Basic geometry of the affine group over Z

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 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 and the Morelli-Wl odarczyk solution of the weak Oda conjecture on the factorization of toric varieties. We then consider {it rational polyhedra}, i.e., finite unions of simplexes in $mathbb R^n$ with rational vertices. Markovs 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 $mathsf{GL}(n,mathbb{Z})ltimes mathbb{Z}^n$-equi-dis-sect-ability of $P$ and $P$ is open. We prove the decidability of the problem whether two rational polyhedra $P,Q$ in $mathbb R^n$ have the same $mathsf{GL}(n,mathbb{Z})ltimes mathbb{Z}^n$-orbit.