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A SageMath Package for Elementary and Sign Vectors with Applications to Chemical Reaction Networks

Marcus S. Aichmayr, Stefan Müller, Georg Regensburger

TL;DR

The paper presents a SageMath package for computing elementary and sign vectors of real subspaces, enabling exact, finite-generation representations via maximal minors and their induced cocircuits in oriented matroids. It then applies these algebraic tools to chemical reaction networks with generalized mass-action kinetics, translating existence and uniqueness of positive complex-balanced equilibria into sign-vector conditions on the stoichiometric and kinetic-order subspaces, and providing algorithmic checks through maximal-minor criteria and degeneracy tests. Key contributions include explicit constructions of elementary vectors from kernel representations, Minty-type solvability tests for linear inequality systems, and practical criteria for CBE existence, uniqueness, and degeneracy, all illustrated on concrete CRN examples. The resulting framework offers a computational route to verifying structural network properties that govern steady-state behavior in CRNs, with direct implementation in SageMath and demonstrated use of the $sign_vector_conditions$ toolkit.

Abstract

We present our SageMath package elementary_vectors for computing elementary and sign vectors of real subspaces. In this setting, elementary vectors are support-minimal vectors that can be determined from maximal minors of a real matrix representing a subspace. By applying the sign function, we obtain the cocircuits of the corresponding oriented matroid, which in turn allow the computation of all sign vectors of a real subspace. As an application, we discuss sign vector conditions for existence and uniqueness of complex-balanced equilibria of chemical reaction networks with generalized mass-action kinetics. The conditions are formulated in terms of sign vectors of two subspaces arising from the stoichiometric coefficients and the kinetic orders of the reactions. We discuss how these conditions can be checked algorithmically, and we demonstrate the functionality of our package sign_vector_conditions in several examples.

A SageMath Package for Elementary and Sign Vectors with Applications to Chemical Reaction Networks

TL;DR

The paper presents a SageMath package for computing elementary and sign vectors of real subspaces, enabling exact, finite-generation representations via maximal minors and their induced cocircuits in oriented matroids. It then applies these algebraic tools to chemical reaction networks with generalized mass-action kinetics, translating existence and uniqueness of positive complex-balanced equilibria into sign-vector conditions on the stoichiometric and kinetic-order subspaces, and providing algorithmic checks through maximal-minor criteria and degeneracy tests. Key contributions include explicit constructions of elementary vectors from kernel representations, Minty-type solvability tests for linear inequality systems, and practical criteria for CBE existence, uniqueness, and degeneracy, all illustrated on concrete CRN examples. The resulting framework offers a computational route to verifying structural network properties that govern steady-state behavior in CRNs, with direct implementation in SageMath and demonstrated use of the toolkit.

Abstract

We present our SageMath package elementary_vectors for computing elementary and sign vectors of real subspaces. In this setting, elementary vectors are support-minimal vectors that can be determined from maximal minors of a real matrix representing a subspace. By applying the sign function, we obtain the cocircuits of the corresponding oriented matroid, which in turn allow the computation of all sign vectors of a real subspace. As an application, we discuss sign vector conditions for existence and uniqueness of complex-balanced equilibria of chemical reaction networks with generalized mass-action kinetics. The conditions are formulated in terms of sign vectors of two subspaces arising from the stoichiometric coefficients and the kinetic orders of the reactions. We discuss how these conditions can be checked algorithmically, and we demonstrate the functionality of our package sign_vector_conditions in several examples.
Paper Structure (8 sections, 7 theorems, 13 equations, 1 algorithm)

This paper contains 8 sections, 7 theorems, 13 equations, 1 algorithm.

Table of Contents

  1. Elementary vectors of a subspace
  2. Solvability of linear inequality systems.
  3. Sign vectors.
  4. Applications to chemical reaction networks
  5. Uniqueness of CBE.
  6. Unique existence of CBE.
  7. Acknowledgments.
  8. Disclosure of Interests.

Key Result

proposition thmcounterproposition

For a matrix $M \in R^{d \times n}$ with rank $d$ and $I \subseteq [n]$ with $\lvert I \rvert = d + 1$, define the vector $v \in R^n$ by The vector $v \in \ker M$ is elementary if $\mathop{\mathrm{rank}}\nolimits M_I = d$.

Theorems & Definitions (11)

  • definition thmcounterdefinition
  • proposition thmcounterproposition: cf. equation (2.1) in Brualdi1995
  • theorem thmcountertheorem: "Minty's Lemma", Theorem 22.6 in Rockafellar1970
  • theorem thmcountertheorem: robust $\delta = \widetilde{\delta} = 0$ theorem, Theorem 46 in Mueller2019
  • proposition thmcounterproposition: Proposition 32 in Mueller2019
  • proposition thmcounterproposition: cf. Proposition 3.1 in Mueller2012
  • corollary thmcountercorollary: cf. Corollary 4 in Mueller2019
  • theorem thmcountertheorem: $\delta = \widetilde{\delta} = 0$ theorem, Theorem 45 in Mueller2019
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • ...and 1 more