A Lower Bound on the Dimension of the $\mathbb{R}$-Disguised Toric Locus of a Reaction Network

Polynomial dynamical systems (i.e. dynamical systems with polynomial right hand side) are ubiquitous in applications, especially as models of reaction networks and interaction networks. The properties of general polynomial dynamical systems can be very difficult to analyze, due to nonlinearity, bifurcations, and the possibility for chaotic dynamics. On the other hand, toric dynamical systems are polynomial dynamical systems that appear naturally as models of reaction networks, and have very robust and stable properties. A disguised toric dynamical system is a polynomial dynamical system generated by a reaction network $\mathcal N$ and some choice of positive parameters, such that (even though it may not be toric with respect to $\mathcal N$) it has a toric realization with respect to some network $\mathcal N'$. Disguised toric dynamical systems enjoy all the robust stability properties of toric dynamical systems. In this paper, we study a larger set of dynamical systems where the rate constants are allowed to take both positive and negative values. More precisely, we analyze the $\mathbb{R}$-disguised toric locus of a reaction network $\mathcal N$, i.e., the subset in the space rate constants (positive or negative) of $\mathcal N$ for which the corresponding polynomial dynamical system is disguised toric. We focus especially on finding a lower bound on the dimension of the $\mathbb{R}$-disguised toric locus.