The Critical Theorem, due to Henry Crapo and Gian-Carlo Rota, has been extended and generalised in many ways. In this paper, we describe properties of the characteristic polynomial of a weighted lattice showing that it has a recursive description. We use this to obtain results on critical exponents of $q$-polymatroids. We prove a Critical Theorem for representable $q$-polymatroids and we provide a lower bound on the critical exponent. We show that certain families of rank-metric codes attain this lower bound.

翻译：临界定理由Henry Crapo和Gian-Carlo Rota提出，随后被以多种方式推广和扩展。本文描述了加权格的特征多项式的性质，证明其具有递归结构。利用这一结果，我们得到了关于$q$-拟阵临界指数的结论，证明了可表示$q$-拟阵的临界定理，并给出了临界指数的下界。我们还证明某些秩度量码族能达到该下界。