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Unimodality problems in Ehrhart theory

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Recent Trends in Combinatorics

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 159))

Abstract

Ehrhart theory is the study of sequences recording the number of integer points in non-negative integral dilates of rational polytopes. For a given lattice polytope, this sequence is encoded in a finite vector called the Ehrhart h ∗-vector. Ehrhart h ∗-vectors have connections to many areas of mathematics, including commutative algebra and enumerative combinatorics. In this survey we discuss what is known about unimodality for Ehrhart h ∗-vectors and highlight open questions and problems.

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References

  1. F. Ardila, C. Benedetti, J. Doker, Matroid polytopes and their volumes. Discret. Comput. Geom. 43(4), 841–854 (2010). doi:10.1007/s00454-009-9232-9. http://dx.doi.org/10.1007/s00454-009-9232-9

    Google Scholar 

  2. C.A. Athanasiadis, h ∗-vectors, Eulerian polynomials and stable polytopes of graphs. Electron. J. Comb. 11(2), Research Paper 6, 13 p. (electronic) (2004/2006)

    Google Scholar 

  3. C.A. Athanasiadis, Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley. J. Reine Angew. Math. 583, 163–174 (2005). doi:10.1515/crll.2005.2005.583.163. http://dx.doi.org/10.1515/crll.2005.2005.583.163

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. M. Beck, B. Braun, Euler-Mahonian statistics via polyhedral geometry. Adv. Math. 244(0), 925–954 (2013). doi:http://dx.doi.org/10.1016/j.aim.2013.06.002. http://www.sciencedirect.com/science/article/pii/S0001870813002065

    Google Scholar 

  6. M. Beck, S. Hoşten, Cyclotomic polytopes and growth series of cyclotomic lattices. Math. Res. Lett. 13(4), 607–622 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Beck, S. Robins, Computing the Continuous Discretely. Undergraduate Texts in Mathematics (Springer, New York, 2007)

    Google Scholar 

  8. M. Beck, A. Stapledon, On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series. Math. Z. 264(1), 195–207 (2010). doi:10.1007/s00209-008-0458-7. http://dx.doi.org/10.1007/s00209-008-0458-7

    Google Scholar 

  9. M. Beck, P. Jayawant, T.B. McAllister, Lattice-point generating functions for free sums of convex sets. J. Comb. Theory A 120(6), 1246–1262 (2013). doi:10.1016/j.jcta.2013.03.007. http://dx.doi.org/10.1016/j.jcta.2013.03.007

    Google Scholar 

  10. M. Beck, B. Braun, M. Köppe, C.D. Savage, Z. Zafeirakopoulos, s-lecture hall partitions, self-reciprocal polynomials, and Gorenstein cones. Ramanujan J. 36(1–2), 123–147 (2015). doi:10.1007/s11139-013-9538-3. http://dx.doi.org/10.1007/s11139-013-9538-3

    Google Scholar 

  11. M. Beck, K. Jochemko, E. McCullough, Manuscript in preparation

    Google Scholar 

  12. U. Betke, P. McMullen, Lattice points in lattice polytopes. Monatsh. Math. 99(4), 253–265 (1985). doi:10.1007/BF01312545. http://dx.doi.org/10.1007/BF01312545

  13. C. Bey, M. Henk, J.M. Wills, Notes on the roots of Ehrhart polynomials. Discret. Comput. Geom. 38(1), 81–98 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Bousquet-Mélou, K. Eriksson, Lecture hall partitions. Ramanujan J. 1(1), 101–111 (1997). doi:10.1023/A:1009771306380. http://dx.doi.org/10.1023/A:1009771306380

    Google Scholar 

  15. P. Brändén, Unimodality, log-concavity, real-rootedness, and beyond. in Handbook of Enumerative Combinatorics (CRC, Boca Raton, 2014). Preprint available at arXiv:1410.6601v1

    Google Scholar 

  16. B. Braun, An Ehrhart series formula for reflexive polytopes. Electron. J. Comb. 13(1), Note 15, 5 p. (electronic) (2006)

    Google Scholar 

  17. B. Braun, R. Davis, Ehrhart series, unimodality, and integrally closed reflexive polytopes to appear in Annals of Combinatorics

