An algorithm for canonical forms of finite subsets of $\mathbb{Z}^d$ up to affinities

In this paper we describe an algorithm for the computation of canonical forms of finite subsets of $\mathbb{Z}^d$, up to affinities over $\mathbb{Z}$. For fixed dimension $d$, this algorithm has worst-case asymptotic complexity $O(n \log^2 n \, s\,\mu(s))$, where $n$ is the number of points in the given subset, $s$ is an upper bound to the size of the binary representation of any of the $n$ points, and $\mu(s)$ is an upper bound to the number of operations required to multiply two $s$-bit numbers. In particular, the problem is fixed-parameter tractable with respect to the dimension $d$. This problem arises e.g. in the context of computation of invariants of finitely presented groups with abelianized group isomorphic to $\mathbb{Z}^d$. In that context one needs to decide whether two Laurent polynomials in $d$ indeterminates, considered as elements of the group ring over the abelianized group, are equivalent with respect to a change of basis.