Complete and computable orbit invariants in the geometry of the affine group over the integers
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Complete and computable orbit invariants in the geometry of the affine group over the integers Daniele Mundici1 Received: 3 February 2019 / Accepted: 18 January 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The subject matter of this paper is the geometry of the affine group over the integers, 𝖦𝖫(n, ℤ) ⋉ ℤn . Turing-computable complete 𝖦𝖫(n, ℤ) ⋉ ℤn-orbit invariants are constructed for rational affine spaces, angles, segments, triangles and ellipses. In rational affine 𝖦𝖫(n, ℚ) ⋉ ℚn-geometry, ellipses are classified by the Clifford–Hasse–Witt invariant, via the Hasse–Minkowski theorem. We classify ellipses in 𝖦𝖫(n, ℤ) ⋉ ℤn-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 ℝn 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 𝖦𝖫(n, ℚ) ⋉ ℚn-equidissectable. The same problem for the continuous 𝖦𝖫(n, ℤ) ⋉ ℤn-equidissectability of P and P′ is open. We prove the decidability of the problem whether two rational polyhedra P and P′ in ℝn have the same 𝖦𝖫(n, ℤ) ⋉ ℤn-orbit. Keywords Affine group over the integers · Klein program · Complete orbit invariant · Turing-computable invariant · 𝖦𝖫(n, ℤ)-orbit · (Farey)regular simplex · Regular complex · Desingularization · Strong Oda conjecture · Hirzebruch–Jung continued fraction algorithm · Rational polyhedron · Conic · Conjugate diameters · Apollonius of Perga · Pappus of Alexandria · Quadratic form · Clifford–Hasse–Witt invariant · Hasse– Minkowski theorem · Markov unrecognizability theorem Mathematics Subject Classification Primary 13A50; Secondary 11D09 · 11E16 · 11H55 · 13A50 · 14G05 · 14H50 · 14L24 · 14M25 · 14R20 · 20H25
In memoriam Roberto Cignoli. * Daniele Mundici [email protected] 1
Department of Mathematics and Computer Science “Ulisse Dini”, University of Florence, Viale Morgagni 67/a, 50134 Florence, Italy
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1 Introduction Klein’s 1872 inaugural lecture at the University of Erlangen contains the following programmatic statement1: As a generalization of geometry arises then the following comprehensive problem […] : Given a manifoldness and a group of transformations of the same; to develop the theory of invariants relating to that group. In the spirit of Klein’s program, in this paper we will construct Turing-computable2 complete invariants of angles, segments, triangles and ellipses in the geometry of the affine group over the integers, 𝖦𝖫(n, ℤ) ⋉ ℤn. Our starting point is the following problem, for arbitrary sets X, X ′ ⊆ ℝn:
Does there exist a map 𝛾 ∈ 𝖦𝖫(n, ℤ) ⋉ ℤn of X onto X � ?
(1)
Otherwise stated: Do X and X ′ have the same 𝖦𝖫(n, ℤ) ⋉ ℤn-orbit ? By a “decision method” for this problem we understand a Turing machine M which, over any input X, X
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