We present our SageMath package elementary_vectors for computing elementary and sign vectors of real subspaces. In this setting, elementary vectors are support-minimal vectors that can be determined from maximal minors of a real matrix representing a subspace. By applying the sign function, we obtain the cocircuits of the corresponding oriented matroid, which in turn allow the computation of all sign vectors of a real subspace. As an application, we discuss sign vector conditions for existence and uniqueness of complex-balanced equilibria of chemical reaction networks with generalized mass-action kinetics. The conditions are formulated in terms of sign vectors of two subspaces arising from the stoichiometric coefficients and the kinetic orders of the reactions. We discuss how these conditions can be checked algorithmically, and we demonstrate the functionality of our package sign_vector_conditions in several examples.

翻译：本文介绍了我们开发的 SageMath 软件包 elementary_vectors，用于计算实子空间的基本向量与符号向量。在此框架下，基本向量是支撑集极小的向量，可通过代表子空间的实矩阵的最大子式确定。通过应用符号函数，我们得到对应有向拟阵的余回路，进而可计算实子空间的所有符号向量。作为应用，我们讨论了具有广义质量作用动力学的化学反应网络中，复杂平衡态存在性与唯一性的符号向量条件。这些条件通过反应计量系数与动力学级数产生的两个子空间的符号向量进行表述。我们探讨了如何通过算法验证这些条件，并通过多个示例展示了 sign_vector_conditions 软件包的功能。