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 $F$ 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 $F\in \mathbb{K}[X, Y ]$ be a polynomial of total degree $D$ defined over a perfect field $\mathbb{K}$ of characteristic zero or greater than $D$. Assuming $F$ separable with respect to $Y$ , we provide an algorithm that computes the singular parts of all Puiseux series of $F$ above $X = 0$ in less than $\tilde{\mathcal{O}}(D\delta)$ operations in $\mathbb{K}$, where $\delta$ is the valuation of the resultant of $F$ and its partial derivative with respect to $Y$. 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 $F$ in $\mathbb{K}[[X]][Y ]$ up to an arbitrary precision $X^N$ with $\tilde{\mathcal{O}}(D(\delta + N ))$ arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by $F$ with $\tilde{\mathcal{O}}(D^3)$ arithmetic operations and, if $\mathbb{K} = \mathbb{Q}$, with $\tilde{\mathcal{O}}((h+1)D^3)$ bit operations using a probabilistic algorithm, where $h$ is the logarithmic heigth of $F$.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 Monodromy via Continuation Methods on Random Riemann Surfaces

International audienceWe consider a Riemann surface $X$ defined by a polynomial $f(x,y)$ of degree $d$, whose coefficients are chosen randomly. Hence, we can suppose that $X$ is smooth, that the discriminant $\delta(x)$ of $f$ has $d(d-1)$ simple roots, $\Delta$, and that $\delta(0) \neq 0$ i.e. the corresponding fiber has $d$ distinct points $\{y_1, \ldots, y_d\}$. When we lift a loop 0 \in \gamma \subset \Ci - \Delta by a continuation method, we get $d$ paths in $X$ connecting $\{y_1, \ldots, y_d\}$, 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 $\Delta$. 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

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

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 $N^6$, where $N$ 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

Computing Puiseux series: a fast divide and conquer algorithm

Let $F ∈ K[X, Y ]$ be a polynomial of total degree D defined over a field K of characteristic zero or greater than D. Assuming F separable with respect to Y , we provide an algorithm that computes all Puiseux series of F above X = 0 in less than $O˜(D δ)$ operations in K, where δ is the valuation of the resultant of F and its partial derivative with respect to Y. To this aim, we use a divide and conquer strategy and replace univariate factorisation by dynamic evaluation. As a first main corollary, we compute the irreducible factors of F in $K[[X]][Y ]$ up to an arbitrary precision X N with $O˜(D(δ + N))$ arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by F with $O˜(D^3)$ arithmetic operations and, if K = Q, with $O˜((h+1) D^3)$ bit operations using probabilistic algorithms, where h is the logarithmic height of F

Modular Composition Modulo Triangular Sets and Applications

International audienceWe generalize Kedlaya and Umans' modular composition algorithm to the multivariate case. As a main application, we give fast algorithms for many operations involving triangular sets (over a finite field), such as modular multiplication, inversion, or change of order. For the first time, we are able to exhibit running times for these operations that are almost linear, without any overhead exponential in the number of variables. As a further application, we show that, from the complexity viewpoint, Charlap, Coley, and Robbins' approach to elliptic curve point counting can be competitive with the better known approach due to Elkies