Representations of torsion-free arithmetic matroids

We study the representability problem for torsion-free arith-metic matroids. After introducing a ‘‘strong gcd property’’ anda new operation called ‘‘reduction’’, we describe and implementan algorithm to compute all essential representations, up toequivalence. As a consequence, we obtain an upper bound tothe number of equivalence classes of representations. In orderto rule out equivalent representations, we describe an efficientway to compute a normal form of integer matrices, up to left-multiplication by invertible matrices and change of sign of thecolumns (we call it the ‘‘signed Hermite normal form’’). Finally,as an application of our algorithms, we disprove two conjecturesabout the poset of layers and the independence poset of a toricarrangement.