Séminaire de géométrie et algèbre effectives

Le séminaire est organisé par Xavier Caruso et Elisa Lorenzo García.

Les exposés ont lieu le vendredi à 10h30 dans la salle 016 du bâtiment 22-23.

Exposés à venir

• Vendredi 16 mars 2018
A new approach for Formal Reduction of Singular Linear Differential Systems Using Eigenrings
In this talk, we will present a new algorithm for the formal reduction of linear differential systems with Laurent series coefficients. We show how to obtain a decomposition of Balser, Jurkat and Lutz using eigenring techniques. We establish structural information on the obtained indecomposable subsystems and retrieve information on their invariants such as ramification. We show why Barkatou's algorithm then perform well on these subsystems. We also give precise estimates of the precision on the power series which is required in each step of the algorithm. We give some examples computed in Maple.
• Vendredi 23 mars 2018
Elif Saçıkara (Sabanci University)
Concatenated Structure of Quasi-Abelian Codes and a Resulting Minimum Distance Bound
For a positive integer l and a group algebra Fq[H], a quasi-abelian(QA)code of index l is a Fq[H]-submodule of Fq[H]^l, where H is an abelian group of order m. The special case H := Z_m, where Z_m is a cyclic group of order m, gives a quasi-cyclic (QC) code of index l and length ml. So, QA codes are natural generalization of QC codes. Ling and Sole showed that QC codes can be decomposed as a direct sum of certain linear codes of length l by applying the Chinese Remainder Theorem, such a method is called the CRT decomposition. Jensen represented a concatenated structure of QC codes and later Guneri-Ozbudak showed that these decompositions are equivalent. In this talk, we present a concatenated structure of QA codes by using the CRT decomposition of QA codes introduced by Jitman and Ling, and we show that both decompositions are equivalent. Concatenated structure also leads to asymtotical goodness and provides a general minimum distance bound, extending the analogue bound for QC codes due to Jensen.
• Vendredi 30 mars 2018
David Lubicz (DGA et IRMAR)
• Quelques améliorations à l'algorithme AGM et ses généralisations en genre plus grand.
Nous décrivons des améliorations à l'algorithme de comptage de points AGM et ses généralisation en genre plus grand. Nous décrivons plus précisément un algorithme qui permet de retrouver efficacement la classe de similitude de l'action du morphisme de Frobenius (ou son dual) sur l'espace cotangent du relevé canonique de la variété abélienne de départ (définie sur un corps fini). Cet algorithme a un intérêt indépendamment de l'AGM. Ivan Pogildiakov (Université de la Polynésie Française et IRMAR)
• Vendredi 6 avril 2018
John Cremona (University of Warwick)
• Vendredi 13 avril 2018
Jade Nardi (Institut de Mathématiques de Toulouse)

