First-Order Tests for Toricity
Résumé
Motivated by problems arising with the symbolic analysis of steady state ideals in Chem-ical Reaction Network Theory, we consider the problem of testing whether the points in acomplex or real variety with non-zero coordinates form a coset of a multiplicative group.That property corresponds to Shifted Toricity, a recent generalization of toricity of the cor-responding polynomial ideal. The key idea is to take a geometric view on varieties ratherthan an algebraic view on ideals. Recently, corresponding coset tests have been proposedfor complex and for real varieties. The former combine numerous techniques from commu-tative algorithmic algebra with Gröbner bases as the central algorithmic tool. The latterare based on interpreted first-order logic in real closed fields with real quantifier eliminationtechniques on the algorithmic side. Here we take a new logic approach to both theories,complex and real, and beyond. Besides alternative algorithms, our approach provides aunified view on theories of fields and helps to understand the relevance and interconnectionof the rich existing literature in the area, which has been focusing on complex numbers,while from a scientific point of view the (positive) real numbers are clearly the relevantdomain in chemical reaction network theory. We apply prototypical implementations ofour new approach to a set of 129 models from the BioModels repository