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Fans, decision problems and generators of free abelian ℓ-groups

  • Daniele Mundici EMAIL logo
Veröffentlicht/Copyright: 28. Januar 2017
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 29 Heft 6

Abstract

Let t1,…,tn be ℓ-group terms in the variables X1,…,Xm. Let t^1,…,t^n be their associated piecewise homogeneous linear functions. Let G be the ℓ-group generated by t^1,…,t^n in the free m-generator ℓ-group 𝒜m. We prove: (i) the problem whether G is ℓ-isomorphic to 𝒜n is decidable; (ii) the problem whether G is ℓ-isomorphic to 𝒜l (l arbitrary) is undecidable; (iii) for m=n, the problem whether {t^1,…,t^n} is a free generating set is decidable. In view of the Baker–Beynon duality, these theorems yield recognizability and unrecognizability results for the rational polyhedron associated to the ℓ-group G. We make pervasive use of fans and their stellar subdivisions.

Keywords: nonsingular fan; desingularization; decision problem; Baker–Beynon duality; Markov unrecognizability theorem
MSC 2010: 06F20; 08B30; 06B25; 52B20; 52B55; 55N10; 57Q05.

Communicated by Manfred Droste

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Received: 2016-12-15
Revised: 2016-12-28
Published Online: 2017-1-28
Published in Print: 2017-11-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. The role of the Rogers–Shephard inequality in the characterization of the difference body
  3. Test vectors for local periods
  4. The local Langlands conjecture for p-adic GSpin4, GSpin6, and their inner forms
  5. A family of irretractable square-free solutions of the Yang–Baxter equation
  6. On Sylow permutable subgroups of finite groups
  7. Using Steinberg algebras to study decomposability of Leavitt path algebras
  8. The ℓ∞-semi-norm on uniformly finite homology
  9. Minimal potential results for Schrödinger equations with Neumann boundary conditions
  10. On a unique continuation for Aharonov–Bohm magnetic Schrödinger equation
  11. Sup-norm bounds for Eisenstein series
  12. Permutohedral structures on E2-operads
  13. On the complete integrability of the periodic quantum Toda lattice
  14. Fans, decision problems and generators of free abelian ℓ-groups
  15. Standard cocycles: Variations on themes of C. Kassel’s and R. Wilson’s
  16. Crossed modules as maps between connected components of topological groups
  17. A local converse theorem for U(2,2)
  18. Probabilistic trace and Poisson summation formulae on locally compact abelian groups
https://doi.org/10.1515/forum-2016-0255
Schlagwörter für diesen Artikel
nonsingular fan; desingularization; decision problem; Baker–Beynon duality; Markov unrecognizability theorem

Artikel in diesem Heft

  1. Frontmatter
  2. The role of the Rogers–Shephard inequality in the characterization of the difference body
  3. Test vectors for local periods
  4. The local Langlands conjecture for p-adic GSpin4, GSpin6, and their inner forms
  5. A family of irretractable square-free solutions of the Yang–Baxter equation
  6. On Sylow permutable subgroups of finite groups
  7. Using Steinberg algebras to study decomposability of Leavitt path algebras
  8. The ℓ∞-semi-norm on uniformly finite homology
  9. Minimal potential results for Schrödinger equations with Neumann boundary conditions
  10. On a unique continuation for Aharonov–Bohm magnetic Schrödinger equation
  11. Sup-norm bounds for Eisenstein series
  12. Permutohedral structures on E2-operads
  13. On the complete integrability of the periodic quantum Toda lattice
  14. Fans, decision problems and generators of free abelian ℓ-groups
  15. Standard cocycles: Variations on themes of C. Kassel’s and R. Wilson’s
  16. Crossed modules as maps between connected components of topological groups
  17. A local converse theorem for U(2,2)
  18. Probabilistic trace and Poisson summation formulae on locally compact abelian groups
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