Skip to main content
SpringerLink
Log in

On Positivity of Ehrhart Polynomials

  • Chapter
  • First Online:
Recent Trends in Algebraic Combinatorics

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 16))

Abstract

Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients and say a polytope is Ehrhart positive if all Ehrhart coefficients are positive (which is not true for all integral polytopes). The main purpose of this chapter is to survey interesting families of polytopes that are known to be Ehrhart positive and discuss the reasons from which their Ehrhart positivity follows. We also include examples of polytopes that have negative Ehrhart coefficients and polytopes that are conjectured to be Ehrhart positive and pose a few relevant questions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Unimodular equivalence is sometimes called integral equivalence, e.g., in [79].

References

  1. F. Ardila, C. Benedetti, J. Doker, Matroid polytopes and their volumes. Discret. Comput. Geom. 43(4), 841–854 (2010). MR 2610473

    Article  MathSciNet  MATH  Google Scholar 

  2. B. Assarf, E. Gawrilow, K. Herr, M. Joswig, B. Lorenz, A. Paffenholz, T.Rehn, Polymake in linear and integer programming (2014), http://arxiv.org/abs/1408.4653

  3. W. Baldoni, M. Vergne, Kostant partitions functions and flow polytopes. Transform. Groups 13(3–4), 447–469 (2008). MR 2452600

    Article  MathSciNet  MATH  Google Scholar 

  4. M.W. Baldoni, M. Beck, C. Cochet, M. Vergne, Maple code for “volume computation for polytopes and partition functions for classical root systems”. https://webusers.imj-prg.fr/~michele.vergne/IntegralPoints.html

  5. G. Balletti, A. Higashitani, Universal inequalities in Ehrhart theory. arXiv:1703.09600

  6. G. Balletti, A.M. Kasprzyk, Three-dimensional lattice polytopes with two interior lattice points. arXiv:1612.08918

  7. M. Beck, Counting lattice points by means of the residue theorem. Ramanujan J. 4(3), 299–310 (2000). MR 1797548

    Google Scholar 

  8. M. Beck, D. Pixton, The Ehrhart polynomial of the Birkhoff polytope. Discret. Comput. Geom. 30(4), 623–637 (2003). MR (2013976)

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Beck, J.A. De Loera, M. Develin, J. Pfeifle, R.P. Stanley, Coefficients and roots of Ehrhart polynomials. Contemp. Math. 374, 15–36 (2005). MR 2134759

    Google Scholar 

  10. M. Beck, S. Robins, in Computing the Continuous Discretely. Integer-Point Enumeration in Polyhedra, 2nd edn. Undergraduate Texts in Mathematics (Springer, New York, 2015). With illustrations by David Austin. MR 3410115

    Book  MATH  Google Scholar 

  11. N. Berline, M. Vergne, Local Euler-Maclaurin formula for polytopes. Mosc. Math. J. 7(3), 355–386, 573 (2007). MR 2343137

    Article  MathSciNet  MATH  Google Scholar 

  12. C. Bey, M. Henk, J. Wills, Notes on the roots of Ehrhart polynomials. Discret. Comput. Geom. 38(1), 81–98 (2007). MR 2322117

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Brändén, in Unimodality, Log-Concavity, Real-Rootedness and Beyond. Handbook of Enumerative Combinatorics; Discrete Math. Appl. (Boca Raton) (CRC Press, Boca Raton, 2015), pp. 437–483. MR 3409348

    Chapter  Google Scholar 

  14. B. Braun, Unimodality problems in Ehrhart theory, in Recent Trends in Combinatorics, vol. 159 (IMA Vol. Math. Appl.) (Sringer, Cham, 2016) pp. 687–711. MR 3526428

    Chapter  MATH  Google Scholar 

  15. B. Braun, R. Davis, L. Solus, Detecting the integer decomposition property and Ehrhart unimodality in reflexive simplices, arXiv:1608.01614

  16. F. Brenti, Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update, Jerusalem combinatorics, vol. 93. Contemporary Mathematics, vol. 178 (American Mathematical Society, Providence, 1994), pp. 71–89. MR 1310575

    Google Scholar 

  17. M. Brion, M. Vergne, An, equivariant Riemann-Roch theorem for complete, simplicial toric varieties. J. Reine Angew. Math. 482, 67–92 (1997). MR 1427657

