Martin Leuner | Searcharxiv

arXiv2019

On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture

In this paper we describe a parallel algorithm for generating all non-isomorphic rank $3$ simple matroids with a given multiplicity vector. We apply our implementation in the HPC version of GAP to generate all rank $3$ simple matroids with at most $14$ atoms and a splitting characteristic polynomial. We have stored the resulting matroids alongside with various useful invariants in a publicly available, ArangoDB-powered database. As a byproduct we show that the smallest divisionally free rank $3$ arrangement which is not inductively free has $14$ hyperplanes and exists in all characteristics distinct from $2$ and $5$. Another database query proves that Terao's freeness conjecture is true for rank $3$ arrangements with $14$ hyperplanes in any characteristic.

math.CO↗

arXiv2013

Fast recognition of alternating groups of unknown degree

We present a constructive recognition algorithm to decide whether a given black-box group is isomorphic to an alternating or a symmetric group without prior knowledge of the degree. This eliminates the major gap in known algorithms, as they require the degree as additional input. Our methods are probabilistic and rely on results about proportions of elements with certain properties in alternating and symmetric groups. These results are of independent interest; for instance, we establish a lower bound for the proportion of involutions with small support.

math.GR↗