Representations of torsion-free arithmetic matroids

Abstract

We study the representability problem for torsion-free arithmetic matroids. After introducing a “strong gcd property” and a new operation called “reduction”, we describe and implement an algorithm to compute all essential representations, up to equivalence. As a consequence, we obtain an upper bound to the number of equivalence classes of representations. In order to rule out equivalent representations, we describe an efficient way to compute a normal form of integer matrices, up to left-multiplication by invertible matrices and change of sign of the columns (we call it the “signed Hermite normal form”). Finally, as an application of our algorithms, we disprove two conjectures about the poset of layers and the independence poset of a toric arrangement.