From reaction networks to "positive algebraic geometry"

1) Wider research context / theoretical framework
Reaction networks with generalized mass-action kinetics naturally give rise to parametrized systems of generalized polynomial equations (with real exponents). Motivated by this, we suggest to extend the standard framework of fewnomial systems by assigning a positive parameter to every monomial (and by considering partitions of the monomials into "classes").

Recently (in 2023), we have shown that the set of positive solutions to parametrized systems of generalized polynomial equations is in one-to-one correspondence with the set of solutions to binomial equations on the "coefficient polytope", depending on monomials in the parameters. This intriguing result establishes the groundwork for a novel approach to "positive algebraic geometry".

2) Hypotheses / research questions / objectives
In the proposed project, we build on our recent results and address significant open problems. In particular, we focus on three problem 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, such as the deficiency one theorem.

3) Approach / methods
(i) We aim to characterize existence and unique existence via Brouwer degree and Hadamard's theorem. (ii) We study multivariate fewnomial systems in a systematic way, in particular, via sign matrices and parameters encoding coefficients, and we aim to obtain a Descartes’ rule of signs. (iii) We aim to extend the "deficiency one theorem", from deficiency one to dependency one and from classical to generalized mass-action kinetics, by combining results for generalized polynomial equations and maps.

4) Level of originality / innovation
Our recent results allow a unified treatment of multivariate polynomial equations with real exponents, positive parameters, and classes. In order to address the proposed problems, we will combine concepts and methods from analysis, polyhedral geometry, and oriented matroids in a novel way. The intended results make significant contributions to real fewnomial theory, and they further extend the applicability of reaction network theory to generalized mass-action systems.

5) Primary researchers involved
The project will build on the fruitful long-term collaboration between the principal investigator Stefan Müller (University of Vienna) and Georg Regensburger (University of Kassel). Additionally, Abhishek Deshpande (IIIT Hyderabad) and Polly Yu (Harvard University) will contribute new perspectives and ensure the comprehensive pursuit of all project goals within the designated timeframe.

We ask for funding for Stefan Müller for four years.