# Máté Telek: Generalizing Descartes' rule of signs to hypersurfaces

**Time: **
Tue 2022-05-17 10.15

**Location: **
3721, Lindstedtsvägen 25, and Zoom

**Video link: **
Meeting ID: 659 3743 5667

**Participating: **
Máté Telek (University of Copenhagen)

### Abstract

The classical Descartes’ rule of signs provides an easily computable upper bound for the number of positive real roots of a univariate polynomial with real coefficients. Generalizations to the multivariate case have focused on systems of n polynomial equations in n variables and on bounding the number of solutions in the positive orthant. In this talk, we take a different perspective and aim to bound the number of connected components in the complement of a hypersurface in the positive orthant. We give conditions based on the geometrical configuration of the exponents and the sign of the coefficients that guarantee that the number of connected components of the complement of the hypersurface where the defining polynomial attains a negative value is at most one or two. Furthermore, we present an application for chemical reaction networks that motivated us to consider this problem.

This is a joint work with Elisenda Feliu.