Descartes' rule for multivariate polynomials
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.
References :
[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.