The team develops new computational methods to obtain exact symbolic representations and certified qualitative information for the solution of certain classes of differential, polynomial and more general functional equations. Such techniques are also translated into software implementations , designed to be used alongside purely numerical methods for the solution of scientific problems.

**Differential and Functional Equations **

Analysis of singularities, asymptotic computations, local solutions, effective differential Galois theory, integrability of Hamiltonian systems.

Effective algebraic analysis of linear functional systems.

#### Algebra and Geometry: symbolic and numeric approaches

Elimination, effective algebraic geometry, numerical linear algebra.

Certified numerical computations : topology of curves and surfaces, series expansions on critical points.

Symbolic-numeric methods for polynomial equations, with applications to semialgebraic optimization; approximate gcd.

Structured matrices: approximate solutions, eigenvalue computation, quasiseparable structure, matrix functions.

Algebraic cryptoanalysis and decoding.

Parametric algebraic models.

#### Software

Maple libraries: *Isolde*, *OreMorphisms*, contributions to *LREtools* (distributed with Maple) ;

Maple Beta programs : regular solutions of differential systems, integrability of Hamiltonian systems, SERRE package;

Software for computer algebra Mathemagix, in collaboration with École Polytechnique and INRIA.

Group seminar.