An Algorithm for Canonical Forms of Finite Subsets of $$\mathbb {Z}^d$$ Z d up to Affinities

Abstract

Abstract. In this paper we describe an algorithm for the computation of canonical forms of finite subsets of Z d , up to affinities over Z. For fixed dimension d, this algorithm has worst-case asymptotic complexity O(n log 2 n s µ(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 µ(s) is an upper bound to the number of operations required to multiply two s-bit numbers.This problem arises e.g. in the context of computation of invariants of finitely presented groups with abelianized group isomorphic to 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 base.