## Section: New Results

### Symbolic numeric computation

#### Solving Toeplitz-block linear systems

Participants : Houssam Khalil, Bernard Mourrain, Michelle Schatzmann [ Univ. Lyon I ] .

Structured matrices appear in various domain, such as scientific computing, signal processing,...Among well-known structured matrices. Toeplitz and Hankel structures have been intensively studied. So far, few investigations have been pursued for the treatment of multi-level structured matrices. We found a relation between Toeplitz matrices and syzygies, and algorithms to compute these syzygies. In following the same way in the univariate case, we gived a relation between bloc-Toeplitz-bloc (TBT) matrices and bivariate syzygies. We are looking for a fast algorithm to compute these syzygies. By computing the SVD of the displacement of a banded TBT matrix we remark that almost of them are very small, which means that we can aproximate this matrix by a small rank matrix. We are trying also to trasform a TBT linear system to another multi-level structured system which was called Cauchy-bloc-Cauchy sytem type, which has a relation with the bivariate interpolation problem.

#### Algorithms for bivariate polynomial absolute factorization

Participants : Guillaume Chèze, André Galligo, Grégoire Lecerf [ CNRS, Univ. Versailles ] .

Absolute factorization stands for factorization over the algebraic closure of the ground field. This is interesting for the applications; e.g. for a multivariate polynomial with rational coefficients it allows us to decompose further than the rational factorization, for instance into polynomials with real coefficients. This is a long standing problem in Computer algebra.

G. Chèze, André Galligo and G. Lecerf are working on an improvement of an approach initiated by Gao using exact modular and p-adic computations. They already obtained nearly optimal complexity bounds. This work is still in progress.

#### Geometry and optimisation

Participants : Nikos Pavlidis [ University of Patras ] , Bernard Mourrain, Michael Vrahatis [ University of Patras ] .

In a work developped in the context of our associate team CALAMATA with Greece, we investigated the application of optimisation methods in geometric problems. We have examined the ability of feedforward neural networks to identify the number of real roots of univariate polynomials, and more generally the ability to approximate semi-algebraic sets. The obtained experimental results indicate that neural networks are capable of performing this task with a high accuracy even when the training set is very small compared to the test set (see [15] ). We are currently investigating applications of Particule Sworm Optimisation methods to geometric and algebraic problems, such are root isolation, 3D reconstruction from distances in Molecular Biology.

#### Tensors, symmetric tensors and rank

Participants : Pierre Common [ I3S ] , Gene Golub [ Stanford Univ. ] , Lek-Heng Lim [ Stanford Univ. ] , Bernard Mourrain.

There are several extensions of the Singular Value Decomposition (SVD) to tensors. In this work, we are interested in the Canonical Decomposition, into a minimal sum of outer products of vectors. In particular, this allows us to define a tensor rank that restricts to the matrix rank for order-2 tensors. This decomposition is essential in the process of Blind Identification of Under-Determined Mixtures (UDM), i.e., linear mixtures with more inputs than observable outputs and appear in many application areas, including Speech, Mobile Communications, Machine Learning, Factor Analysis with k-way arrays (MWA), Biomedical Engineering, Psychometrics, and Chemometrics. In this work, we study various properties of symmetric tensors in relation to a decomposition into a sum of outer product of vectors. A rank-1 order-k tensor is the outer product of k vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of them being symmetric or not. The Rank of a symmetric tensor is the minimal number of rank-1 tsWeensors that are necessary to reconstruct it. The Symmetric Rank is obtained when constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that Rank and Symmetric Rank are generically equal, and that they always exist in an algebraically closed field; in the real field for instance, it is necessary to define several Typical Ranks. The Generic Rank is generally not maximal, contrary to matrices, and is now known for any values of dimension and order. These properties are described in a paper submitted for a journal publication.