    Google Scholar 

  18. W. Bruns, The quest for counterexamples in toric geometry, Commutative algebra and algebraic geometry (CAAG-2010), Ramanujan Math. Soc. Lect. Notes Ser. 17, 45–61 (2013) (Ramanujan Math. Soc., Mysore)

    Google Scholar 

  19. W. Bruns, J. Gubeladze, N.V. Trung, Normal polytopes, triangulations, and Koszul algebras. J. Reine Angew. Math. 485, 123–160 (1997), MR 1442191

    Google Scholar 

  20. A.S. Buch, The saturation conjecture (after A. Knutson and T. Tao). Enseign. Math. (2) 46(1–2), 43–60 (2000) (With an appendix by William Fulton. MR 1769536)

    Google Scholar 

  21. E.R. Canfield, B.D. McKay, The asymptotic volume of the Birkhoff polytope. Online J. Anal. Comb. 4, 4pp (2009). MR 2575172

    Google Scholar 

  22. S.E. Cappell, J.L. Shaneson, Genera of algebraic varieties and counting of lattice points. Bull. Am. Math. Soc. (N.S.) 30(1), 62–69 (1994). MR 1217352

    Article  MATH  Google Scholar 

  23. F. Castillo, Local Ehrhart Positivity, 91, Thesis (Ph.D.)-University of. California, Davis (2017) MR 3731970

    Google Scholar 

  24. F. Castillo, F. Liu, Berline-Vergne valuation and generalized permutohedra. Discret. Comput. Geom. 60(4), 885–908 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  25. F. Castillo, F. Liu, BV-\(\alpha \)-positivity for generalized permutohedra. arXiv:1509.07884v1

  26. F. Castillo, F. Liu, B. Nill, A. Paffenholz, Smooth polytopes with negative Ehrhart coeffcients. J. Comb. Theory Ser. A. 160, 316–331(2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Castillo, F., & Liu, F. (2015). Ehrhart positivity for generalized permutohedra. Discret. Math Theor. Comput. Sci. proc. FPSAC, ’15, 865–876.

    MATH  Google Scholar 

  28. C.S. Chan, D.P. Robbins, D.S. Yuen, On the volume of a certain polytope. Exp. Math. 9(1), 91–99 (2000). MR 1758803

    Article  MathSciNet  MATH  Google Scholar 

  29. H. Conrads, Weighted, projective spaces and reflexive simplices. Manuscripta Math. 107(2), pp. 215–227 (2002). MR 1894741

    Article  MathSciNet  MATH  Google Scholar 

  30. D.A. Cox, C. Haase, T. Hibi, A. Higashitani, Integer decomposition property of dilated polytopes. Electron. J. Comb. 21(4), Paper 4.28, 17pp. (2014). MR 3292265,

    Google Scholar 

  31. J.A. De Loera, D.C. Haws, M. oeppe, Ehrhart polynomials of matroid polytopes and polymatroids. Discret. Comput. Geom. 42(4), 670–702 (2009). MR 2556462

    Google Scholar 

  32. J.A. De Loera, F. Liu, R. Yoshida, A generating function for all semi-magic squares and the volume of the Birkhoff polytope. J. Algebr. Comb. 30(1), 113–139 (2009). MR 2519852 (2010h:52017)

    Google Scholar 

  33. H. Derksen, J. Weyman, On the Littlewood-Richardson polynomials. J. Algebr. 255(2), 247–257 (2002). MR 1935497

    Article  MathSciNet  MATH  Google Scholar 

  34. P. Diaconis, A. Gangolli, in Rectangular Arrays with Fixed Margins. Discrete Probability and Algorithms (Minneapolis, MN, 1993); IMA Volumes in Mathematics and its Applications, vol. 72 (Springer, New York, 1995), pp. 15–41. MR 1380519

    Chapter  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  36. S.E. Fienberg, U.E. Makov, M.M. Meyer, R.J. Steele, Computing the Exact Distribution for a Multi-way Contingency Table Conditional on Its Marginal Totals. Data Analysis from Statistical Foundations (Nova Science Publishers, Huntington, 2001), pp. 145–165. MR 2034512

