Abstract
Much attention has been focused in recent years on the following algebraic problem arising from applications: which chemical reaction networks, when taken with mass-action kinetics, admit multiple positive steady states? The interest behind this question is in steady states that are stable. As a step toward this difficult question, here we address the question of multiple nondegenerate positive steady states. Mathematically, this asks whether certain families of parametrized, real, sparse polynomial systems ever admit multiple positive real roots that are simple. Our main results settle this problem for certain types of small networks, and our techniques may point the way forward for larger networks.
Highlights
This work is motivated by the Nondegeneracy Conjecture from the study of reaction systems [JS17]: if a reaction network admits multiple positive steady states, does it admit multiple nondegenerate positive steady states? Equivalently, for certain families of parametrized sparse-polynomial systems, if one member of the family admits multiple positive roots, does some member admit multiple multiplicity-one positive roots? there has been a great deal of work on characterizing when a network is multistationary, but much less on nondegenerate multistationarity or the stronger condition of bistability [CS18]
A reaction network consists of finitely many reactions
Our work was motivated by the Nondegeneracy Conjecture: Is a network multistationary if and only if it is nondegenerately multistationary? At first, one might think this is so; we would expect to be able to perturb parameters to make a degenerate steady state become nondegenerate
Summary
This work is motivated by the Nondegeneracy Conjecture from the study of reaction systems [JS17]: if a reaction network admits multiple positive steady states, does it admit multiple nondegenerate positive steady states? Equivalently, for certain families of parametrized sparse-polynomial systems, if one member of the family admits multiple positive roots, does some member admit multiple multiplicity-one positive roots? there has been a great deal of work on characterizing when a network is multistationary (surveyed in [JS15]), but much less on nondegenerate multistationarity or the stronger condition of bistability [CS18]. The reason stems from a number of recent results on how a given network’s capacity for multistationarity arises from that of certain smaller networks [BP16, JS13] Is one such “lifting” result, stated informally: if N is a subnetwork of G and both networks have the same number of conservation laws, if N is nondegenerately multistationary, G is too (see Lemma 2.5). We hope to convey that the study of reaction systems leads to interesting problems in real algebraic geometry Algebraic techniques, such as elimination of variables and steady-state parametrizations, have already contributed significantly to recent progress in the field, e.g., [CFMW17, CS18, DDG15, GH02, GHRS16, MD16, Swe17].
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