稠密参数化系统的托里安性
Toric invariance of vertically parametrized systems
AI总结：
本文研究了参数化多项式系统解集是否为托里安结构的问题，提出基于模体理论的极大维托尔斯特征征，为判断解集是否在组件乘法下不变提供必要和充分条件，应用于化学反应网络理论中的绝对浓度鲁棒性和稳态不变性。
摘要：
我们考虑决定参数化多项式系统解集是否在托里安意义下（即允许单项式参数化）的问题。我们关注垂直参数化系统，即稀疏系统，允许相同单项式前系数的线性依赖。我们给出了最大维托尔斯对于所有解集在组件乘法下不变的模体理论特征征。在此基础上，我们提供了解集是有限多个或唯一该托尔斯共集的必要条件和充分条件。本文的动机来自化学反应网络理论，其中托里安结构显著简化了多稳态性的确定；在这里，我们证明这对绝对浓度鲁棒性和稳态不变性同样成立。
英文摘要：
We consider the problem of deciding whether the solution sets of a parametrized polynomial system are toric in the sense that they admit a monomial parametrization. We focus on vertically parametrized systems, which are sparse systems where we allow linear dependencies between coefficients in front of the same monomial. We give a matroid-theoretic characterization of the maximal-dimensional torus for which all solution sets are invariant under componentwise multiplication. Building on this, we provide necessary conditions and sufficient conditions for when the solution sets are unions of finitely many or a unique coset of this torus. The motivation of this work comes from the theory of reaction networks, where toric structure of the steady state system substantially simplifies the determination of multistationarity; here, we show that this is also the case for absolute concentration robustness and steady state invariants.
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