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arXiv · 2607.04810

Disguised complex balance via positive algebraic geometry

Stefan Müller·Abhishek Deshpande·Georg Regensburger
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Abstract

We study dynamical systems arising from reaction networks under mass-action kinetics. For certain choices of the rate constants (parameters), such systems are complex-balanced (vertex-balanced), which guarantees the existence of a unique positive equilibrium. Moreover, this equilibrium is asymptotically stable (admitting a global Lyapunov function) and linearly stable. In a series of recent papers, Craciun and collaborators introduced and studied disguised complex-balanced systems, that is, mass-action systems that are dynamically equal to auxiliary complex-balanced systems and therefore inherit their strong stability properties. Determining the parameter values for which a given system is disguised complex-balanced is a nontrivial algebraic problem. In this work, we show that the defining conditions for disguised complex-balanced equilibria naturally give rise to parametrized systems of polynomial inequalities. Using the framework for positive algebraic geometry developed by M\"uller and Regensburger, we reformulate these systems as binomial equations (on the disguised complex-balanced flux cone). Computing the disguised complex-balanced parameter locus can be viewed as a quantifier-elimination problem, and our approach eliminates the concentrations (state variables) from the problem. We illustrate our results using the running example of a recent paper by Boros et al.

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Stefan Müller, Abhishek Deshpande, Georg Regensburger. 2026-07-06. Disguised complex balance via positive algebraic geometry. https://arxiv.org/abs/2607.04810

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