    Google Scholar 

  18. B. Braun, L. Solus, A shelling of the odd hypersimplex (Preprint), http://arxiv.org/abs/1408.4713

  19. F. Brenti, Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update, in Jerusalem combinatorics ’93. Contemporary Mathematics, vol. 178 (American Mathematical Society, Providence, 1994), pp. 71–89. doi:10.1090/conm/178/01893. http://dx.doi.org/10.1090/conm/178/01893

  20. F. Brenti, V. Welker, The Veronese construction for formal power series and graded algebras. Adv. Appl. Math. 42(4), 545–556 (2009). doi:10.1016/j.aam.2009.01.001. http://dx.doi.org/10.1016/j.aam.2009.01.001

    Google Scholar 

  21. W. Bruns, J. Gubeladze, Unimodular covers of multiples of polytopes. Doc. Math. 7, 463–480 (electronic) (2002)

    Google Scholar 

  22. W. Bruns, J. Gubeladze, Polytopes, Rings, and K-Theory. Springer Monographs in Mathematics (Springer, Dordrecht, 2009). doi:10.1007/b105283. http://dx.doi.org/10.1007/b105283

    Google Scholar 

  23. W. Bruns, J. Herzog, Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics, vol. 39 (Cambridge University Press, Cambridge, 1993)

    Google Scholar 

  24. W. Bruns, T. Römer, h-vectors of Gorenstein polytopes. J. Comb. Theory A 114(1), 65–76 (2007). doi:10.1016/j.jcta.2006.03.003. http://dx.doi.org/10.1016/j.jcta.2006.03.003

    Google Scholar 

  25. M. Chudnovsky, P. Seymour, The roots of the independence polynomial of a clawfree graph. J. Comb. Theory B 97(3), 350–357 (2007). doi:10.1016/j.jctb.2006.06.001. http://dx.doi.org/10.1016/j.jctb.2006.06.001

    Google Scholar 

  26. S. Corteel, S. Lee, C.D. Savage, Enumeration of sequences constrained by the ratio of consecutive parts. Sém. Lothar. Comb. 54A, Art. B54Aa, 12 (2005/2007)

    Google Scholar 

  27. D.A. Cox, C. Haase, T. Hibi, A. Higashitani, Integer decomposition property of dilated polytopes. Electron. J. Comb. 21(4), Paper 4.28, 17 (2014)

    Google Scholar 

  28. R. Davis, Ehrhart series of polytopes related to symmetric doubly-stochastic matrices. Electron. J. Comb. 22(2), P2.17, (electronic) (2015)

    Google Scholar 

  29. E. Ehrhart, Sur les polyèdres rationnels homothétiques à n dimensions. C. R. Acad. Sci. Paris 254, 616–618 (1962)

    MathSciNet  MATH  Google Scholar 

  30. I.M. Gelfand, M.I. Graev, A. Postnikov, Combinatorics of hypergeometric functions associated with positive roots, in The Arnold-Gelfand mathematical seminars (Birkhäuser Boston, Boston, 1997), pp. 205–221

    MATH  Google Scholar 

  31. C. Haase, I.V. Melnikov, The reflexive dimension of a lattice polytope. Ann. Comb. 10(2), 211–217 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. T. Harima, J.C. Migliore, U. Nagel, J. Watanabe, The weak and strong Lefschetz properties for Artinian k-algebras. J. Algebra 262(1), 99–126 (2003). doi:http://dx.doi.org/10.1016/S0021-8693(03)00038-3. http://www.sciencedirect.com/science/article/pii/S0021869303000383

    Google Scholar 

  33. D. Haws, Matroid polytopes: algorithms, theory, and applications, Ph.D Dissertation available at http://arxiv.org/abs/0905.4405

  34. T. Hibi, Flawless O-sequences and Hilbert functions of Cohen-Macaulay integral domains. J. Pure Appl. Algebra 60(3), 245–251 (1989). doi:10.1016/0022-4049(89)90085-6. http://dx.doi.org/10.1016/0022-4049(89)90085-6

    Google Scholar 

  35. T. Hibi, Algebraic Combinatorics on Convex Polytopes (Carslaw Publications, Glebe, 1992)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  37. T. Hibi, A. Higashitani, N. Li, Hermite normal forms and δ-vectors. J. Comb. Theory A 119(6), 1158–1173 (2012). doi:10.1016/j.jcta.2012.02.005. http://dx.doi.org/10.1016/j.jcta.2012.02.005

