Generation modulo the action of a permutation group

International audienceOriginally motivated by algebraic invariant theory, we present an algorithm to enumerate integer vectors modulo the action of a permutation group. This problem generalizes the generation of unlabeled graph up to an isomorphism. In this paper, we present the full development of a generation engine by describing the related theory, establishing a mathematical and practical complexity, and exposing some benchmarks. We next show two applications to effective invariant theory and effective Galois theory

Effective Invariant Theory of Permutation Groups using Representation Theory

Using the theory of representations of the symmetric group, we propose an algorithm to compute the invariant ring of a permutation group. Our approach have the goal to reduce the amount of linear algebra computations and exploit a thinner combinatorial description of the invariant ring.Comment: Draft version, the corrected full version is available at http://www.springer.com

Galois theory, splitting fields and computer algebra

AbstractWe provide some algorithms for dynamically obtaining both a possible representation of the splitting field and the Galois group of a given separable polynomial from its universal decomposition algebra

Evaluation properties of invariant polynomials

AbstractA 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

Homotopy techniques for multiplication modulo triangular sets

International audienceWe study the cost of multiplication modulo triangular families of polynomials. Following previous work by Li et al. (2007), we propose an algorithm that relies on homotopy and fast evaluation-interpolation techniques. We obtain a quasi-linear time complexity for substantial families of examples, for which no such result was known before. Applications are given notably to additions of algebraic numbers in small characteristic

Computability for the absolute Galois group of Q\mathbb{Q}

The absolute Galois group Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$ of the field $\mathbb{Q}$ of rational numbers can be presented as a highly computable object, under the notion of type-2 Turing computation. We formalize such a presentation and use it to address several effectiveness questions about Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$: the difficulty of computing Skolem functions for this group, the arithmetical complexity of various definable subsets of the group, and the extent to which countable subgroups defined by complexity (such as the group of all computable automorphisms of the algebraic closure $\overline{\mathbb{Q}}$) may be elementary subgroups of the overall group

Explicit Methods in Number Theory

These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes effectiveness in rational points on curves and especially on modular curves, modularity, L-functions, and many other topics