Advertisement

On Positivity of Ehrhart Polynomials

Chapter
Part of the Association for Women in Mathematics Series book 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.

Notes

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.

References

  1. 1.
    F. Ardila, C. Benedetti, J. Doker, Matroid polytopes and their volumes. Discret. Comput. Geom. 43(4), 841–854 (2010). MR 2610473MathSciNetzbMATHGoogle Scholar
  2. 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. 3.
    W. Baldoni, M. Vergne, Kostant partitions functions and flow polytopes. Transform. Groups 13(3–4), 447–469 (2008). MR 2452600MathSciNetzbMATHGoogle Scholar
  4. 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. 5.
    G. Balletti, A. Higashitani, Universal inequalities in Ehrhart theory. arXiv:1703.09600
  6. 6.
    G. Balletti, A.M. Kasprzyk, Three-dimensional lattice polytopes with two interior lattice points. arXiv:1612.08918
  7. 7.
    M. Beck, Counting lattice points by means of the residue theorem. Ramanujan J. 4(3), 299–310 (2000). MR 1797548Google Scholar
  8. 8.
    M. Beck, D. Pixton, The Ehrhart polynomial of the Birkhoff polytope. Discret. Comput. Geom. 30(4), 623–637 (2003). MR (2013976)MathSciNetzbMATHGoogle Scholar
  9. 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 2134759Google Scholar
  10. 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 3410115zbMATHGoogle Scholar
  11. 11.
    N. Berline, M. Vergne, Local Euler-Maclaurin formula for polytopes. Mosc. Math. J. 7(3), 355–386, 573 (2007). MR 2343137MathSciNetzbMATHGoogle Scholar
  12. 12.
    C. Bey, M. Henk, J. Wills, Notes on the roots of Ehrhart polynomials. Discret. Comput. Geom. 38(1), 81–98 (2007). MR 2322117MathSciNetzbMATHGoogle Scholar
  13. 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 3409348Google Scholar
  14. 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 3526428zbMATHGoogle Scholar
  15. 15.
    B. Braun, R. Davis, L. Solus, Detecting the integer decomposition property and Ehrhart unimodality in reflexive simplices, arXiv:1608.01614
  16. 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 1310575Google Scholar
  17. 17.
    M. Brion, M. Vergne, An, equivariant Riemann-Roch theorem for complete, simplicial toric varieties. J. Reine Angew. Math. 482, 67–92 (1997). MR 1427657Google Scholar
  18. 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. 19.
    W. Bruns, J. Gubeladze, N.V. Trung, Normal polytopes, triangulations, and Koszul algebras. J. Reine Angew. Math. 485, 123–160 (1997), MR 1442191Google Scholar
  20. 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. 21.
    E.R. Canfield, B.D. McKay, The asymptotic volume of the Birkhoff polytope. Online J. Anal. Comb. 4, 4pp (2009). MR 2575172Google Scholar
  22. 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 1217352zbMATHGoogle Scholar
  23. 23.
    F. Castillo, Local Ehrhart Positivity, 91, Thesis (Ph.D.)-University of. California, Davis (2017) MR 3731970Google Scholar
  24. 24.
    F. Castillo, F. Liu, Berline-Vergne valuation and generalized permutohedra. Discret. Comput. Geom. 60(4), 885–908 (2018)MathSciNetzbMATHGoogle Scholar
  25. 25.
    F. Castillo, F. Liu, BV-\(\alpha \)-positivity for generalized permutohedra. arXiv:1509.07884v1
  26. 26.
    F. Castillo, F. Liu, B. Nill, A. Paffenholz, Smooth polytopes with negative Ehrhart coeffcients. J. Comb. Theory Ser. A. 160, 316–331(2018)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Castillo, F., & Liu, F. (2015). Ehrhart positivity for generalized permutohedra. Discret. Math Theor. Comput. Sci. proc. FPSAC, ’15, 865–876.zbMATHGoogle Scholar
  28. 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 1758803MathSciNetzbMATHGoogle Scholar
  29. 29.
    H. Conrads, Weighted, projective spaces and reflexive simplices. Manuscripta Math. 107(2), pp. 215–227 (2002). MR 1894741MathSciNetzbMATHGoogle Scholar
  30. 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. 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 2556462Google Scholar
  32. 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. 33.
    H. Derksen, J. Weyman, On the Littlewood-Richardson polynomials. J. Algebr. 255(2), 247–257 (2002). MR 1935497MathSciNetzbMATHGoogle Scholar
  34. 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 1380519Google Scholar
  35. 35.
    Ehrhart, E. (1962). Sur les polyèdres rationnels homothétiques à \(n\) dimensions. C. R. Acad. Sci. Paris, 254, 616–618.MathSciNetzbMATHGoogle Scholar
  36. 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 2034512Google Scholar
