Efficiently and Effectively Recognizing Toricity of Steady State Varieties

TL;DR

This paper introduces algorithms for testing whether steady state varieties in biological models are toric or coset structures, using geometric and algebraic methods over complex and real numbers, with practical and theoretical results.

Contribution

It develops new algorithms for recognizing toric and shifted toric varieties, providing practical tests and complexity bounds for biological models.

Findings

Most models exhibit coset structure.

Algorithms produce binomial generators in positive cases.

Single exponential algorithms for toricity testing.

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 Gr\"obner 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.