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

TL;DR

This paper presents an efficient algorithm for computing canonical forms of finite subsets of integer lattices up to affine transformations, with applications in group invariants and polynomial equivalence.

Contribution

It introduces a fixed-parameter tractable algorithm for the problem, with detailed complexity analysis and relevance to algebraic invariants.

Findings

Algorithm has worst-case complexity $O(n \, \log^2 n \, s \, \mu(s))$

Problem is fixed-parameter tractable in dimension $d$

Applicable to invariants of finitely presented groups and polynomial equivalence

Abstract

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.