A priori information about the real solutions to a system of polynomial equations is important for the applications of mathematics, which typically only need the real solutions. The current state of knowledge about this problem, particularly counting or bounding the numbers of solutions, will be the focus of a workshop to be held at the Bernoulli Center in the EPFL (Swiss Federal Technological Institute) in Lausanne, Switzerland the week of April 21 to April 25 in 2008. This is embedded within a semester long program at the Bernoulli Center on real and tropical algebraic geometry. This project will fund the participation of scientists from the United States at this workshop.
It is notoriously difficult to extract any information about real solutions to equations without first solving them completely by finding all complex solutions. Recent breakthroughs have led to new insights about bounds, both upper and lower for the number of real solutions to some structured systems of polynomial equations. There are also new algorithms for only computing the real solutions. At the same time, a better understanding has arisen about average-case behavior, and other scientists have begun to apply these results. This workshop will be small (about 20 participants) and very focused on current developments and future prospects for this important problem of understanding the real solutions to polynomial equations.