Exposés passés

• Vendredi 15 septembre 2017
Victor Cauchois (IRMAR)
• Vendredi 6 octobre 2017
Journée de rentrée de l'équipe
10h45 – 11h30, Codes correcteurs d'erreur (par P. Loidreau)
13h15 – 14h00, Géométrie algébrique réelle (par M.-F. Roy)
14h15 – 15h00, Courbes algébriques (par C. Ritzenthaler)
15h15 – 16h00, Précision p-adique (par X. Caruso)
• Vendredi 13 octobre 2017
Johannes Hoffmann (RWTH Aachen University)
A Constructive Approach to Arithmetics in Ore Localizations
Classical results of Ore from the 1930's tell, that for a domain R, a localization with respect to a multiplicatively closed set S exists if S satisfies left Ore property. We investigate the arithmetics of the localized ring S^{-1}R from both theoretical and practical points of view. We show that effective computations in S^{-1}R require a specialization of the type of S and distill three most frequently occuring types of Ore sets. We provide an implementation of arithmetics over ubiquitous G-algebras in Singular:Plural and discuss algorithmic and theoretic questions in this context.
• Vendredi 20 octobre 2017
Peter Stevenhagen (Université de Leiden)
Ellpitic primitive Roots
• Vendredi 27 octobre 2017
Vincent Neiger (Université de Limoges)
Efficient algorithms for computing univariate relations
In this talk, we will present recent results about the fast computation of univariate relations, including in particular rational interpolants and Hermite-Padé approximants. Such relations are a central tool in computer algebra, arising in situations such as the guessing of linear differential equations, the list-decoding of Reed-Solomon codes, or the computation of normal forms of polynomial matrices. After giving an overview of the problem and of existing fast algorithms, the emphasis will be placed on the main ideas behind the recent algorithmic improvements.
This talk involves joint work with Claude-Pierre Jeannerod, Johan Rosenkilde, Eric Schost, Grigory Solomatov, Gilles Villard, and Vu Thi Xuan.
• Vendredi 10 novembre 2017
Everett Howe (Center for Communications Research)
Constructions of curves of medium genus over F_q with many points
• Vendredi 17 novembre 2017
André Leroy (Université d'Artois)
Sur les factorisations dans les extensions de Ore
On commencera par définir une notion d'évaluation pour les polynômes de Ore à coefficients dans un anneau. On introduira ensuite les transformations pseudo linéaires et on définira les notions de polynômes de Wedderburn et de polynômes complètement réductibles. Après avoir donné plusieurs caractérisations de ces polynômes on s'intéressera à leurs factorisations en particulier dans le cas ou l'anneau de base est un anneau à division. On présentera aussi le comptage des racines généralisant et précisant le résultat de Gordon-Motzkin. On abordera la question de factorisation dans des anneaux généraux et on introduira la notion d'anneaux de polynômes de Ore libre qui semble être le bon moyen d'aborder l'évaluation à plusieurs variables gauches.
• Vendredi 24 novembre 2017
Locally Recoverable Codes with Many Recovery Sets from Fiber Products of Curves
An error-correcting code is said to be locally recoverable if any symbol in a code word can be recovered by accessing a subset of the other symbols. This subset is known as the helper or recovery set for the given symbol. Locally recoverable codes (LRCs) have benefits for distributed storage applications. It may be desirable to have many disjoint recovery sets for each symbol, in case of multiple server failures or to provide many options for recovery. Barg, Tamo, and Vladut recently constructed LRCs with one and two disjoint recovery sets from algebraic curves. This talk presents a generalization of this construction to three or more recovery sets using iterated fiber products of curves. The construction is applied to give including the Suzuki and generalized Giulietti-Korchmaros curves. Codes with arbitrarily many recovery sets are constructed, employing maximal curves from fiber products devised by Van der Geer and Van der Vlugt. This is joint work with Kathryn Haymaker and Gretchen Matthews.
• Vendredi 1er décembre 2017
Ignacio García Marco (I2M, Université d'Aix-Marseille)
Noether Resolutions
Let $$R := K[x_1,\ldots,x_n]$$ be a polynomial ring over an infinite field $$K$$, and let $$I \subset R$$ be a graded ideal. We say that $$A := K[x_1,\ldots,x_d]$$ is a Noether normalization of $$R/I$$ if $$A \hookrightarrow R/I$$ is an integral extension. Whenever $$A$$ is a Noether normalization of $$R/I$$, then $$R/I$$ is a finitely generated $$A$$-module and, thus, it has a finite minimal graded free resolution as $$A$$-module. In this talk we will study this resolution, which we call the Noether resolution of $$R/I$$. When the Krull dimension of $$R/I$$ is 1 or 2 we provide an algorithm for obtaining this resolution that involves the computation of a particular Gröbner basis of $$I$$. When $$R/I$$ is a semigroup ring we will also explain how one can replace the Gröbner basis computation by computations in the underlying semigroup.
• Vendredi 15 décembre 2017
A dichotomic Newton-Puiseux algorithm using dynamic evaluation.
