# Project : galaad

## Section: Scientific Foundations

### Resolution of algebraic systems

The underlying representation behind a geometric model is often of algebraic type. Computing with such models raise algebraic questions, which frequently appear as bottlenecks of the geometric problems. Here are the particular approaches that we develop to handle such questions.

#### Algebraic methods and quotient structure

In order to compute the solutions of a system of polynomial equations in several variables, we analyze and take advantage of the structure of the quotient ring, defined by these polynomials. This raises questions of representing and calculating normal forms in such structures. The numerical and algebraic computations in this context lead us to study new approaches of normal form computations, generalizing the well-known GrÃ¶bner bases.

#### Duality, residues, interpolation

We are interested in the ``effective'' use of duality, that is, the properties of linear forms on the polynomials or quotient rings by ideals. We undertake a detailed study of these tools from an algorithmic perspective, which yields the answer to basic questions in algebraic geometry and brings a substantial improvement on the complexity of resolution of these problems. Our focuses are effective computation of the algebraic residue, interpolation problems, and the relation between coefficients and roots in the case of multivariate polynomials.

#### Structured linear algebra and polynomials

The preceding work lead naturally to the theory of structured matrices. Indeed, the matrices resulting from polynomial problems, such as matrices of resultants or Bezoutians, are structured. Their rows and columns are naturally indexed by monomials, and their structures generalize the Toeplitz matrices to the multivariate case. We are interested in exploiting these properties and the implications in solving polynomial equations [48].

#### Decomposition and factorisation

When solving a system of polynomials equations, a first treatment is to transform it into several simpler subsystems when possible. We are interested in a new type of algorithms that combine the numerical and symbolic aspects, and are simultaneously more effective and reliable. For instance, the (difficult) problem of approximate factorization, the computation of perturbations of the data, which enables us to break up the problem, is studied. More generally, we are working on the problem of decomposing a variety into irreducible components.

#### Deformation and homotopy

The behavior of a problem in the vicinity of a data can be interpreted in terms of deformations. Accordingly, the methods of homotopy consist in introducing a new parameter and in following the evolution of the solutions between a known position and the configuration one seeks to solve. This parameter can also be introduced in a symbolic manner, as in the techniques of perturbation of non-generic situations. We are interested in these methods, in order to use them in the resolution of polynomial equations as well as for new algorithms of approximate factorization.