## Section: Overall Objectives

### Introduction

The main objective of the SALSA project is to solve systems of polynomial equations and inequations. We emphasize on algebraic methods which are more robust and frequently more efficient than purely numerical tools.

Polynomial systems have many applications in various scientific - academic as well as industrial - domains. However much work is yet needed in order to define specifications for the output of the algorithms which are well adapted to the problems.

The variety of these applications implies that our software needs to be robust. In fact, almost all problems we are dealing with are highly numerically unstable, and therefore, the correctness of the result needs to be guaranteed.

Thus, a key target is to provide software which are competitive in terms of efficiency but preserve certified outputs. Therefore, we restrict ourselves to algorithms which verify the assumptions made on the input, check the correctness of possible random choices done during a computation without sacrificing the efficiency. Theoretical complexity for our algorithms is only a preliminary step of our work which culminates with efficient implementations which are designed to solve significant applications.

A consequence of our way of working is that many of our contributions are related to applicative topics such as cryptography, error correcting codes, robotics and signal theory. We have to emphasize that these applied contributions rely on a long-term and global management of the project with clear and constant objectives leading to theoretical and deep advances.