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

TL;DR

This work extends Descartes' rule of signs to multivariate signomials by leveraging the signed support and the Newton polytope geometry to bound the number of negative connected components of $f^{-1}(\mathbb{R}_{<0})$ on $\mathbb{R}^n_{>0}$. It develops a geometric framework based on separating and enclosing hyperplanes, introduces reductions to Newton polytope faces, and presents a recursive algorithm that certifies connectivity under concrete conditions. The methods are applied to reaction networks, proving the parameter region for multistationarity is connected for both weakly and strongly irreversible phosphorylation cycles (with results for specific $m$ and ongoing work toward general $m$). Overall, the paper provides a practical, geometry-driven approach to assess global connectivity properties of semi-algebraic parameter spaces in dynamical systems.

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.

Figures (12)

• Figure 1: Exponent vectors of $f,f_{|A},f_{|B}$ from Example \ref{['Ex::ExRunning']}. Positive exponent vectors are marked by red circles and negative exponent vectors by blue dots. (a) A pair of enclosing hyperplanes of $\sigma_+(f)$. (b) A strict separating hyperplane of $\sigma(f_{|A})$. (c) The Newton polytope of $f_{|B}$ and a non-strict separating hyperplane of $\sigma(f_{|B})$.
• Figure 2: (a) Exponent vectors of $f = 1+ x_1 + x_2 - 4x_1 x_2+ x_1 x_2^2$ from Example \ref{['Ex::Proof24']} and a separating hyperplane of $\sigma(f)$. Positive exponent vectors are depicted by red circles, the negative exponent vector by blue dot. (b),(c) An illustration of the paths from Example \ref{['Ex::Proof24']} and the proof of Proposition \ref{['Prop_NonStrictSepHyp']}.
• Figure 3: (a) A pair of enclosing hyperplanes of the support of the signomial from Example \ref{['Ex:NewEx2']}. (b) Paths as defined in \ref{['Eq::NewExPath']} connecting $x_1,x_2,x_3$ to a point in $f_{|A}^{-1}(\mathbb{R}_{<0}) \cap f_{|B}^{-1}(\mathbb{R}_{<0})$.
• Figure 4: Negative connected components of $f$,$f_{|A}$,$f_{|B}$,$f_{|R}$,$f_{|S}$ from Example \ref{['Ex_ParaFaces']} and Example \ref{['Ex::RSCounterExample']}. For better visibility, the figures show the images of these sets under the coordinate-wise natural logarithm map $\mathbb{R}^2_{>0} \to \mathbb{R}^2, (x,y) \mapsto (\log(x),\log(y) )$.
• Figure 5: Illustration of Example \ref{['Ex_Box']} (a) Negative and positive exponent vectors of $f =-x^{4} y^{4} + 10 \, x^{3} y^{3} - 10 \, x^{4} - 10 \, y^{4} + 7 \, x y + 5 \, x - 1$, blue dots are negative, red circles are positive. The black solid lines are strict enclosing hyperplanes of $\sigma_+(f)$. The gray dashed line separates $\mathop{\mathrm{Conv}}\nolimits((0,4),(4,4))$ from $\mathop{\mathrm{Conv}}\nolimits(\sigma_+(f))$. (b) Negative connected component of $f$. (c) Negative connected component of $f_{|A} =-x^{4} y^{4} + 10 \, x^{3} y^{3} - 10 \, x^{4} + 7 \, x y + 5 \, x$ and $f_{|B}=10 \, x^{3} y^{3} - 10 \, y^{4} + 7 \, x y + 5 \, x - 1$.

Theorems & Definitions (45)

• Example 2.1
• Example 2.2
• Lemma 2.3
• Theorem 2.4
• Proposition 2.5
• Example 2.6
• proof : Proof of Proposition \ref{['Prop_NonStrictSepHyp']}
• Example 2.7
• Proposition 2.8
• proof