Frobenius-Power Ideals and Hyperplane Avoidance for Representable Matroids

Abstract

Let $q=p^k$, where $p$ is prime, and let $M$ be a finite matroid representable over ${\mathbb{F}}_q$. Write ${\chi }_M(t)$ for its characteristic polynomial and $\mbox{decop}(M)$ for the least number of independent sets needed to cover its ground set. We prove that ${\chi }_M(q)>0$ whenever $k\ge\mbox{decop}(M)$. Geometrically, the central hyperplanes determined by any representation of $M$ fail to cover the dual of the ambient vector space. The proof rests on the Frobenius-power ideals $(X_1^{p^s},\dots ,X_n^{p^s})$, $s\ge 1$. Each is preserved by every linear change of coordinates, while nonmembership records the existence of a monomial whose exponent in every variable is bounded. This permits successive normalizations of several invertible systems of linear forms without losing the exponent bounds already obtained. The coefficient form of the Combinatorial Nullstellensatz then produces a common nowhere-zero point. Finally, we test the scope of the theorem. M.~J.~Moghaddamzadeh's unpublished conjecture predicts a stronger statement over prime fields. Projective geometries show that its direct analogue fails over proper extension fields, even under the same numerical inequality.