    Google Scholar 

  37. W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, vol. 131 (Princeton University Press, Princeton, 1993) (The William H Roever Lectures in Geometry). MR 1234037

    Google Scholar 

  38. P. Galashin, S. Hopkins, T. McConville, A. Postnikov, Root system chip-firing I: interval-firing. arXiv:1708.04850

  39. C. Haase, A. Paffenholz, L.C. Piechnik, F. Santos, Existence of unimodular triangulations—positive results, arXiv:1405.1687

  40. J. Haglund, A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants. Adv. Math. 227(5), 2092–2106 (2011). MR 2803796

    Article  MathSciNet  MATH  Google Scholar 

  41. G. Hegedüs, A. Higashitani, A. Kasprzyk, Ehrhart polynomial roots of reflexive polytopes, arXiv:1503.05739

  42. T. Hibi, Distributive lattices, affine semigroup rings and algebras with straightening laws. Commutative Algebra And Combinatorics (Kyoto, 1985). Advanced Studies in Pure Mathematics, vol. 11 (North-Holland, Amsterdam, 1987), pp. 93–109. MR 951198

    Google Scholar 

  43. T. Hibi, Some results on Ehrhart polynomials of convex polytopes. Discret. Math. 83(1), 119–121 (1990). MR 1065691

    Article  MathSciNet  MATH  Google Scholar 

  44. T. Hibi, A lower bound theorem for Ehrhart polynomials of convex polytopes. Adv. Math. 105(2), 162–165 (1994). MR 1275662

    Article  MathSciNet  MATH  Google Scholar 

  45. T. Hibi, Dual polytopes of rational convex polytopes. Combinatorica 12(2), 237–240 (1992). MR 1179260

    Article  MathSciNet  MATH  Google Scholar 

  46. T. Hibi, Star-shaped complexes and Ehrhart polynomials. Proc. Am. Math. Soc. 123(3), 723–726 (1995). MR 1249883

    Article  MathSciNet  MATH  Google Scholar 

  47. T. Hibi, A. Higashitani, A. Tsuchiya, K. Yoshida, Ehrhart polynomials with negative coefficients, arXiv:1506.00467

  48. Hibi, T. (1992). Algebraic Combinatorics of Convex Polytopes. Australia: Carslaw Publications.

    MATH  Google Scholar 

  49. S. Hopkins, A. Postnikov, A positive formula for the Ehrhart-like polynomials from root system chip-firing, arXiv:1803.08472

  50. J.M. Kantor, A. Khovanskii, Une application du théorème de Riemann-Roch combinatoire au polynôme d’Ehrhart des polytopes entiers de \({\bf R}^d\). C. R. Acad. Sci. Paris Sér. I Math. 317(5), 501–507 (1993). MR 1239038

    Google Scholar 

  51. R.C. King, C. Tollu, F. Toumazet, Stretched Littlewood-Richardson and Kostka coefficients, symmetry in physics, in CRM Proceedings and Lecture Notes, vol. 34 (American Mathematical Society, Providence, 2004), pp. 99–112. MR 2056979

    MATH  Google Scholar 

  52. A.N. Kirillov, in Ubiquity of Kostka Polynomials, ed. by A.N. Kirillov, A. Tsuchiya, H. Umemura, Physics and Combinatorics, Proceedings Nagoya 1999. World Scientific (2001), arXiv:math.QA/9912094

  53. A. Knutson, T. Tao, The honeycomb model of GLn(C) tensor products. I. Proof of the saturation conjecture. J. Am. Math. Soc. 12(4), 1055–1090 (1999). MR 1671451

    Google Scholar 

  54. A. Knutson, T. Tao, C. Woodward, The honeycomb model of \({\rm GL}_n(C)\) tensor products. II. Puzzles determine facets of the Littlewood-Richardson cone. J. Am. Math. Soc. 17(1), 19–48 (2004). MR 2015329

    Google Scholar 

  55. B.V. Lidskii, The Kostant function of the system of roots \(A_{n}\). Funktsional. Anal. i Prilozhen. 18(1), 76–77 (1984). MR 739099

    Google Scholar 

  56. F. Liu, A note on lattice-face polytopes and their Ehrhart polynomials. Proc. Am. Math. Soc. 137(10), 3247–3258 (2009). MR 2515395 (2010h:52018)

