A Polyhedral Method for Sparse Systems with Many Positive Solutions

Abstract

We investigate a version of Viro's method for constructing polynomial systems with many positive solutions, based on regular triangulations of the Newton polytope of the system. The number of positive solutions obtained with our method is governed by the size of the largest positively decorable subcomplex of the triangulation. Here, positive decorability is a property that we introduce and which is dual to being a subcomplex of some regular triangulation. Using this duality, we produce large positively decorable subcomplexes of the boundary complexes of cyclic polytopes. As a byproduct we get new lower bounds, some of them being the best currently known, for the maximal number of positive solutions of polynomial systems with prescribed numbers of monomials and variables. We also study the asymptotics of these numbers and observe a log-concavity property.

Parameter regions that give rise to $2 \lfloor \frac{n}{2}\rfloor +1$ positive steady states in the $n$-site phosphorylation system

“…(0, 14), (0, 24), (1,24), (2,13), (2,14), (3,14), (3, 024), (02, 3), (02, 4), (03, 1), (03, 2), (13,2), (13,4), then after rescaling of the k on 's and on 's the distributive sequential 5-site phosphorylation system has at least 5 steady states.…”

“…We developed in this paper both the theoretical setting based on [1,2] and the algorithmic approach that follows from it, to describe multistationarity regions in the space of all parameters for subnetworks of the n-site sequential phosphorylation cycle, where there are up to 2[ n 2 ] + 1 positive steady states with fixed linear conservation constants. We chose to assume that the subnetworks we consider have intermediate species only in the E component, but of course there is a symmetry in the network interchanging E with F , each S i with S n−i , the corresponding intermediates and rate constants, and completely similar results hold if we assume that there are only intermediates in the F component.…”

“…As we saw in Subsection 2.1, the steady states of the systems G J correspond to the positive solutions of the sparse polynomial system (2.4) in two variables. In this section, we briefly recall the general setting from [1,2] to find lower bounds for sparse polynomial systems, that we will use in the proof of Theorems 1.2 and 1.4 in Section 4 and also in Section 5. For detailed examples of this approach, we refer the reader to Section 2 in [14].…”

“…, f d . Our method to obtain a lower bound on the number of positive steady states, based on [1,2], is to restrict our polynomial system (3.1) to subsystems which have a positive solution and then extend these solutions to the total system, via a deformation of the coefficients. The first step is then to find conditions in the coefficient matrix C that guarantee a positive solution to each of the subsystems.…”

On the Maximal Number of Real Embeddings of Spatial Minimally Rigid Graphs

“…This question is related to more general open problems in real algebraic geometry concerning possible gaps between the number of complex and real solutions of an algebraic system depending on its parameters. There exist some upper [29] and lower [3,4] bounds on the number of real positive roots, which take advantage of the structure of polynomials. Regarding applied cases, there is also the famous example on the maximization of the number of real Stewart-Gough Platform configurations [11], using a gradient descent method.…”