    Google Scholar 

  38. T. Hibi, L. Solus, The facets of the r-stable n,k-hypersimplex. To appear in Annals of Combinatorics

    Google Scholar 

  39. A. Higashitani, Counterexamples of the conjecture on roots of Ehrhart polynomials. Discret. Comput. Geom. 47(3), 618–623 (2012). doi:10.1007/s00454-011-9390-4. http://dx.doi.org/10.1007/s00454-011-9390-4

    Google Scholar 

  40. M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes. Ann. Math. (2) 96, 318–337 (1972)

    Google Scholar 

  41. K. Jochemko, On the combinatorics of valuations. PhD Thesis available at http://www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000098742

  42. K. Jochemko, R. Sanyal, Combinatorial positivity of translation-invariant valuations. To appear in J. Eur. Math. Soc. (JEMS)

    Google Scholar 

  43. E. Katz, A. Stapledon, Local h-polynomials, invariants of subdivisions, and mixed Ehrhart theory. Adv. Math. 286, 181–239 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  44. M. Katzman, The Hilbert series of algebras of the Veronese type. Commun. Algebra 33(4), 1141–1146 (2005). doi:10.1081/AGB-200053828. http://dx.doi.org/10.1081/AGB-200053828

    Google Scholar 

  45. G. Kempf, F.F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal Embeddings. I. Lecture Notes in Mathematics, vol. 339 (Springer, Berlin/New York, 1973)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  47. J.C. Lagarias, G.M. Ziegler, Bounds for lattice polytopes containing a fixed number of interior points in a sublattice. Can. J. Math. 43(5), 1022–1035 (1991). doi:10.4153/CJM-1991-058-4. http://dx.doi.org/10.4153/CJM-1991-058-4

    Google Scholar 

  48. T. Lam, A. Postnikov, Alcoved polytopes. I. Discret. Comput. Geom. 38(3), 453–478 (2007). doi:10.1007/s00454-006-1294-3. http://dx.doi.org/10.1007/s00454-006-1294-3

    Google Scholar 

  49. N. Li, Ehrhart h ∗-vectors of hypersimplices. Discret. Comput. Geom. 48(4), 847–878 (2012). doi:10.1007/s00454-012-9452-2. http://dx.doi.org/10.1007/s00454-012-9452-2

  50. J.A. De Loera, D.C. Haws, M. Köppe, Ehrhart polynomials of matroid polytopes and polymatroids. Discret. Comput. Geom. 42(4), 670–702 (2009). doi:10.1007/s00454-008-9080-z. http://dx.doi.org/10.1007/s00454-008-9080-z

    Google Scholar 

  51. J.A. De Loera, J. Rambau, F. Santos, Triangulations. Algorithms and Computation in Mathematics, vol. 25 (Springer, Berlin, 2010). doi:10.1007/978-3-642-12971-1. http://dx.doi.org/10.1007/978-3-642-12971-1. Structures for algorithms and applications

    Google Scholar 

  52. A. McCabe, G.G. Smith, Log-concavity of asymptotic multigraded Hilbert series. Proc. Am. Math. Soc. 141(6), 1883–1892 (2013). doi:10.1090/S0002-9939-2012-11808-8. http://dx.doi.org/10.1090/S0002-9939-2012-11808-8

    Google Scholar 

  53. P. McMullen, On zonotopes. Trans. Am. Math. Soc. 159, 91–109 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  54. E. Miller, B. Sturmfels, Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol. 227 (Springer, New York, 2005)

    Google Scholar 

  55. M. Mustaţǎ, S. Payne, Ehrhart polynomials and stringy Betti numbers. Math. Ann. 333(4), 787–795 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  56. H. Ohsugi, T. Hibi, A normal (0, 1)-polytope none of whose regular triangulations is unimodular. Discret. Comput. Geom. 21(2), 201–204 (1999). doi:10.1007/PL00009415. http://dx.doi.org/10.1007/PL00009415

    Google Scholar 

  57. H. Ohsugi, T. Hibi, Special simplices and Gorenstein toric rings. J. Comb. Theory A 113(4), 718–725 (2006). doi:10.1016/j.jcta.2005.06.002. http://dx.doi.org/10.1016/j.jcta.2005.06.002

    Google Scholar 

  58. S. Payne, Ehrhart series and lattice triangulations. Discret. Comput. Geom. 40(3), 365–376 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  59. A. Postnikov, Permutohedra, associahedra, and beyond. Int. Math. Res. Not. IMRN. 6, 1026–1106 (2009). doi:10.1093/imrn/rnn153. http://dx.doi.org/10.1093/imrn/rnn153

