanding, respectively, Hilbert instability and
stability. To date, it has sufficed to polarize the Hilbert schemes by taking degree m 0—when the answer
is independent of m—and the techniques have involved asymptotic estimates that apply only for +large.
The next stages require making such constructions for fixed small values of m.
I will review this work, focusing on new techniques for checking Hilbert instability and stability for small
m using Gröbner techniques. The first, work of Hassett, Hyeon and Lee, uses Castelnuovo-Mumford regularity
to read off stability with respect to a fixed 1-parameter subgroup from low degree pieces of the associated
Gröbner basis and streamlines many instability calculations. The second, work in progress with Swinarski,
is a first attack on the harder problem of checking low-m stability for curves with many automorphisms by
calculations of state polytopes or Gröbner fans.
9Border bases, perturbations and walk on the Hilbert scheme
Bernard MOURRAIN, INRIA Méditerranée
25/XI 16h15
Solving polynomial equations with approximate coefficients is ubiquitous in many applications. It is also
a challenge from an effective algebraic geometry point of view. To tackle this issue, border basis methods
have been introduced recently. Compared to Grobner basis computation, they yield representations of the
quotient algebra, which are more stable from a numerical point of view.
In this presentation, we try to analysis more precisely this assertion. We first recall the main properties
of these border bases, how they can be characterised, how they can be computed, how the syzygies are
generated and give some examples of Border basis computations. Our motivation is to be able to improve
the numerical quality of a quotient representation, after an approximate computation of a borderbasis at a
given precision. We will describe an explicit Newton-type iteration for this purpose and give effective criteria
to check flatness or the stability of a deformation. The connection with Hilbert scheme of points will be
exploited.
Quelques réflexions incomplètes sur la résolution non-universelle
des équations polynomiales
Luis-Miguel PARDO, Universidad de Cantabria
24/11 11h15
Dans cet exposé, on reviendra sur les méthodes éfficaces de résolution Non–Universelle des équations
polynomiales : l’origine de la question, les résultats positifs et, surtout, les questions ouvertes. L’exposé
contiendra plus de questions que de réponses.
Combinatorial Newton Iteration and Efficient Random
Generation
Bruno SALVY, Inria
27/XI 11h45
We recall the principle of Boltzmann samplers for large combinatorial structures. The construction of
these samplers relies on so-called ”oracles” that compute numerical values of generating series. We show that
such an oracle is realized by a simple Newton iteration. The important point is that using the origin as a
starting point always converges to the desired solution. The proof relies on lifting the Newton iteration as
an iteration on classes of combinatorial structures.
Homotopy Methods for solving systems of polynomial equations
Mike SHUB, Mathematics Department, University of Toronto
28/XI 10h00
10We consider the complexity of homotopy methods for solving systems of polynomial equations, the
condition metric, and the structure of the solution variety. This is joint work with Carlos Beltran, Jean-
Pierre Dedieu, Gregorio Malajovich, Luis Miguel Pardo and Steve Smale.
Evaluation properties of invariant polynomials
X. Dahan,
Éric SCHOST, University of Western Ontario,
J. Wu
26/XI 11h15
A polynomial invariant under the action of a finite group can be rewritten using generators of the invariant
ring. We investigate the complexity aspects of this rewriting process ; we show that evaluation techniques
enable one to reach a polynomial cost.
Sur l’indice et l’ordre de systèmes d’équations
algebro-différentielles ordinaires
Pablo SOLERNÓ, Universidad de Buenos Aires
24/11 17h45
Nous étudions les notions d’indice et d’ordre dans le cas des systèmes quasi-réguliers, quelques estimations
”à la Jacobi” et des conséquences quantitatives.
Computing in intersection theory and intersection rings of flag
bundles and Grassmannians
Mike STILLMAN, Cornell University
24/XI 11h45
The main idea of the talk is to describe some joint work with Dan Grayson, involving intersection rings of
flag bundles, and a package for Macaulay2, “Schubert 2”, which is under development. Schubert 2 is roughly
based on the Maple package Schubert, written by Katz and Stromme.
We start by reviewing intersection theory and describing Grothendieck’s theorem for the intersection ring
of flag bundles. We then show that, with respect to specific monomial orders, their defining ideals have simple
to describe initial ideals, even over the integers. The resulting Groebner bases allow for fast computation in
the intersection rings of flag bundles and Grassmannians. We then show some examples of computations in
enumerative geometry using Schubert2 that take advantage of these methods.