    Article  MathSciNet  MATH  Google Scholar 

  57. F. Liu, Ehrhart polynomials of cyclic polytopes. J. Comb. Theory Ser. A 111(1), 111–127 (2005). MR 2144858 (2006a:05012)

    Article  MathSciNet  MATH  Google Scholar 

  58. F. Liu, Ehrhart polynomials of lattice-face polytopes. Trans. Am. Math. Soc. 360(6), 3041–3069 (2008). MR 2379786 (2009a:52012)

    Article  MathSciNet  MATH  Google Scholar 

  59. F. Liu, Higher integrality conditions, volumes and Ehrhart polynomials. Adv. Math. 226(4), 3467–3494 (2011). MR 2764894 (2012a:52031)

    Article  MathSciNet  MATH  Google Scholar 

  60. F. Liu, L. Solus, On the relationship between Ehrhart unimodality and Ehrhart positivity. Annals of Combinatorics. arXiv:1804.08258

  61. F. Liu, A. Tsuchiya, Stanley’s non-Ehrhart-positive order polytopes, arXiv:1806.08403

  62. B. Lorenz, A. Paffenholz, Smooth reflexive polytopes up to dimension 9 (2016), https://polymake.org/polytopes/paffenholz/www/fano.html

  63. L. Lovász, M.D. Plummer, in Matching Theory. North-Holland Mathematics Studies, vol. 121 (North-Holland Publishing Co., Amsterdam; Akadémiai Kiadó) (Publishing House of the Hungarian Academy of Sciences), Budapest, 1986; Annals of Discrete Mathematics, 29). MR 859549

    Google Scholar 

  64. I.G. Macdonald, Polynomials associated with finite cell-complexes. J. Lond. Math. Soc. 4(2), 181–192 (1971). MR 0298542

    Article  MATH  Google Scholar 

  65. I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edn. Oxford Classic Texts in the Physical Sciences (The Clarendon Press, Oxford University Press, New York, 2015) (With contribution by A. V. Zelevinsky and a foreword by Richard Stanley, Reprint of the 2008 paperback edition [MR1354144]). MR 3443860

    Google Scholar 

  66. P. McMullen, The maximum numbers of faces of a convex polytope. Mathematika 17, 179–184 (1970). MR 0283691

    Article  MathSciNet  MATH  Google Scholar 

  67. P. McMullen, Valuations and Dissections. Handbook of Convex Geometry, vol. A, B (North-Holland, Amsterdam, 1993), pp. 933–988. MR 1243000 (95f:52018)

    Google Scholar 

  68. K. Mészáros, On, product formulas for volumes of flow polytopes. Combinatorial Methods in Topology and Algebra, Springer INdAM Series, vol. 12 (Springer, Cham, 2015), pp. 91–95. MR 3467331

    Google Scholar 

  69. K. Mészáros, On, product formulas for volumes of flow polytopes. Proc. Am. Math. Soc. 143(3), 937–954 (2015). MR 3293712

    Article  MathSciNet  MATH  Google Scholar 

  70. K. Mészáros, A.H. Morales, Volumes and Ehrhart polynomials of flow polytopes, arXiv:1710.00701

  71. K. Mészáros, A.H. Morales, B. Rhoades, The polytope of Tesler matrices. Selecta Math. (N.S.) 23(1), 425–454 (2017). MR 3595898

    Article  MathSciNet  MATH  Google Scholar 

  72. K. Mészáros, A.H. Morales, J. Striker, On flow polytopes, order polytopes, and certain faces of the alternating sign matrix polytope, arXiv:1510.03357

  73. A.H. Morales, Ehrhart polynomials of examples of flow polytopes, https://sites.google.com/site/flowpolytopes/ehrhart

  74. L. Moura, Polyhedral methods in design theory. Computational and Constructive Design Theory, Mathematics Applied, vol. 368 (Kluwer Academic Publishers, Dordrecht, 1996), pp. 227–254. MR 1398195

    Chapter  MATH  Google Scholar 

  75. A. Nijehuis, H. Wilf, Representations of integers by linear forms in nonnegative integers. J Number Theory 4, 98–106 (1972). MR 0288076

