Éric Schost

Lifting techniques for triangular decompositions

We present lifting techniques for triangular decompositions of zero-dimensional varieties, that extend the range of the previous methods. We discuss complexity aspects, and report on a preliminary implementation. Our theoretical results are comforted by these experiments.

Computing Parametric Geometric Resolutions

Abstract Given a polynomial system of n equations in n unknowns that depends on some parameters, we define the notion of parametric geometric resolution as a means to represent some generic solutions in terms of the parameters. The coefficients of this resolution are rational functions of the parameters; we first show that their degree is bounded by the Bézout number dn, where d is a bound on the degrees of the input system. Then we present a probabilistic algorithm to compute a parametric resolution. Its complexity is polynomial in the size of the output and in the complexity of evaluation of the input system. The probability of success is controlled by a quantity polynomial in the Bézout number. We present several applications of this process, notably to computa- tions in the Jacobian of hyperelliptic curves and to questions of real geometry.

Polar varieties and computation of one point in each connected component of a smooth real algebraic set

Let f<inf>1</inf>, ldots, f<inf>s</inf> be polynomials in <b>Q</b>[X<inf>1</inf>, ..., X<inf>n</inf>] that generate a radical ideal and let V be their complex zero-set. Suppose that V is smooth and equidimensional; then we show that computing suitable sections of the polar varieties associated to generic projections of V gives at least one point in each connected component of V ∩ <b>R</b>ⁿ. We deduce an algorithm that extends that of Bank, Giusti, Heintz and Mbakop to non-compact situations. Its arithmetic complexity is polynomial in the complexity of evaluation of the input system, an intrinsic algebraic quantity and a combinatorial quantity.

Polynomial evaluation and interpolation on special sets of points

We give complexity estimates for the problems of evaluation and interpolation on various polynomial bases. We focus on the particular cases when the sample points form an arithmetic or a geometric sequence, and we discuss applications, respectively, to computations with linear differential operators and to polynomial matrix multiplication.

Tellegen's principle into practice

The transposition principle, also called Tellegens principle, is a set of transformation rules for linear programs. Yet, though well known, it is not used systematically, and few practical implementations rely on it. In this article, we propose explicit transposed versions of polynomial multiplication and division but also new faster algorithms for multipoint evaluation, interpolation and their transposes. We report on their implementation in Shoups NTL C++ library.

Sharp estimates for triangular sets

We study the triangular representation of zero-dimensional varieties defined over the rational field (resp. a rational function field). We prove polynomial bounds in terms of intrinsic quantities for the height (resp. degree) of the coefficients of such triangular sets, whereas previous bounds were exponential. We also introduce a rational form of triangular representation, for which our estimates become linear. Experiments show the practical interest of this new representation.

FAST ALGORITHMS FOR COMPUTING ISOGENIES BETWEEN ELLIPTIC CURVES

We survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime. We introduce a new algorithm that computes an isogeny of degree l (l different from the characteristic) in time quasi-linear with respect to l. This is based in particular on fast algorithms for power series expansion of the Weierstrass ℘-function and related functions.

Construction of Secure Random Curves of Genus 2 over Prime Fields

For counting points of Jacobians of genus 2 curves defined over large prime fields, the best known method is a variant of Schoofs algorithm. We present several improvements on the algorithms described by Gaudry and Harley in 2000. In particular we rebuild the symmetry that had been broken by the use of Cantors division polynomials and design a faster division by 2 and a division by 3. Combined with the algorithm by Matsuo, Chao and Tsujii, our implementation can count the points on a Jacobian of size 164 bits within about one week on a PC.

Properness Defects of Projections and Computation of at Least One Point in Each Connected Component of a Real Algebraic Set

Abstract Computing at least one point in each connected component of a real algebraic set is a basic subroutine to decide emptiness of semi-algebraic sets, which is a fundamental algorithmic problem in effective real algebraic geometry. In this article we propose a new algorithm for the former task, which avoids a hypothesis of properness required in many of the previous methods. We show how studying the set of non-properness of a linear projection Π enables us to detect the connected components of a real algebraic set without critical points for Π. Our algorithm is based on this observation and its practical counterpoint, using the triangular representation of algebraic varieties. Our experiments show its efficiency on a family of examples.

On the geometry of polar varieties

We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are necessary to prove the correctness and complexity estimates of these algorithms. Our results form also the geometrical main ingredients for the computational treatment of singular hypersurfaces. In particular, we show the non–emptiness of suitable generic dual polar varieties of (possibly singular) real varieties, show that generic polar varieties may become singular at smooth points of the original variety and exhibit a sufficient criterion when this is not the case. Further, we introduce the new concept of meagerly generic polar varieties and give a degree estimate for them in terms of the degrees of generic polar varieties. The statements are illustrated by examples and a computer experiment.