In recent years, an explosive development of algorithms and software constitutes a breakthrough in the field of polynomial system solving, which makes feasible the solution of several problems that had been intractable up to now, and which provides the means necessary for experimentation and conjecture. These algorithms have created a significant interest for effective algebraic geometry and computer algebra among scientists, engineers, and applied mathematicians. The tools of the field, besides algebraic geometry, call upon several aspects of mathematics and theoretical computer science, ranging from numerical methods and differential equations, to discrete geometry, combinatorics and complexity theory. Important applications include computational geometry, computer-aided design, robotics and machine vision.
The goal of this school is to provide a general introduction to the field, followed by a survey of recent results and the state-of-the-art, in order to conclude with a discussion of open questions as well as suggestions for new directions of research.