    Google Scholar 

  76. M. Øbro, An algorithm for the classification of smooth Fano polytopes, arXiv:0704.0049

  77. I. Pak, Four questions on Birkhoff polytope. Ann. Comb. 4(1), 83–90 (2000). MR 1763951

    Article  MathSciNet  MATH  Google Scholar 

  78. S. Payne, Ehrhart series and lattice triangulations. Discret. Comput. Geom. 40(3), 365–376 (2008). MR 2443289

    Article  MathSciNet  MATH  Google Scholar 

  79. J. Pitman, R.P. Stanley, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron. Discret. Comput. Geom. 27(4), 603–634 (2002). MR 1902680 (2003e:52017)

    Google Scholar 

  80. J.E. Pommersheim, Toric varieties, lattice points and Dedekind sums. Math. Ann. 295(1), 1–24 (1993). MR 1198839

    Google Scholar 

  81. J. Pommersheim, H. Thomas, Cycles representing the Todd class of a toric variety. J. Am. Math. Soc. 17(4), 983–994 (2004). MR 2083474

    Google Scholar 

  82. A. Postnikov, Permutohedra, associahedra, and beyond. Int. Math. Res. Not. IMRN, 6, 1026–1106 (2009). MR 2487491 (2010g:05399)

    Article  MathSciNet  MATH  Google Scholar 

  83. A. Postnikov, V. Reiner, L. Williams, Faces of generalized permutohedra. Doc. Math. 13, 207–273 (2008). MR 2520477 (2010j:05425)

    Google Scholar 

  84. A. Postnikov, R.P. Stanley, Acyclic flow polytopes and Kostant’s partition function. http://math.mit.edu/~rstan/transparencies/flow.ps

  85. A.V. Pukhlikov, A.G. Khovanskii, The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes. Algebra i Analiz 4(4), 188–216 (1992). MR 1190788

    Google Scholar 

  86. E. Rassart, A polynomiality property for Littlewood-Richardson coefficients. J. Comb. Theory Ser. A 107(2), 161–179 (2004). MR 2078884

    Article  MathSciNet  MATH  Google Scholar 

  87. M.H. Ring, A. Schürmann, Local formulas for Ehrhart coefficients from lattice tiles, arXiv:1709.10390

  88. Rodriguez-Villegas, F. R. (2002). On the zeros of certain polynomials. Proc. Am. Math. Soc., 130, 2251–2254.

    Article  MathSciNet  MATH  Google Scholar 

  89. J.R. Schmidt, A.M. Bincer, The Kostant partition function for simple Lie algebras. J. Math. Phys. 25(8), 2367–2373 (1984). MR 751517

    Article  MathSciNet  MATH  Google Scholar 

  90. P.R. Scott, On, convex lattice polygons. Bull. Aust. Math. Soc. 15(3), 395–399 (1976). MR 0430960

    Article  MathSciNet  MATH  Google Scholar 

  91. L. Solus, Simplices for numeral systems. Trans. Am. Math. Soc. arXiv:1706.00480 (to appear)

  92. R.P. Stanley, A zonotope associated with graphical degree sequences, in Applied Geometry and Discrete Mathematics. DIMACS Series Discrete Mathematics & Theoretical Computer Science, vol. 4 (American Mathematical Society, Providence, 1991), pp. 555–570. MR 1116376 (92k:52020)

    Google Scholar 

  93. R.P. Stanley, Decompositions of rational convex polytopes. Ann. Discret. Math. 6, 333–342 (1980). Combinatorial mathematics, optimal designs and their applications (Proceedings of the Symposium Combinatorial Mathematics and Optimal Design, Colorado State University, Fort Collins, Colorado, 1978). MR 593545

    Google Scholar 

  94. R.P. Stanley, Enumerative Combinatorics, vol. 1, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 49 (Cambridge University Press, Cambridge, 2012). MR 2868112

    Google Scholar 

  95. R.P. Stanley, Hilbert functions of graded algebras. Adv. Math. 28(1), 57–83 (1978). MR 0485835

    Article  MathSciNet  MATH  Google Scholar 

  96. R.P. Stanley, in Enumerative Combinatorics, vol. 2. Cambridge Studies in Advanced Mathematics, vol. 62 (Cambridge University Press, Cambridge, 1999) (With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin). MR 1676282

    Google Scholar 

  97. R.P. Stanley, Lectures on lattice points in polytopes (2010), http://math.mit.edu/~rstan/transparencies/ehrhart1.pdf (2010)

