Efficiently and Effectively Recognizing Toricity of Steady State Varieties

Abstract

We consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a multiplicative group or, more generally, a coset of a multiplicative group. For the coset case, we study the notion of shifted toric varieties which generalizes the notion of toric varieties. This requires a geometric view on the varieties rather than an algebraic view on the ideals. We present algorithms and computations on 129 models from the BioModels repository testing for group and coset structures over both the complex numbers and the real numbers. Our methods over the complex numbers are based on Gröbner basis techniques and binomiality tests. Over the real numbers we use first-order characterizations and employ real quantifier elimination. In combination with suitable prime decompositions and restrictions to subspaces it turns out that almost all models show coset structure. Beyond our practical computations, we give upper bounds on the asymptotic worst-case complexity of the corresponding problems by proposing single exponential algorithms that test complex or real varieties for toricity or shifted toricity. In the positive case, these algorithms produce generating binomials. In addition, we propose an asymptotically fast algorithm for testing membership in a binomial variety over the algebraic closure of the rational numbers.

arXiv commentWe made the presentation clearer and fixed many small flaws and typos. A database with our computations is now available as ancillary file

References in corpus (8)

• Extended Detailed Balance for Systems with Irreversible Reactions
• Satisfiability Checking meets Symbolic Computation (Project Paper)
• Thermodynamics in the Limit of Irreversible Reactions
• Identifying the Parametric Occurrence of Multiple Steady States for some Biological Networks
• Decompositions of Binomial Ideals
• Detecting Binomiality
• Multistationarity in the space of total concentrations for systems that admit a monomial parametrization
• Better Answers to Real Questions

Cited by in corpus (6)

• Algorithmic Reduction of Biological Networks With Multiple Time Scales
• A Linear Algebra Approach for Detecting Binomiality of Steady State Ideals of Reversible Chemical Reaction Networks
• Parametric Toricity of Steady State Varieties of Reaction Networks
• First-Order Tests for Toricity
• On the Number of Real Types of Univariate Polynomials
• Testing Binomiality of Chemical Reaction Networks Using Comprehensive Gröbner Systems