Frédéric Bihan: Bounds for the number of positive solutions of sparse polynomial systems

Descartes' rule of signs is a fundamental result in real algebraic geometry. It gives, as a corollary, a bound on the number of positive roots of any real univariate polynomial in terms of its number of monomials. Various generalizations of this bound are described in the book Fewnomials by Askold Khovanskii (1980). Since then many substantial improvements have been obtained in the case of (square) polynomial systems, some of them giving sharp bounds. In this talk we will review recent results and open problems in this topic.