Frédéric Bihan (Université Savoie Mont Blanc): Multivariate generalizations of Descartes' rule of signs

Abstract: Descartes rule of signs is a fondamental result in real algebraic geometry. It gives a bound on the number of positive roots of any real univariate polynomial taking into account of the signs +,- or 0 of its coefficients. Moreover, this bound is sharp. Only very recently the first generalizations of Descartes' rule for the number of positive solutions of real polynomial systems have been discovered. We will explain these generalizations. It turns out that oriented matroid theory play a central role here.