We consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a multiplicative group or, more generally, a coset of a multiplicative group. For the coset case, we study the notion of shifted toric varieties which generalizes the notion of toric varieties. This requires a geometric view on the varieties rather than an algebraic view on the ideals. We present algorithms and computations on 129 models from the BioModels repository testing for group and coset structures over both the complex numbers and the real numbers. Our methods over the complex numbers are based on Gr\"obner basis techniques and binomiality tests. Over the real numbers we use first-order characterizations and employ real quantifier elimination. In combination with suitable prime decompositions and restrictions to subspaces it turns out that almost all models show coset structure. Beyond our practical computations, we give upper bounds on the asymptotic worst-case complexity of the corresponding problems by proposing single exponential algorithms that test complex or real varieties for toricity or shifted toricity. In the positive case, these algorithms produce generating binomials. In addition, we propose an asymptotically fast algorithm for testing membership in a binomial variety over the algebraic closure of the rational numbers.

中文翻译：

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高效且有效地识别稳态品种的复数性

我们考虑测试具有非零坐标的复数或实数种类中的点是否形成乘法群或更一般地乘法群的陪集的问题。对于陪集案例，我们研究了转移复曲面变体的概念，它概括了复曲面变体的概念。这需要对变体的几何观点，而不是对理想的代数观点。我们展示了来自 BioModels 存储库的 129 个模型的算法和计算，用于测试复数和实数上的群和陪集结构。我们在复数上的方法基于 Gr\"obner 基础技术和二项式检验。在实数上，我们使用一阶特征并采用实量词消除。结合适当的素数分解和对子空间的限制，结果证明几乎所有模型都显示陪集结构。除了我们的实际计算之外，我们通过提出单指数算法来测试复曲面或移位复曲面的复数或实数变体，从而给出相应问题的渐近最坏情况复杂度的上限。在积极的情况下，这些算法产生生成二项式。此外，我们提出了一种渐近快速算法，用于在有理数的代数闭包上测试二项式变体的隶属度。我们通过提出测试复曲面或移位复曲面的复数或实数变体的单指数算法，给出了相应问题的渐近最坏情况复杂度的上限。在积极的情况下，这些算法产生生成二项式。此外，我们提出了一种渐近快速算法，用于在有理数的代数闭包上测试二项式变体的隶属度。我们通过提出测试复曲面或移位复曲面的复数或实数变体的单指数算法，给出了相应问题的渐近最坏情况复杂度的上限。在积极的情况下，这些算法产生生成二项式。此外，我们提出了一种渐近快速算法，用于在有理数的代数闭包上测试二项式变体的隶属度。