Motivated by problems from Chemical Reaction Network Theory, we investigate whether steady state ideals of reversible reaction networks are generated by binomials. We take an algebraic approach considering, besides concentrations of species, also rate constants as indeterminates. This leads us to the concept of unconditional binomiality, meaning binomiality for all values of the rate constants. This concept is different from conditional binomiality that applies when rate constant values or relations among rate constants are given. We start by representing the generators of a steady state ideal as sums of binomials, which yields a corresponding coefficient matrix. On these grounds we propose an efficient algorithm for detecting unconditional binomiality. That algorithm uses exclusively elementary column and row operations on the coefficient matrix. We prove asymptotic worst case upper bounds on the time complexity of our algorithm. Furthermore, we experimentally compare its performance with other existing methods.

中文翻译：

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用于检测可逆化学反应网络的稳态理想二项式的线性代数方法

受化学反应网络理论问题的启发，我们研究了可逆反应网络的稳态理想是否由二项式产生。我们采用代数方法，除物种浓度外，还将速率常数视为不确定因素。这将我们引向了无条件二项式的概念，即所有速率常数值的二项式。这个概念与条件二项式不同，条件二项式在给定速率常数值或速率常数之间的关系时适用。我们首先将稳态理想的生成器表示为二项式的总和，从而产生相应的系数矩阵。基于这些理由，我们提出了一种检测无条件二项性的有效算法。该算法仅对系数矩阵使用基本的列和行操作。我们证明了算法时间复杂度的渐近最坏情况上限。此外，我们通过实验将其性能与其他现有方法进行了比较。