Adrien Poteaux

Computing monodromy groups defined by plane algebraic curves

We present a symbolic-numeric method to compute the monodromy group of a plane algebraic curve viewed as a ramified covering space of the complex plane. Following the definition, our algorithm is based on analytic continuation of algebraic functions above paths in the complex plane. Our contribution is three-fold : first of all, we show how to use a minimum spanning tree to minimize the length of paths ; then, we propose a strategy that gives a good compromise between the number of steps and the truncation orders of Puiseux expansions, obtaining for the first time a complexity result about the number of steps; finally, we present an efficient numerical-modular algorithm to compute Puiseux expansions above critical points,which is a non trivial task.

Complexity bounds for the rational Newton-Puiseux algorithm over finite fields

We carefully study the number of arithmetic operations required to compute rational Puiseux expansions of a bivariate polynomial F over a finite field. Our approach is based on the rational Newton-Puiseux algorithm introduced by D. Duval. In particular, we prove that coefficients of F may be significantly truncated and that certain complexity upper bounds may be expressed in terms of the output size. These preliminary results lead to a more efficient version of the algorithm with a complexity upper bound that improves previously published results. We also deduce consequences for the complexity of the computation of the genus of an algebraic curve defined over a finite field or an algebraic number field. Our results are practical since they are based on well established subalgorithms, such as fast multiplication of univariate polynomials with coefficients in a finite field.

Good reduction of puiseux series and complexity of the Newton-Puiseux algorithm over finite fields

In [12], we sketched a numeric-symbolic method to compute Puiseux series with floating point coefficients. In this paper, we address the symbolic part of our algorithm. We study the reduction of Puiseux series coefficients modulo a prime ideal and prove a good reduction criterion sufficient to preserve the required information, namely Newton polygon trees. We introduce a convenient modification of Newton polygons that greatly simplifies proofs and statements of our results. Finally, we improve complexity bounds for Puiseux series calculations over finite fields, and estimate the bit-complexity of polygon tree computation.

Modular Composition Modulo Triangular Sets and Applications

We 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.

Hierarchical Spline Approximation of the Signed Distance Function

We present a method to approximate the signed distance function of a smooth curve or surface by using polynomial splines over hierarchical T-meshes (PHT splines). In particular, we focus on closed parametric curves in the plane and implicitly defined surfaces in space.

Improving Complexity Bounds for the Computation of Puiseux Series over Finite Fields

Let K be a field of characteristic p with q elements and FΕ in L[X,Y] be a polynomial with p> deg_Y(F) and total degree d. In [40], we showed that rational Puiseux series of F above X=0 could be computed with an expected number of O~(d5+d3log q) arithmetic operations in L. In this paper, we reduce this bound to O~(d4+d2log q) using Hensel lifting and changes of variables in the Newton-Puiseux algorithm that give a better control of the number of steps. The only asymptotically fast algorithm required is polynomial multiplication over finite fields. This approach also allows to test the irreducibility of F in L[[X]][Y] with O(d3) operations in K. Finally, we describe a method based on structured bivariate multiplication [34] that may speed up computations for some input.

An Algorithm for Converting Nonlinear Differential Equations to Integral Equations with an Application to Parameter Estimation from Noisy Data

This paper provides a contribution to the parameter estimation methods for nonlinear dynamical systems. In such problems, a major issue is the presence of noise in measurements. In particular, most methods based on numerical estimates of derivations are very noise sensitive. An improvement consists in using integral equations, acting as noise filtering, rather than differential equations. Our contribution is a pair of algorithms for converting fractions of differential polynomials to integral equations. These algorithms rely on an improved version of a recent differential algebra algorithm. Their usefulness is illustrated by an application to the problem of estimating the parameters of a nonlinear dynamical system, from noisy data.

Computing monodromy via continuation methods on random Riemann surfaces

We 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 ?(x) of f has d(d?1) simple roots, ?, and that ?(0)?0, i.e. the corresponding fiber has d distinct points {y1,?,yd}. When we lift a loop 0???C?? by a continuation method, we get d paths in X connecting {y1,?,yd}, 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.

Approximate Implicitization of Space Curves

The process of implicitization generates an implicit representation of a curve or surface from a given parametric one. This process is potentially interesting for applications in Computer Aided Design, where the robustness and efficiency of intersection algorithm can be improved by simultaneously considering implicit and parametric representations. This paper gives an brief survey of the existing techniques for approximate implicitization of hyper surfaces. In addition it describes a framework for the approximate implicitization of space curves.

An Equivalence Theorem For Regular Differential Chains

This paper provides new equivalence theorems for regular chains and regular differential chains, which are generalizations of Ritts characteristic sets. These theorems focus on regularity properties of elements of residue class rings defined by these chains, which are revealed by resultant computations. New corollaries to these theorems have quite simple formulations.