# Project : galaad

## Section: Other Grants and Activities

### Bilateral actions

#### Associated team CALAMATA

Participants : Ioannis Emiris, Athanasios Kakargias, Bernard Mourrain [contact person], Nikos Pavlidis, Jean-Pierre Técourt, Monique Teillaud, Elias Tsigaridas, Michael Vrahatis.

The Team of Geometric and Algebraic Algorithms at the National University of Athens, Greece, has been associated with GALAAD since 2003. See its web site.

This bilateral collaboration is entitled CALAMATA (CALculs Algebriques, MATriciels et Applications). The Greek team ( http://www.di.uoa.gr/~erga/) is headed by Ioannis Emiris.

The focus of this project is the solution of polynomial systems by matrix methods. Our approach leads naturally to problems in structured and sparse matrices. Real root isolation, either of one univariate polynomial or of a polynomial system, is of special interest, especially in applications in geometric modeling, CAGD or nonlinear computational geometry. We are interested in computational geometry, actually, in what concerns curves and surfaces. The framework of this work has been the European project ECG.

In 2004, 4 members of the Greek team visited INRIA, either for week-long visits or for longer visits (from one to 3 months). Two INRIA researchers visited Athens for one week. We also mention the participation of members of both teams in international or national conferences: the final workshop of ECG in Paris, the conference on Algebraic geometry and geometric modeling in Nice, and the conference in honor of Daniel Lazard in Paris.

#### NSF-INRIA collaboration

Participants : Laurent Busé, André Galligo, Mohamed Elkadi, Bernard Mourrain [contact person], Jean-Pierre Técourt.

The objective of this collaboration between GALAAD and the Geometric Modeling group at Rice University in Houston, Texas (USA) is to investigate techniques from Effective Algebraic Geometry in order to solve some of the key problems in Geometric Modeling and Computational Biology. The two groups have similar interests and complementary strengths. Effective Algebraic Geometry is the branch of Algebraic Geometry that pursues concrete algorithms rather than abstract proofs. It deals mainly with practical methods for representing polynomial curves and surfaces along with robust techniques for solving systems of polynomial equations. Many applications in Geometric Modeling and Computational Biology require fast robust methods for solving systems of polynomial equations. Here we concentrate our collective efforts on solving standard problems such as implicitization, inversion, intersection, and detection of singularities for rational curves and surfaces. To aid in modeling, we shall also investigate some novel approaches to representing shape. In contemporary Computational Biology, many problems can be reduced to solving large systems of low degree polynomial equations. We plan to apply our polynomials solvers together with new tools for analyzing complex shapes to help study these currently computationally intractable problems.

#### NCSU-GALAAD collaboration

Participant : Olivier Ruatta.

Agnes Szanto works at the department of mathematics of the North Carolina State University.

This is a project on overdetermined algebraic systems funded by a U.S. grant of one year. Olivier Ruatta went to Raleigh in February 2004 for 18 days and Agnes Szanto came at Sophia Antipolis in May 2004 for 13 days. This collaboration is prolongated and is partially supported by a NFS grant obtained by A. Szanto which include founds for this work.

The objective of this investigation is to develop and implement highly efficient algorithms for the solution of over-constrained polynomial systems with finitely many, possibly multiple roots over the complex numbers, when the input is given with inexact coefficients. We refer to such problems as inexact degenerate systems. Both ``resultant based'' and analytic iterative methods are considered to tackle this problem using the large number of already existing works. The researchers addresses the problem of the definition of ``nearly consistent'' systems, with computational methods generalizing the S.V.D. of the linear case. The complexity is one of the central issues of this research since we want efficient methods.The first results obtained are related to overdetermined Weierstrass iteration in [28][34] which had been presented in different conferences.