Adrien Poteaux

Associate professor (Maître de conférences)
CFHP team, CRISTAL

Centre de Recherche en Informatique, Signal et Automatique de Lille (CRIStAL)
Batiment M3, bureau 330 b
Université Lille I,
59655 Villeneuve d'Ascq,
France.

Tel: +33 3 20 33 77 10
Fax: +33 3 28 77 85 37
E-mail: surname-with-no-space
		    arobase univ-lille1.fr

Publications

bibtex file

PhD thesis (in french)

I made my PhD thesis at the university of Limoges, supervised by Moulay Barkatou and Marc Rybowicz. Here is my thesis, Calcul de développements de Puiseux et application au calcul du groupe de monodromie d'une courbe algébrique plane. I defended my thesis on October 15th 2008. You can find the slides here.

2015

2014

2013

2012

2011

2010

2009

2008

2007

Posters

Code

Here I give several files .mpl including some Maple code I implemented during my thesis and postdocs. This includes all the algorithms described in my PhD, but also the algorithms explained in our SNC'09 paper.

All these algorithms have been written using Maple 11, and do not have any other pretentions than to be prototypes one. For instance, computations in finite extentsion above finite fields do not work well in Maple. In a more general way, we do not use fast algorithms for basic operations or polynomial multiplication. Finally, the numerical part of the algorithms need some more work: we do not have any bound on the numerical errors that may occur during the computations. Thus, even if the algorithms described in the thesis have been thought to be certificated, they are not for the moment since we do not have provable error bounds. Nethertheless, these algorithms have a good practical numerical behaviour.

A modular-numeric Newton-Puiseux algorithm

All files

The second chapter of our PhD thesis is dedicated to the description of a new modular-numeric algorithm to compute numerical approximations. In essence, we first compute the Puiseux series modulo a prime number p, which is chosen such that the structure of these series is preserved by modular reduction. The file puiseuxmodp.mpl is dedicated to the computation of the Puiseux series above a finite field, whereas the file choixp.mpl includes the algorithms to find a good prime number p. Then, we use the structure of the Puiseux series to conduct the computation of approximate coefficients of the series. These last algorithms are given in the file numpuiseux.mpl. Finally, we provide a maple file that illustrates the use of these algorithms on a (very simple) example; the different options of the algorithms are given in the .mpl files.

A symbolic-numeric algorithm to compute monodromy groups of plane algebraic curves

All files (the monodromy algorithm need Puiseux files to be used)

We propose in the third chapter of our thesis a new strategy to compute monodromy groups of a plane algebraic curve. The file monodromy.mpl gives a Maple implementation of the algorithm we describe in this chapter.

Analytic continuation and monodromy on random Riemann surfaces

All files

In our SNC'09 paper, we focused on random Riemann surfaces, i.e. Riemann surfaces defined by bivariate polynomials with random coefficients. To guess the behaviour of such surface, we first studied some monodromy groups. These groups have been computed with the algorithm given above. As the computation of such groups can take time, due mainly to the number of critical points to consider, we limited our study to polynomials of total degree between 7 and 10. We give here some of these examples we have computed. The first example is of total degree 7. It includes a picture of the Euclidean minimal spanning tree we follow, the monodromy group, and an approximation of the fibers above critical points. Examples 2 to 5 are of degree 7 for the first three ones, and 9 for example 5. Here we only give monodromy groups. Example 6 and 7 are two degree 10 examples, for which we give their monodromy group. Finally, example 8 and example 9 are of degree 10. These two files give monodromy groups and approximations of the fibers above critical points.

After a careful study of these data, we developped a strategy base on analytic continuation to find the subsets of the fiber above 0 which correspond to the factors of the bivariate polynomial. A first implementation of this strategy has been done in Maple. You can find the code here

Talks (in french or english)

