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.

我们研究了无扭转算术拟阵的可表示性问题。引入“强大的gcd属性”和称为“归约”的新操作后，我们描述并实现了一种算法，以计算所有必要的表示形式，直至等效。结果，我们获得了表示的等价类数的上限。为了排除等效表示，我们描述了一种有效的方法来计算整数矩阵的正则形式，直到可逆矩阵和列符号的变化（我们称其为“有符号Hermite正规形式”）到左乘法为止。最后，作为我们算法的应用，我们证明了关于层状波和复曲面排列的独立性波的两个猜想。