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Semi-algebraic sets of f-vectors

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Abstract

Polytope theory has produced a great number of remarkably simple and complete characterization results for face-number sets or f-vector sets of classes of polytopes. We observe that in most cases these sets can be described as the intersection of a semi-algebraic set with an integer lattice. Such semi-algebraic sets of lattice points have not received much attention, which is surprising in view of a close connection to Hilbert's Tenth problem, which deals with their projections.

We develop proof techniques in order to show that, despite the observations above, some f-vector sets are not semi-algebraic sets of lattice points. This is then proved for the set of all pairs (f1, f2) of 4-dimensional polytopes, the set of all f-vectors of simplicial d-polytopes for d ≥ 6, and the set of all f-vectors of general d-polytopes for d ≥ 6. For the f-vector set of all 4-polytopes this remains open.

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References

  1. D. W. Barnette, The projection of the f-vectors of 4-polytopes onto the (E, S)-plane, Discrete Mathematics 10 (1974), 201–216.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. W. Barnette and J. R. Reay, Projections of f-vectors of four-polytopes, Journal of Combinatorial Theory. Series A 15 (1973), 200–209.

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Basu, R. Pollack and M.-F. Roy, Algorithms in Real Algebraic Geometry, Algorithms and Combinatorics, Vol. 10, Springer-Verlag, Heidelberg, 2006.

  4. L. J. Billera and C. W. Lee, Sufficiency of McMullen's conditions for f-vectors of simpli-cial polytopes, Bulletin of the American Mathematical Society 2 (1980), 181–185.

    Article  MATH  Google Scholar 

  5. L. J. Billera and C. W. Lee, A proof of the sufficiency of McMullen's conditions for f-vectors of simplicial polytopes, Journal of Combinatorial Theory. Series A 31 (1981), 237–255.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Bjorner and S. Linusson, The number of k-faces of a simple d-polytope, Discrete & Computational Geometry 21 (1999), 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  7. G. Blind and R. Blind, Gaps in the numbers of vertices of cubical polytopes, I, Discrete & Computational Geometry 11 (1994), 351–356.

    Article  MathSciNet  MATH  Google Scholar 

  8. G. Blind and R. Blind, The cubical d-polytopes with fewer than 2 d+1 faces, Discrete & Computational Geometry 13 (1995), 321–345.

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Brinkmann and G. M. Ziegler, Small f-vectors of 3-spheres and of 4-polytopes, Mathematics of Computation 87 (2018), 2955–2975.

    MathSciNet  MATH  Google Scholar 

  10. M. Davis, Hilbert's tenth problem is unsolvable, American Mathematical Monthly 80 (1973), 233–269.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Eckhoff, Combinatorial properties of f-vectors of convex polytopes, Normat 54 (2006), 146–159.

    MathSciNet  Google Scholar 

  12. B. Griinbaum, Convex Polytopes, Graduate Texts in Mathematics, Vol. 221, Springer, New York, 2003.

  13. A. Hoppner and G. M. Ziegler, A census of Sag-vectors of 4-polytopes, in Polytopes—Combinatorics and Computation (Oberewolfach, 1997), DMV Seminars, Vol. 29, Birkhauser, Basel, 2000, pp. 105–110.

    MATH  Google Scholar 

  14. J. P. Jones, Universal diophantine equation, Journal of Symbolic Logic 47 (1982), 549–571.

    Article  MathSciNet  MATH  Google Scholar 

  15. T. Kusunoki and S. Murai, The numbers of edges of 5-polytopes with a given number of vertices, Annals of Combinatorics 23 (2019), 89–101.

    Article  MathSciNet  MATH  Google Scholar 

  16. Y. V. Matijasevič, Enumerable sets are Diophantine, Doklady Akademii Nauk SSSR 29 (1970), 279–282; English translation: Soviet Mathematics. Doklady 11 (1970), 354–357.

    Google Scholar 

  17. Y. V. Matiyasevich, Hilbert's Tenth Problem, Foundations of Computing Series, MIT Press, Cambridge, MA, 1993.

    Google Scholar 

  18. H. Miyata, Studies on classifications and constructions of combinatorial structures related to oriented matroids, Ph.D. thesis, University of Tokyo, 2011.

    Google Scholar 

  19. A. Paffenholz and A. Werner, Constructions for 4-polytopes and the cone of flag vectors, in Algebraic and Geometric Combinatorics, Contemporary Mathematics, Vol. 423, American Mathematical Society, Providence, RI, 2006, pp. 283–303.

    Article  MathSciNet  MATH  Google Scholar 

  20. G. Pineda-Villavicencio, J. Ugon and D. Yost, The excess degree of a polytope, SIAM Journal on Discrete Mathematics 32(3) (2018), 2011–2046.

    Article  MathSciNet  MATH  Google Scholar 

  21. H. Sjöberg and G. M. Ziegler, Characterizing face and flag vector pairs for polytopes, Discrete & Computational Geometry, to appear, https://doi.org/10.1007/s00454-018-0044-7.

  22. R. P. Stanley, The number of faces of simplicial convex polytopes, Advances in Mathematics 35 (1980), 236–238.

    Article  MathSciNet  MATH  Google Scholar 

  23. E. Steinitz, Über die Eulerschen Polyederrelationen, Archiv der Mathematik und Physik 11 (1906), 86–88.

    MATH  Google Scholar 

  24. Z.-W. Sun, Further results on Hilbert's Tenth Problem, preprint, https://doi.org/abs/1704.03504v3.

  25. A. Werner, Linear constraints on face numbers of polytopes, Ph.D. thesis, TU Berlin, 2009, https://doi.org/10.14279/depositonce-2183.

    MATH  Google Scholar 

  26. G. M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, Vol. 152, Springer, New York, 1995.

  27. G. M. Ziegler, Additive structures on f-vector sets of polytopes, Advances in Geometry, to appear, https://doi.org/10.1515/advgeom-2018-0025.

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Acknowledgements

Thanks to Isabella Novik and to an anonymous referee for insightful comments.

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Authors and Affiliations

  1. Institut für Mathematik, FU Berlin, Arnimallee 2, 14195, Berlin, Germany

    Hannah Sjöberg & Günter M. Ziegler

Authors
  1. Hannah Sjöberg
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  2. Günter M. Ziegler
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Correspondence to Hannah Sjöberg.

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Dedicated to Anders Björner on the occasion of his 70th birthday

Research supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics.”

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Sjöberg, H., Ziegler, G.M. Semi-algebraic sets of f-vectors. Isr. J. Math. 232, 827–848 (2019). https://doi.org/10.1007/s11856-019-1888-0

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  • DOI: https://doi.org/10.1007/s11856-019-1888-0

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