11On the art of guessing
Joris VAN DER HOEVEN, département de Mathématiques, université Paris
Sud
24/XI 14h30
Current computer algebra systems usually work in a shell mode : the user asks a question and the system
hopefully gives an answer. To what extent would it be possible to discover additionnal mathematical structure
in the user’s problem in an automatic fashion ? For instance, if we encounter 2.094395102393195492308428922
in the output of a numerical computation, the system might suggest its replacement by 2π/3. Even though
guessing such addional relations was not explicitly requested by the user, it may lead to interesting insights
and does not necessarily lead to a big increase of the overall computation time.
In our talk, we will address several classical guessing algorithms and present a few new ones. We will
start with rational number and function recovery, the LLL-algorithm and Pade-Hermite approximation. We
will next turn our attention to guessing possible asymptotic expansions for sequences and possible relations
at singularities of analytic functions. If time permits it, we will also discuss some guessing techniques for
symbolic expressions.
Calcul certifié du pgcd approché
Jean-Claude YAKOUBSOHN, université Paul Sabatier, Toulouse III
En collaboration avec Guillaume Chèze, André Galligo et Bernard Mourrain
28/XI 11h15
Le calcul du pgcd approché peut se ramener à la résolution d’un problème de minimisation d’une fraction
rationnelle de deux variables. Cette formulation a été donnée par Karmarkar et Lakshman dans leur papier
“On approximate gcds of univariate polynomials”, J. Symbolic Comput., 1998, 26, 653-666.
Nous proposons un calcul certifié de ce problème de minimisation. Nous illustrons cette certification dans
les cas où les polynômes sont représentés dans la base des monômes et dans la base de Bernstein.
123 Participants
Norbert A’CAMPO, université de Bâle
Bernd BANK, Humboldt Universität zu Berlin
Dave BAYER, Barnard College, Columbia University
Carlos BELTRÀN, Dept. of Mathematics, University of Toronto
Jeremy BERTHOMIEU, LIX
Alexandre BENOIT, INRIA-MSR
Manuel BODIRSKY, LIX
Paola BOITO, Institut de Mathématiques, Université Paul Sabatier, Toulouse
Alin BOSTAN, INRIA Paris-Rocquencourt
Jérôme BRACHAT, INRIA Sophia-Antipolis
Laurent BUSÉ, INRIA Sophia Antipolis
Antonio CAFURE, Universidad Nacional de General Sarmiento
Jacques CALMET, Universität Karlsruhe (TH)
Frédéric CHYZAK, INRIA Paris - Rocquencourt
Jean-Pierre DEDIEU, Institut de Mathématiques, Université Paul Sabatier, Toulouse
Xavier DAHAN, Universite de Kyushu, Fukuoka, Japon
Daouda Niang DIATTA, Université de Limoges
Michel DEMAZURE, professeur des universités retraité
Clémence DURVYE, université de Versailles
Michel FLIESS, LIX, Ecole polytechnique & Projet ALIEN, INRIA
Anne FREDET, IUT de Saint-Denis
André GALLIGO, université de Nice
Marc GIUSTI, LIX
Joos HEINTZ, Universidad de Buenos Aires and CONICET, Argentina and Universidad de Cantabria,
Santander, Spain
Pierre-Vincent KOSELEFF, UPMC-PARIS SALSA-INRIA
Teresa KRICK, Université de Buenos Aires
Romain LEBRETON, LIX
Bernard MALGRANGE, Académie des Sciences
Grégoire LECERF, université de Versailles
Guillermo MATERA, Universidad Nacional de General Sarmiento
Marc MEZZAROBBA, INRIA Rocquencourt (Équipe Algorithms)
Jean MOULIN OLLAGNIER, LIX, École polytechnique et Université Paris XII
Teo MORA, Universita’ di Genova
Guillermo MORENO-SOCIAS, université de Versailles
Ian MORRISON, Fordham University
Guillaune MOROZ, INRIA/LIP6
Bernard MOURRAIN, INRIA Méditerranée
François OLLIVIER, LIX, École polytechnique
Luis Miguel PARDO, Universidad de Cantabria
Michel PETITOT, LIFL, université Lille I
Adrien POTEAUX, INRIA Galaad - Universite de Nice Sophia Antipolis
Bruno SALVY, Inria
Eric SCHOST, University of Western Ontario
Alexandre SEDOGLAVIC, LIFL, USTL
Mike SHUB, Mathematics Department, University of Toronto
Pablo SOLERNÓ, Universidad de Buenos Aires
Michèle SORIA, LIP6, université Paris 6
Mike STILLMAN, Cornell University
13Philippe TREBUCHET, Paris 6
Annick VALIBOUZE, Université Pierre et marie Curie (UPMC)
Giuseppe VALLA, Universita di Genova
Joris VAN DER HOEVEN, département de Mathématiques, université Paris Sud
Jacques-Arthur WEIL, XLIM, Université de Limoges
Jean-Claude YAKOUBSON, université Paul Sabatier, Toulouse III
14You can also read
