Geometry of the signed support of a multivariate polynomial and Descartes' rule of signs

TL;DR

This paper explores the geometric structure of multivariate polynomials' signed support to determine conditions under which the negative set is connected, extending Descartes' rule of signs and applying it to reaction network analysis.

Contribution

It generalizes Descartes' rule of signs to multivariate polynomials and provides a recursive algorithm to verify connectedness of the negative set based on Newton polytope geometry.

Findings

Conditions ensuring at most one connected component of {f<0}

A recursive algorithm for verifying connectedness

Application to reaction networks showing connected parameter regions

Abstract

We investigate the signed support, that is, the set of the exponent vectors and the signs of the coefficients, of a multivariate polynomial $f$. We describe conditions on the signed support ensuring that the semi-algebraic set, denoted as $\{ f < 0 \}$, containing points in the positive real orthant where $f$ takes negative values, has at most one connected component. These results generalize Descartes' rule of signs in the sense that they provide a bound which is independent of the values of the coefficients and the degree of the polynomial. Based on how the exponent vectors lie on the faces of the Newton polytope, we give a recursive algorithm that verifies a sufficient condition for the set $\{ f < 0 \}$ to have one connected component. We apply the algorithm to reaction networks in order to prove that the parameter region of multistationarity of a ubiquitous network comprising phosphorylation cycles is connected.