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Abstract

Motivated by problems arising with the symbolic analysis of steady state
ideals in Chemical Reaction Network Theory, we consider the problem of testing
whether the points in a complex 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 corresponding polynomial
ideal. The key idea is to take a geometric view on varieties rather than an
algebraic view on ideals. Recently, corresponding coset tests have been
proposed for complex and for real varieties. The former combine numerous
techniques from commutative algorithmic algebra with Gr\"obner bases as the
central algorithmic tool. The latter are based on interpreted first-order logic
in real closed fields with real quantifier elimination techniques 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 a
unified view on theories of fields and helps to understand the relevance and
interconnection of 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 relevant domain in chemical reaction
network theory. We apply prototypical implementations of our new approach to a
set of 129 models from the BioModels repository.