    Google Scholar 

  60. C.D. Savage, M.J. Schuster, Ehrhart series of lecture hall polytopes and Eulerian polynomials for inversion sequences. J. Comb. Theory A 119(4), 850–870 (2012). doi:10.1016/j.jcta.2011.12.005. http://dx.doi.org/10.1016/j.jcta.2011.12.005

    Google Scholar 

  61. C.D. Savage, M. Visontai, The s-Eulerian polynomials have only real roots. Trans. Am. Math. Soc. 367(2), 1441–1466 (2015). doi:10.1090/S0002-9947-2014-06256-9. http://dx.doi.org/10.1090/S0002-9947-2014-06256-9

    Google Scholar 

  62. J. Schepers, L. van Langenhoven, Unimodality questions for integrally closed lattice polytopes. Ann. Comb. 17(3), 571–589 (2013). doi:10.1007/s00026-013-0185-6. http://dx.doi.org/10.1007/s00026-013-0185-6

    Google Scholar 

  63. R.P. Stanley, Hilbert functions of graded algebras. Adv. Math. 28(1), 57–83 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  64. R.P. Stanley, Decompositions of rational convex polytopes. Ann. Discret. Math. 6, 333–342 (1980). Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978)

    Google Scholar 

  65. R.P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry, in Graph Theory and Its Applications: East and West (Jinan, 1986). Annals of the New York Academy of Sciences, vol. 576 (New York Academy of Sciences, New York, 1989), pp. 500–535. doi:10.1111/j.1749-6632.1989.tb16434.x. http://dx.doi.org/10.1111/j.1749-6632.1989.tb16434.x

    Google Scholar 

  66. R.P. Stanley, A monotonicity property of h-vectors and h ∗-vectors. Eur. J. Comb. 14(3), 251–258 (1993). doi:10.1006/eujc.1993.1028. http://dx.doi.org/10.1006/eujc.1993.1028

    Google Scholar 

  67. R.P. Stanley, Combinatorics and Commutative Algebra. Progress in Mathematics, vol. 41, 2nd edn. (Birkhäuser, Boston, 1996)

    Google Scholar 

  68. R.P. Stanley, Enumerative Combinatorics, vol. 2. Cambridge Studies in Advanced Mathematics, vol. 62 (Cambridge University Press, Cambridge, 1999). doi:10.1017/CBO9780511609589. http://dx.doi.org/10.1017/CBO9780511609589. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin

  69. A. Stapledon, Inequalities and Ehrhart δ-vectors. Trans. Am. Math. Soc. 361(10), 5615–5626 (2009). doi:10.1090/S0002-9947-09-04776-X. http://dx.doi.org/10.1090/S0002-9947-09-04776-X

    Google Scholar 

  70. N.L. White, A unique exchange property for bases. Linear Algebra Appl. 31, 81–91 (1980). doi:10.1016/0024-3795(80)90209-8. http://dx.doi.org/10.1016/0024-3795(80)90209-8

    Google Scholar 

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Acknowledgements

The author is partially supported by the National Security Agency through award H98230-13-1-0240. Thanks to Christos Athanasiadis, Matthias Beck, Robert Davis, Jesus De Loera, David Haws, Takayuki Hibi, Akihiro Higashitani, Carla Savage, and Liam Solus for their helpful comments and suggestions.

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  1. University of Kentucky, Lexington, KY, 40506–0027, USA

    Benjamin Braun

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Editors and Affiliations

  1. Dept of Math, Statistics & Comp.Sc, Macalester College, St. Paul, Minnesota, USA

    Andrew Beveridge

  2. Dept. of Mathematics, University of South Carolina, Columbia, South Carolina, USA

    Jerrold R. Griggs

  3. Dept. of Mathematics, Iowa State University, Ames, Iowa, USA

    Leslie Hogben

  4. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

    Gregg Musiker

  5. School of Mathematics & Computer Sc, Georgia Institute of Technology, Atlanta, Georgia, USA

    Prasad Tetali

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Braun, B. (2016). Unimodality problems in Ehrhart theory. In: Beveridge, A., Griggs, J., Hogben, L., Musiker, G., Tetali, P. (eds) Recent Trends in Combinatorics. The IMA Volumes in Mathematics and its Applications, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-24298-9_27

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