The Abstract

	Annals of Combinatorics 10 (2006) 211-217 The Reflexive Dimension of a Lattice Polytope Christian Haase and Ilarion V. Melnikov Department of Mathematics and Computer Science, Freie Universit¨at Berlin, D-14195 Berlin, Germany christian.haase@math.fu-berlin.de Particle Theory Group, Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA lmel@theory.uchicago.edu Received July 2, 2004 AMS Subject Classification: 52B20, 14M25, 83E30 Abstract. The reflexive dimension refldim(P) of a lattice polytope P is the minimal integer d so that P is the face of some d-dimensional reflexive polytope. We show that refldim(P) is fi nite for every P, and give bounds for refldim(kP) in terms of refldim(P) and k. Keywords: reflexive polytopes References 1. A. Barvinok, A Course in Convexity, Graduate Studies in Mathematics, Vol. 54, Amer. Math. Soc., Providence, RI, 2002. 2. V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994) 493–535. 3. W. Fulton, Introduction to Toric Varieties, Ann. of Math. Stud., Vol. 131, Princeton University Press, 1993. 4. C. Haase, Reflexive polytopes in dimension 2 and 3, and the numbers 12 and 24, In: Integer Points in Polyhedra, Geometry, Number Theory, Algebra, Optimization, A. Barvinok, M. Beck, C. Haase, and B. Reznick Eds., Contemporary Mathematics, Vol. 374, AMS, xi 191. 5. T. Hibi, Dual polytopes of rational convex polytopes, Combinatorica 12 (2) (1992) 237–240. 6. M. Kreuzer and H. Skarke, Classification of reflexive polyhedra in three dimensions, Adv. Theor. Math. Phys. 2 (4) (1998) 847–864. 7. M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four dimensions, Adv. Theor. Math. Phys. 4 (6) (2000) 1209–1230. 8. M. Kreuzer and H. Skarke, PALP: a package for analysing lattice polytopes with applications to toric geometry, Comput. Phys. Comm. 157 (1) (2004) 87–106. ArXiv:math.SC/ 0204356, available at http://tph16.tuwien.ac.at/~kreuzer/CY/. 9. J.C. Lagarias and G.M. Ziegler, Bounds for lattice polytopes containing a fixed number of interior points in a sublattice, Canad. J. Math. 43 (5) (1991) 1022–1035. 10. D.R. Morrison and M.R. Plesser, Summing the instantons: quantum cohomology and mirror symmetry in toric varieties, Nuclear Phys. B 440 (1-2) (1995) 279–354. 11. B. Nill, Gorenstein toric Fano varieties, Manuscripta Math. 116 (2) (2005) 183–210. 12. B. Nill, Volume and lattice points of reflexive simplices, preprint, arXiv:math.AG/ 0412480, 2004. 13. M.A. Perles, J.M. Wills, and J. Zaks, On lattice polytopes having interior lattice points, Elem. Math. 37 (2) (1982) 44–46. 14. O. Pikhurko, Lattice points in lattice polytopes, Mathematika 48 (2001) (1-2) 15–24 (2003). 15. N. Sloane, The On-Line Encyclopedia of Integer Sequences, Notices Amer. Math. Soc. 50 (8) (2003) 912–915. Available at http://www.research.att.com/~njas/sequences/. 16. M.D. Vose, Egyptian fractions, Bull. London Math. Soc. 17 (1) (1985) 21–24. Get the DVI \| PS \| PDF file of this abstract. back

Annals of Combinatorics 10 (2006) 211-217

The Reflexive Dimension of a Lattice Polytope

Christian Haase and Ilarion V. Melnikov

Department of Mathematics and Computer Science, Freie Universit¨at Berlin, D-14195 Berlin, Germany
christian.haase@math.fu-berlin.de

Particle Theory Group, Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA
lmel@theory.uchicago.edu

Received July 2, 2004

AMS Subject Classification: 52B20, 14M25, 83E30

Abstract. The reflexive dimension refldim(P) of a lattice polytope P is the minimal integer d so that P is the face of some d-dimensional reflexive polytope. We show that refldim(P) is fi nite for every P, and give bounds for refldim(kP) in terms of refldim(P) and k.

Keywords: reflexive polytopes

References

1. A. Barvinok, A Course in Convexity, Graduate Studies in Mathematics, Vol. 54, Amer. Math. Soc., Providence, RI, 2002.

2. V.V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994) 493–535.

3. W. Fulton, Introduction to Toric Varieties, Ann. of Math. Stud., Vol. 131, Princeton University Press, 1993.

4. C. Haase, Reflexive polytopes in dimension 2 and 3, and the numbers 12 and 24, In: Integer Points in Polyhedra, Geometry, Number Theory, Algebra, Optimization, A. Barvinok, M. Beck, C. Haase, and B. Reznick Eds., Contemporary Mathematics, Vol. 374, AMS, xi 191.

5. T. Hibi, Dual polytopes of rational convex polytopes, Combinatorica 12 (2) (1992) 237–240.

6. M. Kreuzer and H. Skarke, Classification of reflexive polyhedra in three dimensions, Adv. Theor. Math. Phys. 2 (4) (1998) 847–864.

7. M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four dimensions, Adv. Theor. Math. Phys. 4 (6) (2000) 1209–1230.

8. M. Kreuzer and H. Skarke, PALP: a package for analysing lattice polytopes with applications to toric geometry, Comput. Phys. Comm. 157 (1) (2004) 87–106. ArXiv:math.SC/ 0204356, available at http://tph16.tuwien.ac.at/~kreuzer/CY/.

9. J.C. Lagarias and G.M. Ziegler, Bounds for lattice polytopes containing a fixed number of interior points in a sublattice, Canad. J. Math. 43 (5) (1991) 1022–1035.

10. D.R. Morrison and M.R. Plesser, Summing the instantons: quantum cohomology and mirror symmetry in toric varieties, Nuclear Phys. B 440 (1-2) (1995) 279–354.

11. B. Nill, Gorenstein toric Fano varieties, Manuscripta Math. 116 (2) (2005) 183–210.

12. B. Nill, Volume and lattice points of reflexive simplices, preprint, arXiv:math.AG/ 0412480, 2004.

13. M.A. Perles, J.M. Wills, and J. Zaks, On lattice polytopes having interior lattice points, Elem. Math. 37 (2) (1982) 44–46.

14. O. Pikhurko, Lattice points in lattice polytopes, Mathematika 48 (2001) (1-2) 15–24 (2003).

15. N. Sloane, The On-Line Encyclopedia of Integer Sequences, Notices Amer. Math. Soc. 50 (8) (2003) 912–915. Available at http://www.research.att.com/~njas/sequences/.

16. M.D. Vose, Egyptian fractions, Bull. London Math. Soc. 17 (1) (1985) 21–24.

Get the DVI | PS | PDF file of this abstract.

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