206 research outputs found
Using approximate roots for irreducibility and equi-singularity issues in K[[x]][y]
We provide an irreducibility test in the ring K[[x]][y] whose complexity is
quasi-linear with respect to the valuation of the discriminant, assuming the
input polynomial F square-free and K a perfect field of characteristic zero or
greater than deg(F). The algorithm uses the theory of approximate roots and may
be seen as a generalization of Abhyankhar's irreducibility criterion to the
case of non algebraically closed residue fields. More generally, we show that
we can test within the same complexity if a polynomial is pseudo-irreducible, a
larger class of polynomials containing irreducible ones. If is
pseudo-irreducible, the algorithm computes also the valuation of the
discriminant and the equisingularity types of the germs of plane curve defined
by F along the fiber x=0.Comment: 51 pages. Title modified. Slight modifications in Definition 5 and
Proposition 1
Computing Puiseux series : a fast divide and conquer algorithm
Let be a polynomial of total degree defined over
a perfect field of characteristic zero or greater than .
Assuming separable with respect to , we provide an algorithm that
computes the singular parts of all Puiseux series of above in less
than operations in , where
is the valuation of the resultant of and its partial derivative with
respect to . To this aim, we use a divide and conquer strategy and replace
univariate factorization by dynamic evaluation. As a first main corollary, we
compute the irreducible factors of in up to an
arbitrary precision with arithmetic
operations. As a second main corollary, we compute the genus of the plane curve
defined by with arithmetic operations and, if
, with bit operations
using a probabilistic algorithm, where is the logarithmic heigth of .Comment: 27 pages, 2 figure
On the complexity of computing with zero-dimensional triangular sets
We study the complexity of some fundamental operations for triangular sets in
dimension zero. Using Las-Vegas algorithms, we prove that one can perform such
operations as change of order, equiprojectable decomposition, or quasi-inverse
computation with a cost that is essentially that of modular composition. Over
an abstract field, this leads to a subquadratic cost (with respect to the
degree of the underlying algebraic set). Over a finite field, in a boolean RAM
model, we obtain a quasi-linear running time using Kedlaya and Umans' algorithm
for modular composition. Conversely, we also show how to reduce the problem of
modular composition to change of order for triangular sets, so that all these
problems are essentially equivalent. Our algorithms are implemented in Maple;
we present some experimental results
Computing the equisingularity type of a pseudo-irreducible polynomial
Germs of plane curve singularities can be classified accordingly to their equisingularity type. For singularities over C, this important data coincides with the topological class. In this paper, we characterise a family of singularities, containing irreducible ones, whose equisingularity type can be computed in quasi-linear time with respect to the discriminant valuation of a Weierstrass equation
Computing Monodromy via Continuation Methods on Random Riemann Surfaces
International audienceWe consider a Riemann surface defined by a polynomial of degree , whose coefficients are chosen randomly. Hence, we can suppose that is smooth, that the discriminant of has simple roots, , and that i.e. the corresponding fiber has distinct points . When we lift a loop 0 \in \gamma \subset \Ci - \Delta by a continuation method, we get paths in connecting , hence defining a permutation of that set. This is called monodromy. Here we present experimentations in Maple to get statistics on the distribution of transpositions corresponding to loops around each point of . Multiplying families of ''neighbor'' transpositions, we construct permutations and the subgroups of the symmetric group they generate. This allows us to establish and study experimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups. Assuming that these two conjectures are true, we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization, under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions.On considere une surface de Riemann dont l'equation f(x,y)=0 est un polynome dont les coefficients sont des variables aleatoires Gaussiennes standards, ainsi que sa projection p sur l'axe des x. Puis on etudie et calcule des generateurs du groupe de monodromie correspondant a p
Accompagnement et pratiques pédagogiques dans l’enseignement supérieur
Recherche et formation : En regard de ton expérience de chercheuse et d’enseignante dans l’enseignement supérieur, quelle est ton interprétation du terme « accompagnement », qui émerge d’ailleurs hors du champ de l’enseignement supérieur ? Comment interprètes-tu l’usage de ce terme quand il s’agit de l’orientation, de la formation, de l’insertion des étudiants ? Comment expliques-tu la présence de cette notion dans le vocabulaire institutionnel et aussi dans le vocabulaire pédagogique des ens..
Algorithm for connectivity queries on real algebraic curves
We consider the problem of answering connectivity queries on a real algebraic
curve. The curve is given as the real trace of an algebraic curve, assumed to
be in generic position, and being defined by some rational parametrizations.
The query points are given by a zero-dimensional parametrization. We design an
algorithm which counts the number of connected components of the real curve
under study, and decides which query point lie in which connected component, in
time log-linear in , where is the maximum of the degrees and
coefficient bit-sizes of the polynomials given as input. This matches the
currently best-known bound for computing the topology of real plane curves. The
main novelty of this algorithm is the avoidance of the computation of the
complete topology of the curve.Comment: 10 pages, 2 figure
Intégrité génomique et repeuplements chez la truite commune du versant méditerranéen
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