  98. R.P. Stanley, Log-concave and unimodal sequences in algebra, combinatorics, and geometry. Graph theory and its applications: East and West (Jinan, 1986). Ann. N Y Acad. Sci. 576, 500–535 (1989). MR 1110850

    Article  MathSciNet  MATH  Google Scholar 

  99. R.P. Stanley, On the Hilbert function of a graded Cohen-Macaulay domain. J. Pure Appl. Algebr. 73(3), 307–314 (1991). MR 1124790

    Article  MathSciNet  MATH  Google Scholar 

  100. R.P. Stanley, Positivity of Ehrhart polynomial coefficients. MathOverflow, https://mathoverflow.net/q/200574(version:2015-03-20)

  101. R.P. Stanley, Two enumerative results on cycles of permutations. Eur. J. Comb. 32(6), 937–943 (2011). MR 2821562

    Article  MathSciNet  MATH  Google Scholar 

  102. R.P. Stanley, Two poset polytopes. Discret. Comput. Geom. 1(1), 9–23 (1986). MR 824105

    Article  MathSciNet  MATH  Google Scholar 

  103. A. Stapledon, Inequalities and Ehrhart \(\delta \)-vectors. Trans. Am. Math. Soc. 361(10), 5615–5626 (2009). MR 2515826

    Google Scholar 

  104. R. Steinberg, A general Clebsch-Gordan theorem. Bull. Am. Math. Soc. 67, 406–407 (1961). MR 0126508

    Article  MathSciNet  MATH  Google Scholar 

  105. J. Treutlein, Lattice polytopes of degree 2. J. Comb. Theory Ser. A 117(3), 354–360 (2010). MR 2592905

    Article  MathSciNet  MATH  Google Scholar 

  106. V.A. Yemelichev, M.M. Kovalëv, M.K. Kravtsov, in Polytopes, Graphs and Optimisation (Cambridge University Press, Cambridge, 1984), Translated from the Russian by G. H. Lawden, MR 744197

    Google Scholar 

  107. D. Zeilberger, Proof of a conjecture of Chan, Robbins, and Yuen. Electron. Trans. Numer. Anal. 9, 147–148 (1999). Orthogonal polynomials: numerical and symbolic algorithms (Leganés, 1998). MR 1749805

    Google Scholar 

  108. G.M. Ziegler, Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152 (Springer, New York, 1995). MR 1311028

    Book  MATH  Google Scholar 

Download references

Acknowledgements

The author is partially supported by a grant from the Simons Foundation #426756. The writing was completed when the author was attending the program “Geometric and Topological Combinatorics” at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester, and she was partially supported by the NSF grant DMS-1440140.

The author would like to thank Gabriele Balletti, Ben Braun, Federico Castillo, Ron King, Akihiro Higashitani, Sam Hopkins, Thomas McConville, Karola Mészáros, Alejandro Morales, Benjamin Nill, Andreas Paffenholz, Alex Postnikov, Richard Stanley, Liam Solus, and Akiyoshi Tsuchiya for valuable discussions and helpful suggestions. The author is particularly grateful to Federico Castillo and Alejandro Morales for their help in putting together some figures and data used in this chapter. Finally, the author is thankful to the two anonymous referees for their careful reading of this chapter and various insightful comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, University of California, One Shields Avenue, Davis, CA, 95616, USA

    Fu Liu

Authors
  1. Fu Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fu Liu .

Editor information

Editors and Affiliations

  1. Mathematical Sciences Research Institute , Berkeley, CA, USA

    Hélène Barcelo

  2. Department of Mathematics, Pomona College, Claremont, CA, USA

    Gizem Karaali

  3. Department of Mathematics, Dartmouth College, Hanover, NH, USA

    Rosa Orellana

Rights and permissions

Reprints and permissions

Copyright information

© 2019 The Author(s) and the Association for Women in Mathematics

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Liu, F. (2019). On Positivity of Ehrhart Polynomials. In: Barcelo, H., Karaali, G., Orellana, R. (eds) Recent Trends in Algebraic Combinatorics. Association for Women in Mathematics Series, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-030-05141-9_6

Download citation

Publish with us

Policies and ethics

Search

Navigation