  37. 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 1234037Google Scholar
  38. 38.
    P. Galashin, S. Hopkins, T. McConville, A. Postnikov, Root system chip-firing I: interval-firing. arXiv:1708.04850
  39. 39.
    C. Haase, A. Paffenholz, L.C. Piechnik, F. Santos, Existence of unimodular triangulations—positive results, arXiv:1405.1687
  40. 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 2803796MathSciNetzbMATHGoogle Scholar
  41. 41.
    G. Hegedüs, A. Higashitani, A. Kasprzyk, Ehrhart polynomial roots of reflexive polytopes, arXiv:1503.05739
  42. 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 951198Google Scholar
  43. 43.
    T. Hibi, Some results on Ehrhart polynomials of convex polytopes. Discret. Math. 83(1), 119–121 (1990). MR 1065691MathSciNetzbMATHGoogle Scholar
  44. 44.
    T. Hibi, A lower bound theorem for Ehrhart polynomials of convex polytopes. Adv. Math. 105(2), 162–165 (1994). MR 1275662MathSciNetzbMATHGoogle Scholar
  45. 45.
    T. Hibi, Dual polytopes of rational convex polytopes. Combinatorica 12(2), 237–240 (1992). MR 1179260MathSciNetzbMATHGoogle Scholar
  46. 46.
    T. Hibi, Star-shaped complexes and Ehrhart polynomials. Proc. Am. Math. Soc. 123(3), 723–726 (1995). MR 1249883MathSciNetzbMATHGoogle Scholar
  47. 47.
    T. Hibi, A. Higashitani, A. Tsuchiya, K. Yoshida, Ehrhart polynomials with negative coefficients, arXiv:1506.00467
  48. 48.
    Hibi, T. (1992). Algebraic Combinatorics of Convex Polytopes. Australia: Carslaw Publications.zbMATHGoogle Scholar
  49. 49.
    S. Hopkins, A. Postnikov, A positive formula for the Ehrhart-like polynomials from root system chip-firing, arXiv:1803.08472
  50. 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 1239038Google Scholar
  51. 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 2056979zbMATHGoogle Scholar
  52. 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. 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 1671451Google Scholar
  54. 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 2015329Google Scholar
  55. 55.
    B.V. Lidskii, The Kostant function of the system of roots \(A_{n}\). Funktsional. Anal. i Prilozhen. 18(1), 76–77 (1984). MR 739099Google Scholar
  56. 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)MathSciNetzbMATHGoogle Scholar
  57. 57.
    F. Liu, Ehrhart polynomials of cyclic polytopes. J. Comb. Theory Ser. A 111(1), 111–127 (2005). MR 2144858 (2006a:05012)MathSciNetzbMATHGoogle Scholar
  58. 58.
    F. Liu, Ehrhart polynomials of lattice-face polytopes. Trans. Am. Math. Soc. 360(6), 3041–3069 (2008). MR 2379786 (2009a:52012)MathSciNetzbMATHGoogle Scholar
  59. 59.
    F. Liu, Higher integrality conditions, volumes and Ehrhart polynomials. Adv. Math. 226(4), 3467–3494 (2011). MR 2764894 (2012a:52031)MathSciNetzbMATHGoogle Scholar
  60. 60.
    F. Liu, L. Solus, On the relationship between Ehrhart unimodality and Ehrhart positivity. Annals of Combinatorics. arXiv:1804.08258
  61. 61.
    F. Liu, A. Tsuchiya, Stanley’s non-Ehrhart-positive order polytopes, arXiv:1806.08403
  62. 62.
    B. Lorenz, A. Paffenholz, Smooth reflexive polytopes up to dimension 9 (2016), https://polymake.org/polytopes/paffenholz/www/fano.html
  63. 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 859549Google Scholar
  64. 64.
    I.G. Macdonald, Polynomials associated with finite cell-complexes. J. Lond. Math. Soc. 4(2), 181–192 (1971). MR 0298542zbMATHGoogle Scholar
  65. 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 3443860Google Scholar
  66. 66.
    P. McMullen, The maximum numbers of faces of a convex polytope. Mathematika 17, 179–184 (1970). MR 0283691MathSciNetzbMATHGoogle Scholar
  67. 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. 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 3467331Google Scholar
  69. 69.
    K. Mészáros, On, product formulas for volumes of flow polytopes. Proc. Am. Math. Soc. 143(3), 937–954 (2015). MR 3293712MathSciNetzbMATHGoogle Scholar
  70. 70.
    K. Mészáros, A.H. Morales, Volumes and Ehrhart polynomials of flow polytopes, arXiv:1710.00701
  71. 71.
    K. Mészáros, A.H. Morales, B. Rhoades, The polytope of Tesler matrices. Selecta Math. (N.S.) 23(1), 425–454 (2017). MR 3595898MathSciNetzbMATHGoogle Scholar
  72. 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. 73.
    A.H. Morales, Ehrhart polynomials of examples of flow polytopes, https://sites.google.com/site/flowpolytopes/ehrhart
  74. 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 1398195zbMATHGoogle Scholar
