From reaction networks to "positive algebraic geometry"

Reaction networks are a modeling framework used in several fields of chemistry, in areas of biology such as ecology and epidemiology, and even in economics and engineering. In particular, every polynomial dynamical system (with integer exponents) and even every power-law dynamical system (with real exponents) can be written as a reaction network with (generalized) mass-action kinetics (MAK or GMAK). Assuming MAK, the rate of a reaction is the product of a rate constant and a monomial in the concentrations of the reactants. As a consequence, a network of several reactions leads to a polynomial dynamical system (for nonnegative variables). Its steady states are the zeros of multivariate polynomials, as studied in real algebraic geometry. Assuming GMAK (allowing real exponents), one obtains power-law dynamical systems. In previous work, we have focused on special positive steady states determined by the underlying network. In the proposed project, we turn to general steady states. In abstract terms, we study positive zeros of parametrized systems of generalized polynomial equations (with real exponents). Indeed, we have just established the groundwork for a novel approach to positive algebraic geometry. First, we have identified the crucial geometric objects of generalized multivariate polynomials, namely the coefficient polytope and two linear subspaces representing monomial differences and dependencies. Second, we have shown that (parametrized systems of generalized) polynomial equations can be written as binomial (!) equations on the coefficient polytope, depending on monomials in the parameters. In the proposed project, we build on our recent results and address important (and challenging) open problems. In particular, we focus on three areas: (i) existence and unique existence of solutions, for given or for all parameters, (ii) upper bounds for the number of solutions, as studied in real fewnomial theory, and (iii) unification and extension of classical results for reaction networks. To address the proposed problems, we will combine concepts and methods from analysis, polyhedral geometry, and oriented matroids. The intended results make significant contributions to real fewnomial theory, and they further extend the applicability of reaction network theory (from MAK to GMAK). Moreover, since we allow real exponents, we can consider small perturbations of both coefficients and exponents. As a cross-cutting issue, we will investigate the robustness of all intended results.