The excluded minors for GF(5)-representable matroids on ten elements
Nick Brettell
TL;DR
The paper establishes that there are exactly $2128$ excluded minors for the class of $GF(5)$-representable matroids on $10$ elements, extending the prior $564$ known for at most $9$ elements. It develops and applies the Hydra-$i$ partial-field framework to propagate $GF(5)$-representability across successive Hydra layers, using seeds $ abla_i$-stabilizers and splicing techniques to efficiently enumerate candidates. A significant part of the work is dedicated to handling exceptional $12$- and $13$-element cases via detailed structural analysis of quad-flower and nest configurations, identifying precisely which of these can occur as representable minors and which cannot. The results not only push forward the finite exclusion-minor program for $GF(5)$-representability but also illuminate the connections to $3$-regular matroids and related $k$-regular families, suggesting both practical enumeration strategies and deeper structural insights for minor-closed classes over finite fields.
Abstract
Mayhew and Royle (2008) showed that there are 564 excluded minors for the class of GF(5)-representable matroids having at most 9 elements. We enumerate the excluded minors for GF(5)-representable matroids having 10 elements: there are precisely 2128 such excluded minors. In the process we find, for each $i \in \{2,3,4\}$, the excluded minors for the class of $\mathbb{H}_i$-representable matroids having at most 10 elements, and the excluded minors for the class of $\mathbb{H}_5$-representable matroids having at most 13 elements.