Advertisement

Publications mathématiques de l'IHÉS

, Volume 124, Issue 1, pp 99–163 | Cite as

Relative Stanley–Reisner theory and Upper Bound Theorems for Minkowski sums

  • Karim A. Adiprasito
  • Raman Sanyal
Article

Abstract

In this paper we settle two long-standing questions regarding the combinatorial complexity of Minkowski sums of polytopes: We give a tight upper bound for the number of faces of a Minkowski sum, including a characterization of the case of equality. We similarly give a (tight) upper bound theorem for mixed facets of Minkowski sums. This has a wide range of applications and generalizes the classical Upper Bound Theorems of McMullen and Stanley.

Our main observation is that within (relative) Stanley–Reisner theory, it is possible to encode topological as well as combinatorial/geometric restrictions in an algebraic setup. We illustrate the technology by providing several simplicial isoperimetric and reverse isoperimetric inequalities in addition to our treatment of Minkowski sums.

Keywords

Simplicial Complex Local Cohomology Simplicial Polytopes Local Cohomology Module Bound Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Adi15]
    K. A. Adiprasito, Toric chordality, preprint, arXiv:1503.06640.
  2. [AB12]
    K. A. Adiprasito and B. Benedetti, Subdivisions, shellability, and collapsibility of products, Combinatorica, February 2012, to appear, available at arXiv:1202.6606.
  3. [ABG83]
    K. A. Adiprasito, A. Björner and A. Goodarzi, Face numbers of sequentially Cohen–Macaulay complexes and Betti numbers of componentwise linear ideals, preprint, arXiv:1502.01183.
  4. [ABPS15]
    K. A. Adiprasito, P. Brinkmann, A. Padrol and R. Sanyal, Mixed faces and colorful depth, 2015, in preparation. Google Scholar
  5. [BKL86]
    D. Barnette, P. Kleinschmidt and C. W. Lee, An upper bound theorem for polytope pairs, Math. Oper. Res., 11 (1986), 451–464. MathSciNetCrossRefzbMATHGoogle Scholar
  6. [BL80]
    L. J. Billera and C. W. Lee, Sufficiency of McMullen’s conditions for \(f\)-vectors of simplicial polytopes, Bull. Am. Math. Soc., New Ser., 2 (1980), 181–185. MathSciNetCrossRefzbMATHGoogle Scholar
  7. [BL81]
    L. J. Billera and C. W. Lee, The numbers of faces of polytope pairs and unbounded polyhedra, Eur. J. Comb., 2 (1981), 307–322. MathSciNetCrossRefzbMATHGoogle Scholar
  8. [Bjö80]
    A. Björner, Shellable and Cohen-Macaulay partially ordered sets, Trans. Am. Math. Soc., 260 (1980), 159–183. MathSciNetCrossRefzbMATHGoogle Scholar
  9. [Bjö03]
    A. Björner, Nerves, fibers and homotopy groups, J. Comb. Theory, Ser. A, 102 (2003), 88–93. MathSciNetCrossRefzbMATHGoogle Scholar
  10. [Bjö07]
    A. Björner, A comparison theorem for \(f\)-vectors of simplicial polytopes, Pure Appl. Math. Q., 3 (2007), 347–356. MathSciNetCrossRefzbMATHGoogle Scholar
  11. [BWW05]
    A. Björner, M. L. Wachs and V. Welker, Poset fiber theorems, Trans. Am. Math. Soc., 357 (2005), 1877–1899. MathSciNetCrossRefzbMATHGoogle Scholar
  12. [Bor48]
    K. Borsuk, On the imbedding of systems of compacta in simplicial complexes, Fundam. Math., 35 (1948), 217–234. MathSciNetzbMATHGoogle Scholar
  13. [BM71]
    H. Bruggesser and P. Mani, Shellable decompositions of cells and spheres, Math. Scand., 29 (1971), 197–205, 1972. MathSciNetCrossRefzbMATHGoogle Scholar
  14. [BH93]
    W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. zbMATHGoogle Scholar
  15. [Buc43]
    R. C. Buck, Partition of space, Am. Math. Mon., 50 (1943), 541–544. MathSciNetCrossRefzbMATHGoogle Scholar
  16. [CD95]
