Séminaire de Géométrie
École polytechnique – Centre de Mathématiques Laurent Schwartz
10h30-12h00 – Oliver Lorscheid (IMPA)
"Descartes, Newton and hyperfields"
Abstract : Descartes bounds the number of positive roots of a polynomial with real coefficients by the number of sign changes in the coefficients, Newton bounds the number of roots of a polynomial over a nonarchimedean field in terms of its Newton polygon.
In this talk, we review these two classical theorems and explain how they follow by the same principle from assertions about the multiplicities of roots for polynomials over the sign and the tropical hyperfield, respectively. This is joint work with Matthew Baker.