Puiseux series (generalisation of Taylor series above critical points) are a fundamental object in the theory of plane algebraic curves. This talk will focus on the arithmetic complexity of the well-known Newton Puiseux algorithm and its variants. Denoted $$D$$ the total degree of the input bivariate polynomial, Duval proved a $$O(D^8)$$ complexity result. This has been improved to $$\tilde O(D^5)$$ by truncating powers of $$X$$ during the computation and introducing fast multiplication.
After providing the tools of the Newton-Puiseux algorithm, we will first present some results that enable to reduce the total number of recursive calls of the algorithm from $$O(D^2)$$ to $$\tilde O(D)$$, leading to a complexity in $$\tilde O(D^4)$$. Finally, we will present a new divide and conquer algorithm that reduces the complexity to $$\tilde O(D^3)$$.
This work begun during my PhD, under the supervision of Marc Rybowicz; the recent ameliorations started from a collaboration with Marc in 2011. The new divide and conquer algorithm is a collaboration with Martin Weimann. This paper is dedicated to Marc Rybowicz, who sadly passed away in November 2016.
• Vendredi 19 janvier 2018
Growth of torsion of elliptic curves.
One of the main goals in the theory of elliptic curves is to characterize the possible torsion structures over a given number field, or over all number fields of a given degree. One of the milestone in this subject was the characterization of the rational case given by Mazur in 1978. Later, the quadratic case was obtained by Kamienny, Kenku and Momose in 1992. For greater degree a complete answer for this problem is still open, although there have been some advances in the last years.
The purpose of this talk is to shed light on how the torsion group of an elliptic curve defined over the rationals grows upon base change.
This is an ongoing project partially joint with A. Lozano-Robledo, F. Najman and J. M. Tornero.
• Vendredi 26 janvier 2018
Chloe Martindale (Technical University of Eindhoven)
Isogeny graphs of abelian varieties and applications to the discrete logarithm problem.
An isogeny graph is a graph whose vertices correspond to abelian varieties (with some structure) and whose edges correspond to isogenies of a certain degree. The structure of isogeny graphs for elliptic curves was first studied by David Kohel in his PhD thesis and has since been an essential tool in algorithmic number theory (for example to compute the endomorphism ring of an ordinary elliptic curve over a finite field) and cryptography (for example in the post-quantum cryptographic protocol SIDH). We present a structure theorem for isogeny graphs of ordinary abelian varieties, and explain how, under some heuristic assumptions, this can be applied to attacking the discrete logarithm problem for genus 3 curves and possibly some families of elliptic curves.
• Vendredi 2 février 2018
Marius Vuille (École Polytechnique Fédérale de Lausanne)
Computing cyclic isogenies between principally polarized abelian varieties over finite fields.
The explicit computation of isogenies between abelian varieties over finite fields is an interesting problem that finds its applications not only in transporting instances of the Discrete Logarithm Problem, but can also be used to compute endomorphism rings or even in point counting. So far, only isogenies with kernel a maximal isotropic subgroup of the \ell - torsion have been described and implemented. We describe an algorithm to compute isogenies with cyclic kernels, provided they exist.
• Vendredi 9 février 2018
Frédéric Bihan (Université de Savoie)
Croissance du volume mixte pour les polytopes convexes et une règle de Cramer pour les systèmes polynomiaux.
Si P_1,...,P_n, Q_1,...,Q_n sont des polytopes convexes à sommets dans le réseau des points entiers de R^n et que P_i est inclus dans Q_i pour tout i, alors le volume mixte de (P_1,...,P_n) est majoré par celui de (Q_1,...,Q_n). Dans un travail récent avec Ivan Soprunov (Cleveland University), nous caractérisons les paires de n-uplets de polytopes pour lesquelles cette inégalité est stricte. Le lien entre volume mixte et systèmes polynomiaux est le théorème de Bernstein-Kouchnirenko qui prédit que le nombre de solutions dans le tore complexe d'un système polynomial de polytopes de Newton P_1,...,P_n est génériquement égal au volume mixte de ces polytopes. On utilise un de nos critères pour obtenir une règle de Cramer pour les systèmes polynomiaux qui généralise celle connue dans le cas linéaire.
• Vendredi 16 février 2018
Fabien Pazuky (Université de Copenhagen)
Courbes elliptiques et isogénies.
Deux courbes elliptiques E et E' définies sur un corps de nombres K sont isomorphes sur la clôture algébrique de K si et seulement si elles ont le même j-invariant. On se pose la question suivante : comment est-ce que ce j-invariant varie dans les classes d'isogénies ? On montre une majoration de la différence des hauteurs des j-invariants de courbes elliptiques isogènes et on en déduit plusieurs conséquences, notamment une borne explicite sur la hauteur des polynômes modulaires et un contrôle des formules de Vélu. Si le temps le permet, on fera de plus une remarque sur les rangs de Mordell-Weil des courbes elliptiques.
• Du lundi 19 au vendredi 23 février 2018
Conférence : Méthodes numériques pour les courbes algébriques

Voir par ici.