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

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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.

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Notes

  1. See page 219 in his paper “A comparative review of recent researches in geometry,” Bull. New York Math. Soc., 2(10) (1893) 215–249, https://projecteuclid.org/euclid.bams/1183407629.

  2. For background on decidable (= recursive = computable = Turing-decidable) sets and problems we refer to [7] and [23].

  3. In the sense that two objects have the same invariant iff they have the same \({\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n\)-orbit.

  4. Regular simplexes are said to be “unimodular” in [12, Sect. 7] and [17].

  5. Regular cones are said to be “nonsingular” in [9, p. 29] and [19, p. 15].

  6. If indeed arc length or the circular functions are more elementary than the invariant \({\mathsf {angle}}\) in \({\mathsf {GL}}(n,{\mathbb {Z}})\ltimes {\mathbb {Z}}^n\)-geometry.

  7. This follows from the positive solution of the low-dimensional strong Oda conjecture on the decomposition of birational toric maps. See [8, Sect. 6, pp. 181–183] for a formulation in terms of two-dimensional fans. Passing to affine correspondents we immediately obtain (15).

  8. Also known as an “axial affine transformation.”

  9. Pappus [21, Book VIII, Sect. XVII, Proposition 14] constructs the axes of an ellipse from any given pair of conjugate semi-diameters. For more recent constructions see [20, p. 69 ff].

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Acknowledgements

The author is grateful to the referee for many valuable insights and suggestions which led to a thorough revision of the original manuscript.

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  1. Department of Mathematics and Computer Science “Ulisse Dini”, University of Florence, Viale Morgagni 67/a, 50134, Florence, Italy

    Daniele Mundici

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  1. Daniele Mundici
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Correspondence to Daniele Mundici.

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In memoriam Roberto Cignoli.

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Mundici, D. Complete and computable orbit invariants in the geometry of the affine group over the integers. Annali di Matematica 199, 1843–1871 (2020). https://doi.org/10.1007/s10231-020-00945-y

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