Section: Scientific Foundations
Algebraic Geometric Computing
The underlying representation behind the geometric model that we consider are often of algebraic type. Computing with such models raise algebraic questions, which frequently appear as bottlenecks of the geometric problems.
In order to compute the solutions of a system of polynomial equations in several variables, we analyse and take advantage of the structure of the quotient ring, defined by these polynomials. This raises questions of representing and computing normal forms in such quotient structures. The numerical and algebraic computations in this context lead us to study new approaches of normal form computations, generalizing the well-known Gröbner bases. We are also 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.
We are also interested in subdivision methods, which are able to localise efficiently the real roots of polynomial equations. The specificities of these methods are local behavior, fast convergence properties and robustness. Key problems are related to the analysis of multiple points.
An important issue in analysing these methods is how to obtain good complexity bounds by exploiting the structure of the problem. Many algebraic problems can be reformulated in terms of linear algebra questions. Thus, it is not surprising to see that complexity analysis of our methods leads 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 their implications in solving polynomial equations.
When solving a system of polynomials equations, a first treatment is to transform it into several simpler subsystems when possible. The problem of decomposition and factorisation is thus also important. 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.