On generalizing Descartes' rule of signs to hypersurfaces

TL;DR

This paper extends Descartes' rule of signs to multivariate polynomials with real exponents, providing bounds on the number of connected components of their hypersurfaces' complements in the positive orthant based on geometric and sign conditions.

Contribution

It introduces a new approach to generalize Descartes' rule of signs to multivariate cases, focusing on topological bounds rather than solution counting.

Findings

Upper bounds on connected components where the polynomial is negative

Conditions based on exponents' geometry and coefficient signs

Complete coverage of cases with bounds equal to one

Abstract

We give partial generalizations of the classical Descartes' rule of signs to multivariate polynomials (with real exponents), in the sense that we provide upper bounds on the number of connected components of the complement of a hypersurface in the positive orthant. In particular, 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 where the polynomial attains a negative value is at most one or two. Our results fully cover the cases where such an upper bound provided by the univariate Descartes' rule of signs is one. This approach opens a new route to generalize Descartes' rule of signs to the multivariate case, differing from previous works that aim at counting the number of positive solutions of a system of multivariate polynomial equations.