    R. M. Charney and M. W. Davis, Strict hyperbolization, Topology, 34 (1995), 329–350. MathSciNetCrossRefzbMATHGoogle Scholar
  17. [CLS11]
    D. A. Cox, J. B. Little and H. K. Schenck, Toric Varieties, Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, 2011. zbMATHGoogle Scholar
  18. [Dav08]
    M. W. Davis, The Geometry and Topology of Coxeter Groups, London Mathematical Society Monographs Series, vol. 32, Princeton University Press, Princeton, 2008. zbMATHGoogle Scholar
  19. [dLRS10]
    J. A. de Loera, J. Rambau and F. Santos, Triangulations, Algorithms and Computation in Mathematics, vol. 25, Springer, Berlin, 2010, Structures for algorithms and applications. zbMATHGoogle Scholar
  20. [Duv96]
    A. M. Duval, Algebraic shifting and sequentially Cohen-Macaulay simplicial complexes, Electron. J. Comb., 3 (1996), 14, research paper r21. MathSciNetzbMATHGoogle Scholar
  21. [Eis95]
    D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150, Springer, New York, 1995. zbMATHGoogle Scholar
  22. [EC95]
    I. Z. Emiris and J. F. Canny, Efficient incremental algorithms for the sparse resultant and the mixed volume, J. Symb. Comput., 20 (1995), 117–149. MathSciNetCrossRefzbMATHGoogle Scholar
  23. [FW07]
    K. Fukuda and C. Weibel, \(f\)-Vectors of Minkowski additions of convex polytopes, Discrete Comput. Geom., 37 (2007), 503–516. MathSciNetCrossRefzbMATHGoogle Scholar
  24. [FW10]
    K. Fukuda and C. Weibel, A linear equation for Minkowski sums of polytopes relatively in general position, Eur. J. Comb., 31 (2010), 565–573. MathSciNetCrossRefzbMATHGoogle Scholar
  25. [FS97]
    W. Fulton and B. Sturmfels, Intersection theory on toric varieties, Topology, 36 (1997), 335–353. MathSciNetCrossRefzbMATHGoogle Scholar
  26. [Geo08]
    R. Geoghegan, Topological Methods in Group Theory, Graduate Texts in Mathematics, vol. 243, Springer, New York, 2008. zbMATHGoogle Scholar
  27. [Grä87]
    H.-G. Gräbe, Generalized Dehn-Sommerville equations and an upper bound theorem, Beitr. Algebra Geom., 25 (1987), 47–60. MathSciNetzbMATHGoogle Scholar
  28. [GS93]
    P. Gritzmann and B. Sturmfels, Minkowski addition of polytopes: computational complexity and applications to Gröbner bases, SIAM J. Discrete Math., 6 (1993), 246–269. MathSciNetCrossRefzbMATHGoogle Scholar
  29. [Hib91]
    T. Hibi, Quotient algebras of Stanley-Reisner rings and local cohomology, J. Algebra, 140 (1991), 336–343. MathSciNetCrossRefzbMATHGoogle Scholar
  30. [Hoc77]
    M. Hochster, Cohen-Macaulay rings, combinatorics, and simplicial complexes, in Ring Theory, II, Proc. Second Conf., Univ. Oklahoma, Norman, Okla, 1975, Lecture Notes in Pure and Appl. Math., vol. 26, pp. 171–223, Dekker, New York, 1977. Google Scholar
  31. [HR74]
    M. Hochster and J. L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Adv. Math., 13 (1974), 115–175. MathSciNetCrossRefzbMATHGoogle Scholar
  32. [Hov78]
    A. G. Hovanskiĭ, Newton polyhedra, and the genus of complete intersections, Funkc. Anal. Prilozh., 12 (1978), 51–61. MathSciNetGoogle Scholar
  33. [ILL+07]
    S. B. Iyengar, G. J. Leuschke, A. Leykin, C. Miller, E. Miller, A. K. Singh and U. Walther, Twenty-Four Hours of Local Cohomology, Graduate Studies in Mathematics, vol. 87, American Mathematical Society, Providence, 2007. zbMATHGoogle Scholar
  34. [JMR83]
    W. Julian, R. Mines and F. Richman, Alexander duality, Pac. J. Math., 106 (1983), 115–127. MathSciNetCrossRefzbMATHGoogle Scholar
  35. [Kal91]
    G. Kalai, The diameter of graphs of convex polytopes and \(f\)-vector theory, in Applied Geometry and Discrete Mathematics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, pp. 387–411, Amer. Math. Soc., Providence, 1991. Google Scholar
