## Section: Scientific Foundations

### Solving Systems over the Reals and Applications.

Participants : Mohab Safey El Din, Daniel Lazard, Elias Tsigaridas, Pierre-Jean Spaenlehauer, Aurélien Greuet, Simone Naldi.

We will 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.