Descartes' rule is one of the first main result of algebraic geometry. Formulated in 1637 in the appendix entitled "La Géométrie" of the famous "Discours de la Méthode" [S], this rule gives an upper bound of the number of positive roots of a polynomial of one real-valued variable. The formulation of a similar rule for multi-variate polynomials has gone through new developements with the works of Frédéric Bihan and his collaborators who propose in [BD] a new upper bound for the number of roots with positive coordinates.
[BD] F. Bihan et A. Dickenstein, Descartes’ rule of signs for polynomial systems supported on circuits, to appear in International Mathematics Research Notices, 2017.
[S] D. J. Struik (ed.), A source book in mathematics, 1200-1800, Source Books in the History of the Sciences. Cambridge, Mass. : Harvard University Press, XIV, 427 p., 1969.
Contact : Frédéric Bihan | Laboratoire de Mathématiques | Université Savoie Mont Blanc & CNRS.