  36. [KKT15]
    M. I. Karavelas, C. Konaxis and E. Tzanaki, The maximum number of faces of the Minkowski sum of three convex polytopes, J. Comput. Geom., 6 (2015), 21–74. MathSciNetzbMATHGoogle Scholar
  37. [KT11]
    M. I. Karavelas and E. Tzanaki, The maximum number of faces of the Minkowski sum of two convex polytopes, preprint, (2011), Discrete Comput. Geom., to appear, available at doi: 10.1007/s00454-015-9726-6.
  38. [KT15]
    M. I. Karavelas and E. Tzanaki, A geometric approach for the upper bound theorem for Minkowski sums of convex polytopes, in 31st International Symposium on Computational Geometry, LIPIcs, Leibniz Int. Proc. Inform., vol. 34, pp. 81–95, Schloss Dagstuhl. Leibniz-Zent. Inform, Wadern, 2015. Google Scholar
  39. [Kat12]
    E. Katz, Tropical intersection theory from toric varieties, Collect. Math., 63 (2012), 29–44. MathSciNetCrossRefzbMATHGoogle Scholar
  40. [KK79]
    B. Kind and P. Kleinschmidt, Schälbare Cohen-Macauley-Komplexe und ihre Parametrisierung, Math. Z., 167 (1979), 173–179. MathSciNetCrossRefzbMATHGoogle Scholar
  41. [Kle64]
    V. Klee, On the number of vertices of a convex polytope, Can. J. Math., 16 (1964), 701–720. MathSciNetCrossRefzbMATHGoogle Scholar
  42. [Lat91]
    J-C. Latombe, Robot Motion Planning, vol. 25, Kluwer Academic Publishers, Boston, 1991. CrossRefzbMATHGoogle Scholar
  43. [MPP11]
    B. Matschke, J. Pfeifle and V. Pilaud, Prodsimplicial-neighborly polytopes, Discrete Comput. Geom., 46 (2011), 100–131. MathSciNetCrossRefzbMATHGoogle Scholar
  44. [McM70]
    P. McMullen, The maximum numbers of faces of a convex polytope, Mathematika, 17 (1970), 179–184. MathSciNetCrossRefzbMATHGoogle Scholar
  45. [McM04]
    P. McMullen, Triangulations of simplicial polytopes, Beitr. Algebra Geom., 45 (2004), 37–46. MathSciNetzbMATHGoogle Scholar
  46. [MW71]
    P. McMullen and D. W. Walkup, A generalized lower-bound conjecture for simplicial polytopes, Mathematika, 18 (1971), 264–273. MathSciNetCrossRefzbMATHGoogle Scholar
  47. [MNS11]
    E. Miller, I. Novik and E. Swartz, Face rings of simplicial complexes with singularities, Math. Ann., 351 (2011), 857–875. MathSciNetCrossRefzbMATHGoogle Scholar
  48. [MS05]
    E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227, Springer, New York, 2005. zbMATHGoogle Scholar
  49. [Min11]
    H. Minkowski, Theorie der konvexen körper, insbesondere Begründung ihres Oberflächenbegriffs, Gesammelte Abh. Hermann Minkowski, 2 (1911), 131–229. Google Scholar
  50. [Miy89]
    M. Miyazaki, Characterizations of Buchsbaum complexes, Manuscr. Math., 63 (1989), 245–254. MathSciNetCrossRefzbMATHGoogle Scholar
  51. [Mot57]
    T. S. Motzkin, Comonotone curves and polyhedra, Bull. Am. Math. Soc., 63 (1957), 35. MathSciNetCrossRefGoogle Scholar
  52. [MRTT53]
    T. S. Motzkin, H. Raiffa, G. L. Thompson and R. M. Thrall, The double description method, in Contributions to the Theory of Games, Annals of Mathematics Studies, vol. 28, pp. 51–73, Princeton University Press, Princeton, 1953. Google Scholar
  53. [Nov03]
    I. Novik, Remarks on the upper bound theorem, J. Comb. Theory, Ser. A, 104 (2003), 201–206. MathSciNetCrossRefzbMATHGoogle Scholar
  54. [Nov05]
    I. Novik, On face numbers of manifolds with symmetry, Adv. Math., 192 (2005), 183–208. MathSciNetCrossRefzbMATHGoogle Scholar
  55. [NS09]
    I. Novik and E. Swartz, Applications of Klee’s Dehn–Sommerville relations, Discrete Comput. Geom., 42 (2009), 261–276. MathSciNetCrossRefzbMATHGoogle Scholar
  56. [NS12]
