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.

中文翻译：

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Toricity 的一阶检验

受化学反应网络理论中稳态理想符号分析问题的启发，我们考虑测试具有非零坐标的复数或实数中的点是否形成乘法群的陪集的问题。该属性对应于移动复曲面，这是相应多项式理想复曲面的最新推广。关键思想是对变体采取几何观点，而不是对理想采取代数观点。最近，已经提出了针对复变体和实变体的相应陪集检验。前者结合了交换算法代数中的众多技术，以 Gr\"obner 基为中心算法工具。后者基于实闭域中的解释一阶逻辑，在算法方面具有实量词消除技术。在这里，我们对复杂的和真实的理论以及其他理论采用了一种新的逻辑方法。除了替代算法之外，我们的方法还提供了关于领域理论的统一观点，并有助于理解该领域现有丰富文献的相关性和相互联系，这些文献一直关注复数，而从科学的角度来看（积极的）实数显然是化学反应网络理论的相关领域。我们将新方法的原型实现应用于 BioModels 存储库中的一组 129 个模型。它一直专注于复数，而从科学的角度来看，（正）实数显然是化学反应网络理论的相关领域。我们将新方法的原型实现应用于 BioModels 存储库中的一组 129 个模型。它一直专注于复数，而从科学的角度来看，（正）实数显然是化学反应网络理论的相关领域。我们将新方法的原型实现应用于 BioModels 存储库中的一组 129 个模型。