## Section: Scientific Foundations

### Symbolic-numeric computation

Either in geometric or algebraic problems, symbolic and numeric computation are closely intertwined. Our aim is to exploit the complementarity of these domains, in order to develop controlled methods, as explained now.

#### Certification

The numerical problems are often approached locally. However in many situations, it is significant to give global answers, making it possible to certify calculations. The symbolic-numeric approach combining the algebraic and analytical aspects, intends to address these local-global problems. Especially, we focus on certification of geometric predicates that are essential for the analysis of geometrical structures [30] .

#### Approximation

The sequence of geometric constructions, if treated in an exact way, often leads to a rapid complexification of the problems. It is then significant to be able to approximate these objects while controlling the quality of approximation. Subdivision techniques based on the algebraic formulation of our problems are exploited in order to control the approximation, while locating interesting features such as singularities. This approach combines geometrical, algebraic and algorithmic aspects.

#### Degeneracies and arithmetic

According to an engineer in CAGD,
the problems of singularities obey the following rule:
less than 20% of the treated cases are
singular, but more than 80% of time is necessary to develop a code
allowing to treat them correctly. Degenerated cases are thus critical from both
theoretical and practical perspectives.
To resolve these difficulties, in addition to the
qualitative studies and classifications, we study methods of *perturbations* of symbolic systems, or adaptive methods based on
exact arithmetics. For example, we work on the computation of the
sign of expressions, and on approaches combining modular and approximate
computations, which speed up the exact answer
[25] .