    I. Novik and E. Swartz, Face numbers of pseudomanifolds with isolated singularities, Math. Scand., 110 (2012), 198–222. MathSciNetCrossRefzbMATHGoogle Scholar
  57. [PS93]
    P. Pedersen and B. Sturmfels, Product formulas for resultants and Chow forms, Math. Z., 214 (1993), 377–396. MathSciNetCrossRefzbMATHGoogle Scholar
  58. [Rei76]
    G. A. Reisner, Cohen-Macaulay quotients of polynomial rings, Adv. Math., 21 (1976), 30–49. MathSciNetCrossRefzbMATHGoogle Scholar
  59. [RS72]
    C. P. Rourke and B. J. Sanderson, Introduction to Piecewise-Linear Topology, Ergebnisse Series, vol. 69, Springer, New York, 1972. CrossRefzbMATHGoogle Scholar
  60. [San09]
    R. Sanyal, Topological obstructions for vertex numbers of Minkowski sums, J. Comb. Theory, Ser. A, 116 (2009), 168–179. MathSciNetCrossRefzbMATHGoogle Scholar
  61. [Sch81]
    P. Schenzel, On the number of faces of simplicial complexes and the purity of Frobenius, Math. Z., 178 (1981), 125–142. MathSciNetCrossRefzbMATHGoogle Scholar
  62. [Sch82]
    P. Schenzel, Dualisierende Komplexe in der lokalen Algebra und Buchsbaum-Ringe, Lecture Notes in Mathematics, vol. 907, Springer, Berlin, 1982, with an English summary. CrossRefzbMATHGoogle Scholar
  63. [Sch93]
    R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. CrossRefzbMATHGoogle Scholar
  64. [Sta75]
    R. P. Stanley, The upper bound conjecture and Cohen-Macaulay rings, Stud. Appl. Math., 54 (1975), 135–142. MathSciNetCrossRefzbMATHGoogle Scholar
  65. [Sta87]
    R. P. Stanley, Generalized \(H\)-vectors, intersection cohomology of toric varieties, and related results, in Commutative Algebra and Combinatorics, Adv. Stud. Pure Math., vol. 11, Kyoto, 1985, pp. 187–213, North-Holland, Amsterdam, 1987. Google Scholar
  66. [Sta93]
    R. P. Stanley, A monotonicity property of \(h\)-vectors and \(h^{*}\)-vectors, Eur. J. Comb., 14 (1993), 251–258. MathSciNetCrossRefzbMATHGoogle Scholar
  67. [Sta96]
    R. P. Stanley, Combinatorics and Commutative Algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Boston, 1996. zbMATHGoogle Scholar
  68. [ST10]
    R. Steffens and T. Theobald, Combinatorics and genus of tropical intersections and Ehrhart theory, SIAM J. Discrete Math., 24 (2010), 17–32. MathSciNetCrossRefzbMATHGoogle Scholar
  69. [Ste26]
    J. Steiner, Einige Gesetze über die Theilung der Ebene und des Raumes, J. Reine Angew. Math., 1 (1826), 349–364. MathSciNetCrossRefGoogle Scholar
  70. [Stu02]
    B. Sturmfels, Solving Systems of Polynomial Equations, CBMS Regional Conference Series in Mathematics, vol. 97, Conference Board of the Mathematical Sciences, Washington, 2002. zbMATHGoogle Scholar
  71. [Swa05]
    E. Swartz, Lower bounds for \(h\)-vectors of \(k\)-CM, independence, and broken circuit complexes, SIAM J. Discrete Math., 18 (2005), 647–661. MathSciNetCrossRefzbMATHGoogle Scholar
  72. [Wei12]
    C. Weibel, Maximal \(f\)-vectors of Minkowski sums of large numbers of polytopes, Discrete Comput. Geom., 47 (2012), 519–537. MathSciNetCrossRefzbMATHGoogle Scholar
  73. [Zee66]
    E. C. Zeeman, Seminar on Combinatorial Topology, Institut des Hautes Etudes Scientifiques, Paris, 1966. zbMATHGoogle Scholar
  74. [Zie95]
    G. M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, Springer, New York, 1995, Revised edition, 1998; seventh updated printing 2007. zbMATHGoogle Scholar

Copyright information

© IHES and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Einstein Institute for MathematicsHebrew University of JerusalemJerusalemIsrael
  2. 2.Fachbereich Mathematik und InformatikFreie Universität BerlinBerlinGermany

Personalised recommendations