## Section: Research Program

### Solving Systems over the Reals and Applications.

Participants : Mohab Safey El Din, Elias Tsigaridas, Daniel Lazard, Thi Xuan Vu.

We shall develop algorithms for solving polynomial systems over complex/real numbers. Again, the goal is to extend significantly the range of reachable applications using algebraic techniques based on Gröbner bases and dedicated linear algebra routines. Targeted application domains are global optimization problems, stability of dynamical systems (e.g. arising in biology or in control theory) and theorem proving in computational geometry.

The following functionalities shall be requested by the end-users:

*(i)* deciding the emptiness of the real solution set of systems
of polynomial equations and inequalities,

*(ii)* quantifier
elimination over the reals or complex numbers,

*(iii)*
answering
connectivity queries for such real solution sets.

We will focus on these functionalities.

We will develop algorithms based on the so-called critical point
method to tackle systems of equations and inequalities (problem *(i)*) . These techniques are based on solving 0-dimensional
polynomial systems encoding "critical points" which are defined by
the vanishing of minors of Jacobian matrices (with polynomial
entries). Since these systems are highly structured, the expected
results of Objective 1 and 2 may allow us to obtain dramatic
improvements in the computation of Gröbner bases of such
polynomial systems. This will be the foundation of practically fast
implementations (based on singly exponential algorithms)
outperforming the current ones based on the historical Cylindrical
Algebraic Decomposition (CAD) algorithm (whose complexity is doubly
exponential in the number of variables). We will also develop
algorithms and implementations that allow us to analyze, at least
locally, the topology of solution sets in some specific
situations. A long-term goal is obviously to obtain an analysis of
the global topology.