@online{Rahkooy_arXiv2105.10853,
TITLE = {Parametric Toricity of Steady State Varieties of Reaction Networks},
AUTHOR = {Rahkooy, Hamid and Sturm, Thomas},
LANGUAGE = {eng},
URL = {https://arxiv.org/abs/2105.10853},
EPRINT = {2105.10853},
EPRINTTYPE = {arXiv},
YEAR = {2021},
ABSTRACT = {We study real steady state varieties of the dynamics of chemical reaction<br>networks. The dynamics are derived using mass action kinetics with parametric<br>reaction rates. The models studied are not inherently parametric in nature.<br>Rather, our interest in parameters is motivated by parameter uncertainty, as<br>reaction rates are typically either measured with limited precision or<br>estimated. We aim at detecting toricity and shifted toricity, using a framework<br>that has been recently introduced and studied for the non-parametric case over<br>both the real and the complex numbers. While toricity requires that the variety<br>specifies a subgroup of the direct power of the multiplicative group of the<br>underlying field, shifted toricity requires only a coset. In the non-parametric<br>case these requirements establish real decision problems. In the presence of<br>parameters we must go further and derive necessary and sufficient conditions in<br>the parameters for toricity or shifted toricity to hold. Technically, we use<br>real quantifier elimination methods. Our computations on biological networks<br>here once more confirm shifted toricity as a relevant concept, while toricity<br>holds only for degenerate parameter choices.<br>},
}