Complete and computable orbit invariants in the geometry of the affine group over the integers

Abstract

The subject matter of this paper is the geometry of the affine group over the integers, $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ . Turing-computable complete $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -orbit invariants are constructed for rational affine spaces, angles, segments, triangles and ellipses. In rational affine $${\mathsf {GL}}(n,{\mathbb {Q}})\ltimes {\mathbb {Q}}^n$$ -geometry, ellipses are classified by the Clifford–Hasse–Witt invariant, via the Hasse–Minkowski theorem. We classify ellipses in $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -geometry combining results by Apollonius of Perga and Pappus of Alexandria with the Hirzebruch–Jung continued fraction algorithm. We then consider rational polyhedra, i.e., finite unions of simplexes in $${\mathbb {R}}^n$$ with rational vertices. Markov’s unrecognizability theorem for combinatorial manifolds states the undecidability of the problem whether two rational polyhedra P and $$P'$$ are continuously $${\mathsf {GL}}(n,{\mathbb {Q}})\ltimes {\mathbb {Q}}^n$$ -equidissectable. The same problem for the continuous $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -equidissectability of P and $$P'$$ is open. We prove the decidability of the problem whether two rational polyhedra P and $$P'$$ in $${\mathbb {R}}^n$$ have the same $${\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n$$ -orbit.