MOURRAIN , Bernard

Semi-algebraic sets occur naturally when dealing with implicit models and boolean operations between them. In this work we present an algorithm to efficiently and in a certified way compute the connected components of semi-algebraic sets given by intersection or union of conjunctions of bi-variate equalities and inequalities. For any given...

In this chapter, we review applications of Polynomial Optimization techniques to Geometric Modeling problems. We present examples of topical problems in Geometric Modeling, illustrate their solution using Polynomial Optimization Tools, report some experimental results and analyse the behavior of the methods, showing what are their strengths and...

In this paper, we are interested in the use of duality in effective computations on polynomials. We represent the elements of the dual of the algebra R of polynomials over the field K as formal series in K[[d]] in differential operators. We use the correspondence between ideals of R and vector spaces of K[[d]], stable by derivation and closed...

We analyze the decomposition problem of multivariate polynomial-exponential functions from truncated series and present new algorithms to compute their decomposition. Using the duality between polynomials and formal power series, we first show how the elements in the dual of an Artinian algebra correspond to polynomial-exponential functions....

In this tutorial paper, we first discuss the motivation of doing symbolic-numeric computation, with the aim of developing efficient and certified polynomial solvers. We give a quick overview of fundamental algebraic properties, used to recover the roots of a polynomial system, when we know the multiplicative structure of its quotient algebra....

We study of the punctual Hilbert scheme from an algorithmic point of view. We first present algorithms, which allow to compute the inverse system of an isolated point. We define the punctual Hilbert scheme as a subvariety of a Grassmannian variety and provide explicit equations defining it. Then we localised our study to the algebraic variety...

École thématique

Given a multi-index sequence $σ$, we present a new efficient algorithm to compute generators of the linear recurrence relations between the terms of $σ$. We transform this problem into an algebraic one, by identifying multi-index sequences, multivariate formal power series and linear functionals on the ring of multivariate polynomials. In this...