The Critical Theorem for q-Polymatroids
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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.
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