Descartes, Newton and hyperfields

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 clasical theorems and explain how they follow by the same principle from assertions about roots of polynomials over the hyperfield of signs and the tropical hyperfield, respectively. This is joint work with Matthew Baker.