Section: New Results

Algebraic Geometric Analysis

A sparse flat extension theorem for moment matrices

Participant : Bernard Mourrain.

We prove a generalization of the flat extension theorem of Curto and Fialkow for truncated moment matrices. It applies to moment matrices indexed by an arbitrary set of monomials and its border, assuming that this set is connected to 1. When formulated in a basis-free setting, this gives an equivalent result for truncated Hankel operators. Applications to real radical computation and tensor decomposition are considered.

This work was done in collaboration with Monique Laurent, CWI, Nertherland, and published in [19] .

Differential invariants of a Lie group action: syzygies on a generating set

Participant : Evelyne Hubert.

Given a group action, known by its infinitesimal generators, we exhibit a complete set of syzygies on a generating set of differential invariants. For that we elaborate on the reinterpretation of Cartan's moving frame by Fels and Olver (1999). This provides constructive tools for exploring algebras of differential invariants.

This research on the algebraic structure of differential invariants was initiated with a view towards differential elimination. Yet differential invariants are essential means of characterizing a form independently form a group action (such as the traditional rigid motions).

This work has been published in [18] .

On the total order of reducibility of a pencil of algebraic plane curves

Participant : Laurent Busé.

In this work, the problem of bounding the number of reducible curves in a pencil of algebraic plane curves is addressed. Unlike most of the previous related works, each reducible curve of the pencil is here counted with its appropriate multiplicity. It is proved that this number of reducible curves, counted with multiplicity, is bounded by d²-1 where d is the degree of the pencil. Then, a sharper bound is given by taking into account the Newton's polygon of the pencil.

A preprint version of this work, done in collaboration with Guillaume Chèze, Univ. Toulouse and submitted for publication, is available at http://hal.archives-ouvertes.fr/hal-00348561/en/ .

Noether's forms for the study of non-composite rational functions and their spectrum

Participant : Laurent Busé.

In this work, the spectrum and the decomposability of a multivariate rational function are studied by means of the effective Noether's irreducibility theorem given by Ruppert. With this approach, some new effective results are obtained. In particular, we show that the reduction modulo p of the spectrum of a given integer multivariate rational function r coincides with the spectrum of the reduction of r modulo p for p a prime integer greater or equal to an explicit bound. This bound is given in terms of the degree, the height and the number of variables of r . With the same strategy, we also study the decomposability of r modulo p . Some similar explicit results are also provided for the case of polynomials with coefficients in a polynomial ring.

A preprint version of this work, done in collaboration with Guillaume Chèze, Univ. Toulouse and Salah Najib, Univ. Lille and submitted for publication, is available at http://hal.inria.fr/inria-00395839/en/ .

Analysis of intersection of quadrics through signature sequences

Participant : Bernard Mourrain.

We present an efficient method for classifying the morphology of the intersection curve of two quadrics (QSIC) in PR3, 3D real projective space; here, the term morphology is used in a broad sense to mean the shape, topological, and algebraic properties of a QSIC, including singularity, reducibility, the number of connected components, and the degree of each irreducible component, etc. There are in total 35 different QSIC morphologies with non-degenerate quadric pencils. For each of these 35 QSIC morphologies, through a detailed study of the eigenvalue curve and the index function jump we establish a characterizing algebraic condition expressed in terms of the Segre characteristics and the signature sequence of a quadric pencil. We show how to compute a signature sequence with rational arithmetic in order to determine the morphology of the intersection curve of any two given quadrics. Two immediate applications of our results are the robust topological classification of QSIC in computing B-rep surface representation in solid modeling and the derivation of algebraic conditions for collision detection of quadric primitives

This work is done in collaboration with Changhe Tu, Shandong University, Wenping Wang, Hong Kong University, Jiaye, Wang, Shandong University and is published in [21] .

On the distribution of the solutions of systems of polynomial equations

Participant : André Galligo.

In a work done in collaboration with Carlos D'Andrea, Univ. Barcelona and Martin Sombra, Univ. Bordeaux, we generalize a celebrated result, due to P. Erdös and P. Turán, on the distribution of roots of univariate polynomials to the sparse multivariate case.

In relation with probabilistic algorithms for monodromy computation, we consider K2 independent copies of the random walk on the symmetric group S_N starting from the identity and generated by the products of either independent uniform transpositions or independent uniform successive transpositions. At any time , let G_n be the subgroup of S_N generated by the K positions of the chains. In the uniform transposition model, we prove that there is a cut-off phenomenon at time Nln(N)/(2K) for the non-existence of fixed point of G_n and for the transitivity of G_n , thus showing that these properties occur before the chains have reach equilibrium. In the uniform successive transposition model, a transition for the non-existence of fixed point of G_n appears at time of order (at least for K3 ), but there is no cut-off phenomenon. In the latter model, we recover a cut-off phenomenon for the non-existence of fixed point at a time proportional to N by allowing the number K to be proportional to ln(N) . The main tools of the proofs are spectral analysis and coupling techniques.

A preprint of this work, done in collaboration with Laurent Miclo, Univ. Toulouse and submitted for publication, is available at http://hal.archives-ouvertes.fr/hal-00384188/en/ .