  • Calcul de groupe de monodromie d'une courbe algébrique plane :
    • Séminaire de calcul formel de Limoges, November 15th 2005
    • Rencontres Poitiers-Limoges, May 4th 2006
    • Journées Nationales de Calcul Formel 2007, Marseille, January 28th 2007
    • École Jeunes Chercheurs en Informatique et Mathématique, Nancy, March 22th 2007
    • Séminaire ALGO, INRIA Rocquencourt, April 23th 2007
    • Séminaire de calcul formel et complexité de Rennes, June 1st 2007
  • Calcul symbolique-numérique des développements de Puiseux au-dessus des points critiques : partie modulaire, Journées Gecko, November 18th 2007, INRIA Sophia Antipolis
  • Travaux sur les courbes algébriques planes, Séminaire des doctorants de Limoges, February 1st 2008
  • Calcul de développements de Puiseux et application au calcul de groupe de monodromie, Séminaire GALAAD, INRIA Sophia Antipolis, October 3rd 2008
  • Calcul de développements de Puiseux : un nouvel algorithme symbolique-numérique, JNCF 2008, Luminy, October 22th 2008
  • Computing Numerical approximations of Puiseux expansions, Saga Kick-off meeting 2008, November 20th 2008
  • Computing Puiseux Series and Motivations, Symbolic Computation Lab seminar, London, Ontario, June 5th 2009
  • Computing Puiseux expansions: a new symbolic-numeric algorithm, Applications of Computer Algebra, Montréal, June 26th 2009
  • Distance computation via approximation of supported functions, SAGA Workshop, Auron, March 15th 2010
  • Approximation of curves by Quadratically Supported Curves, Seminar, Santander, April 9th 2010
  • Puiseux series computation: a modular-numeric approach, Matesco seminar, Santander, April 23th 2010
  • Composition modulaire et applications, JNCF 2010, Luminy, May 6th 2010
  • Opérations modulo un ensemble triangulaire en un temps quasi linéaire, Séminaire calcul formel - Limoges, September 16th 2010
  • Distance computation via QSC approximation, SAGA fall school, Kolympari, October 7th 2010
  • Symbolic-numerical computation of Puiseux series, Séminaire calcul formel SALSA, Paris, November 5th 2010
  • Composition modulaire modulo un ensemble triangulaire et application au problème du comptage de points d'une courbe elliptique, Séminaire de cryptologie - GREYC, Caen, November 25th 2010
  • Calculs rapides modulo un ensemble triangulaire et applications, Séminaire LFANT - Institut Mathématiques de Bordeaux, January 27th 2011
  • Calcul symbolique-numérique des séries de Puiseux, RAIM 2011, Perpignan, February 10th 2011
  • Sur la complexité du calcul modulo un ensemble triangulaire zéro-dimensionnel, Séminaire BIPOP-CASYS - Grenoble, February 17th 2011
  • Calcul du groupe de monodromie d'une courbe algébrique plane, Séminaire de théorie des nombres de Toulouse, February 22th 2011
  • Un algorithme symbolique-numérique pour calculer au-dessus des points d'une courbe algébrique plane, Séminaire de Calcul Formel du LIFL, Lille, February 24th 2011
  • Calcul symbolique-numérique de séries de Puiseux, Séminaire Vegas-Caramel, Loria, Nancy, March 3rd 2011
  • Puiseux series computation: avoiding computations over Q, Symbolic Computation seminar, Linz, January 9th 2012
  • Distance computation via QSC approximation, Applied Geometry seminar, Linz, January 20th 2012
  • Sur la complexité du calcul modulo un ensemble triangulaire zéro-dimensionnel, Computer Algebra working groupe, Lille, February 15th 2012
  • Calcul pratique des séries de Puiseux, Séminaire de théorie des nombres de Grenoble, April 4th 2012
  • Towards a fast Newton-Puiseux algorithm ?, Symbolic Computation seminar, Linz, July 4th 2012
  • Calcul pratique des séries de Puiseux. Groupe de Travail de l'équipe de Calcul Formel, Lille, September 17th 2012
  • Puiseux series computation: towards a usable implementation. Journées ANR Biotempo, Paris, January 30th 2013
  • Calcul de racines de systèmes creux, un article de Bernd Sturmfels. Partie 1 : Problème, définitions et preuve du cas binomial. Groupe de Travail de l'équipe de Calcul Formel, Lille, May 27th 2013
  • Calcul de racines de systèmes creux, un article de Bernd Sturmfels. Partie 2 : Séries de Puiseux et prolongement analytique. Groupe de Travail de l'équipe de Calcul Formel, Lille, June 3rd 2013
  • On the complexity of computations modulo zero-dimensional triangular sets, Gröbner Bases, Resultants and Linear Algebra workshop, Hagenberg, Austria, September 5th 2013
  • Towards fast Puiseux series computation, Computer algebra and polynomials Workshop, Linz, Austria, November 25th 2013
  • Évaluation et interpolation rapide, Groupe de Travail de l'équipe de Calcul Formel, Lille, May 20th 2014
  • About the Abel map computation, XXVèmes rencontres arithmétiques de Caen, Tatihou, July 1st 2014
  • Practical computation of Puiseux series, Séminaire Mathématique-Physique, Institut Mathématique de Bourgogne, Dijon, December 17th 2014
  • Étude de singulariés d'une courbe algébrique plane : améliorations de l'algorithme de Newton-Puiseux, Séminaire de Théorie des Nombres, Caen, May 15th 2015
  • Sur le calcul efficace des séries de Puiseux, JNCF 2015, Cluny, November 3rd 2015