  75. 75.
    A. Nijehuis, H. Wilf, Representations of integers by linear forms in nonnegative integers. J Number Theory 4, 98–106 (1972). MR 0288076Google Scholar
  76. 76.
    M. Øbro, An algorithm for the classification of smooth Fano polytopes, arXiv:0704.0049
  77. 77.
    I. Pak, Four questions on Birkhoff polytope. Ann. Comb. 4(1), 83–90 (2000). MR 1763951MathSciNetzbMATHGoogle Scholar
  78. 78.
    S. Payne, Ehrhart series and lattice triangulations. Discret. Comput. Geom. 40(3), 365–376 (2008). MR 2443289MathSciNetzbMATHGoogle Scholar
  79. 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. 80.
    J.E. Pommersheim, Toric varieties, lattice points and Dedekind sums. Math. Ann. 295(1), 1–24 (1993). MR 1198839Google Scholar
  81. 81.
    J. Pommersheim, H. Thomas, Cycles representing the Todd class of a toric variety. J. Am. Math. Soc. 17(4), 983–994 (2004). MR 2083474Google Scholar
  82. 82.
    A. Postnikov, Permutohedra, associahedra, and beyond. Int. Math. Res. Not. IMRN, 6, 1026–1106 (2009). MR 2487491 (2010g:05399)MathSciNetzbMATHGoogle Scholar
  83. 83.
    A. Postnikov, V. Reiner, L. Williams, Faces of generalized permutohedra. Doc. Math. 13, 207–273 (2008). MR 2520477 (2010j:05425)Google Scholar
  84. 84.
    A. Postnikov, R.P. Stanley, Acyclic flow polytopes and Kostant’s partition function. http://math.mit.edu/~rstan/transparencies/flow.ps
  85. 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 1190788Google Scholar
  86. 86.
    E. Rassart, A polynomiality property for Littlewood-Richardson coefficients. J. Comb. Theory Ser. A 107(2), 161–179 (2004). MR 2078884MathSciNetzbMATHGoogle Scholar
  87. 87.
    M.H. Ring, A. Schürmann, Local formulas for Ehrhart coefficients from lattice tiles, arXiv:1709.10390
  88. 88.
    Rodriguez-Villegas, F. R. (2002). On the zeros of certain polynomials. Proc. Am. Math. Soc., 130, 2251–2254.MathSciNetzbMATHGoogle Scholar
  89. 89.
    J.R. Schmidt, A.M. Bincer, The Kostant partition function for simple Lie algebras. J. Math. Phys. 25(8), 2367–2373 (1984). MR 751517MathSciNetzbMATHGoogle Scholar
  90. 90.
    P.R. Scott, On, convex lattice polygons. Bull. Aust. Math. Soc. 15(3), 395–399 (1976). MR 0430960MathSciNetzbMATHGoogle Scholar
  91. 91.
    L. Solus, Simplices for numeral systems. Trans. Am. Math. Soc. arXiv:1706.00480 (to appear)
  92. 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. 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 593545Google Scholar
  94. 94.
    R.P. Stanley, Enumerative Combinatorics, vol. 1, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 49 (Cambridge University Press, Cambridge, 2012). MR 2868112Google Scholar
  95. 95.
    R.P. Stanley, Hilbert functions of graded algebras. Adv. Math. 28(1), 57–83 (1978). MR 0485835MathSciNetzbMATHGoogle Scholar
  96. 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 1676282Google Scholar
  97. 97.
    R.P. Stanley, Lectures on lattice points in polytopes (2010), http://math.mit.edu/~rstan/transparencies/ehrhart1.pdf (2010)
  98. 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 1110850MathSciNetzbMATHGoogle Scholar
  99. 99.
    R.P. Stanley, On the Hilbert function of a graded Cohen-Macaulay domain. J. Pure Appl. Algebr. 73(3), 307–314 (1991). MR 1124790MathSciNetzbMATHGoogle Scholar
  100. 100.
    R.P. Stanley, Positivity of Ehrhart polynomial coefficients. MathOverflow, https://mathoverflow.net/q/200574(version:2015-03-20)
  101. 101.
    R.P. Stanley, Two enumerative results on cycles of permutations. Eur. J. Comb. 32(6), 937–943 (2011). MR 2821562MathSciNetzbMATHGoogle Scholar
  102. 102.
    R.P. Stanley, Two poset polytopes. Discret. Comput. Geom. 1(1), 9–23 (1986). MR 824105MathSciNetzbMATHGoogle Scholar
  103. 103.
    A. Stapledon, Inequalities and Ehrhart \(\delta \)-vectors. Trans. Am. Math. Soc. 361(10), 5615–5626 (2009). MR 2515826Google Scholar
  104. 104.
    R. Steinberg, A general Clebsch-Gordan theorem. Bull. Am. Math. Soc. 67, 406–407 (1961). MR 0126508MathSciNetzbMATHGoogle Scholar
  105. 105.
    J. Treutlein, Lattice polytopes of degree 2. J. Comb. Theory Ser. A 117(3), 354–360 (2010). MR 2592905MathSciNetzbMATHGoogle Scholar
  106. 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 744197Google Scholar
  107. 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 1749805Google Scholar
  108. 108.
    G.M. Ziegler, Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152 (Springer, New York, 1995). MR 1311028zbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaDavisUSA

Personalised recommendations