🤖 AI Summary
This paper addresses the critical exponent problem for $q$-matroids. We establish a recursive analytical framework for the weighted lattice characteristic polynomial, yielding the first critical theorem for representable $q$-matroids. By characterizing the lattice structure and recurrence relations of their characteristic polynomials, we derive a tight lower bound on the critical exponent and prove its achievability—specifically, optimal rank-metric codes (e.g., MRD codes) attain this bound precisely. The work integrates weighted lattice theory, combinatorial structures of $q$-matroids, and rank-metric code analysis. It unifies the algebraic characterization of critical phenomena in $q$-analogues and provides the first criticality-determination tool for $q$-matroids grounded in the recursion of characteristic polynomials. This advances the interdisciplinary study of $q$-analogue combinatorics and coding theory by introducing a novel methodological paradigm.
📝 Abstract
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.
