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First-Order Tests for Toricity

Hamid Rahkooy 1, 2 Thomas Sturm 2, 1, 3
1 MOSEL - Proof-oriented development of computer-based systems
LORIA - FM - Department of Formal Methods
2 VERIDIS - Modeling and Verification of Distributed Algorithms and Systems
MPII - Max-Planck-Institut für Informatik, Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods
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.
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https://hal.archives-ouvertes.fr/hal-03017907
Contributor : Thomas Sturm Connect in order to contact the contributor
Submitted on : Saturday, November 21, 2020 - 5:15:06 PM
Last modification on : Saturday, October 16, 2021 - 11:26:09 AM

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Hamid Rahkooy, Thomas Sturm. First-Order Tests for Toricity. 2020. ⟨hal-03017907⟩

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