Abstract
An expository account is presented describing the use of methods of commutative algebra to solve problems concerning the enumeration of faces of convex polytopes. Assuming only basic knowledge of vector spaces and polynomial rings, the enumeration theory of Stanley is developed to the point where one can see how the Upper Bound Theorem for spheres is proved. A briefer account is then given of the extension of these techniques which yielded the proof of the necessity of McMullen’s conjectured characterization of the f-vectors of convex polytopes. The latter account includes a glimpse of the application of these methods to the study of integer solutions to systems of linear inequalities.
Prepared for the XI International Symposium on Mathematical Programming, Bonn, August 23–27, 1982. Supported in part by the National Science Foundation under Grant MCS81-02353.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Bibliography
M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison- Wesley, Reading, Massachusetts, 1969.
D. W. Barnette, A proof of the lower bound conjecture for convex polytopes, Pacific J. Math. 46 (1973), 349–354.
L. J. Billera and C. W. Lee, Sufficiency of McMullen’s conditions for f-vectors of simplicial polytopes, Bull. Amer. Math. Soc. (New Series) 2 (1980), 181–185.
L. J. Billera and C. W. Lee, A proof of the sufficiency of McMullen’s conditions for f-vectors of simplicial convex polytopes, Jour. Combinatorial Theory (A) 31 (1981), 237–255.
L. J. Billera and C. W. Lee, The numbers of faces of polytope pairs and unbounded polyhedra, Europ. J. Combin. 2 (1981), 307–332.
A. Björner, The minimum number of faces of a simple polyhedron, Europ. J. Combin. 1 (1980), 27–31.
H. Bruggesser and P. Mani, Shellable decompositions of cells and spheres, Math. Scand. 29 (1971), 197–205.
G. Clements and B. Lindström, A generalization of a combinatorial theorem of Macauly, J. Combinatorial Theory 1 (1969), 230–238.
G. Danaraj and V. Klee, Which spheres are shellable?, Ann. Discrete Math. 2 (1978), 32–52.
V. I. Danilov, The geometry of toric varieties, Russian Math. Surveys 33:2 (1978), 97–154; translated from Uspekhi Mat. Nauk. 33: 2 (1978), 85–134.
B. GrĂĽnbaum, Convex Polytopes, Wiley Interscience, New York, 1967.
P. J. Hilton and S. Wylie, Homology Theory, Cambridge Univ. Press, 1960.
M. Höchster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Annals of Math. 96 (1972), 318–337.
M. Höchster, Cohen-Macaulay rings, combinatorics, and simplicial complexes, in Ring Theory II (Proc. Second Oklahoma Conference), B. R. McDonald and R. Morris, eds., Marcel Dekker, New York and Basel, 1977, 171–223.
I. Kaplansky, Commutative Rings, rev. ed., Univ. of Chicago Press, 1974.
V. Klee, A combinatorial analogue of Poincare’s duality theorem, Can. Jour. Math. 16 (1964), 517–531.
V. Klee, Polytope pairs and their relationship to linear programming, Acta Math. 133 (1974), 1–25.
C. W. Lee, Bounding the numbers of faces of polytope pairs and simple polyhetra, Ann. Discrete Math, (to appear).
C. W. Lee, Two combinatorial properties of a class of simplicial polytopes, Research Report RC 8957, IBM, Yorktown Heights, New York, 1981.
C. W. Lee, Characterizing the numbers of faces of a simplicial convex polytope, in Convexity and Related Combinatorial Geometry, D. C. Kay and M. Breen, eds., Marcel Dekker, New York and Basel, 1982.
F. S. Macaulay, The Algebraic Theory of Modular Systems, Cambridge Tracts in Mathematics and Mathematical Physics, No. 19, Cambridge Univ. Press, London, 1916.
F. S. Macaulay, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927), 531–555.
P. McMullen, The maximum numbers of faces of a convex polytope, Mathematika 17 (1970), 179–184.
P. McMullen, The minimum number of faces of a convex polytope, J. London Math. Soc. (2) 3 (1971), 350–354.
P. McMullen, The numbers of faces of simplicial polytopes, Israel J. Math. 9 (1971), 559–570.
P. McMullen and G. C. Shephard, Convex Polytopes and the Upper Bound Conjecture, London Math. Soc. Lecture Notes Series, Vol. 3, 1971.
P. McMullen and D. W. Walkup, A generalized lower-bound conjecture for simplicial polytopes, Mathematika 18 (1971), 264–273.
T. S. Motzkin, Comonotone curves and polyhedra, Abstract 111, Bull. Amer. Math. Soc. 63 (1957), p. 35.
L. S. Pontryagin, Foundations of Combinatorial Topology, Graylock Press, Rochester, N.Y., 1952.
G. Reisner, Cohen-Macaulay quotients of polynomial rings, Advances in Math. 21 (1976), 30–49.
P. Schenzel, On the number of faces of simplicial complexes and the purity of Frobenius, Math. Z. 178 (1981), 125–142.
W. Smoke, Dimension and multiplicity for graded algebras, J. Algebra 21 (1972), 149–173.
D. M. Y. Sommerville, The relations connecting the angle-sums and volume of a polytope in space of n dimensions, Proc. Roy. Soc. London, ser. A 115 (1927), 103– 119.
R. Stanley, Linear homogeneous diophantine equations and magic labelings of graphs, Duke Math. J. 40 (1973), 607–632.
R. Stanley, Combinatorial reciprocity theorems, Advances in Math. 14 (1974), 194–253.
R. Stanley, Cohen-Macaulay rings and constructible polytopes, Bull. Am. Math. Soc. 81 (1975) 133–135.
R. Stanley, The upper bound conjecture and Cohen-Macaulay rings, Studies in Applied Math. 54 (1975), 135–142.
R. Stanley, Magic labelings of graphs, symmetric magic squares, systems of parameters, and Cohen-Macaulay rings, Duke Math. J. 43 (1976), 511–531.
R. Stanley, Cohen-Macaulay complexes, in Higher Combinatorics, M. Aiger, ed., Reidel, Dordrecht and Boston, 1977, 51–62.
R. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), 57–83.
R. Stanley, Combinatorics and invariant theory, in Relations between Combinatorics and Other Parts of Mathematics, D. K. Ray-Chaudhuri, ed., Proceedings of Symposia in Pure Mathematics, Vol. XXXIV, American Mathematical Society, Providence, 1979, 345–355.
R. Stanley, Decompositions of rational convex polytopes, Annals of Discrete Math. 6 (1980), 333–342.
R. Stanley, The number of faces of a simplicial convex polytope, Advances in Math. 35 (1980), 236–238.
R. Stanley, Interactions between commutative algebra and combinatorics, Report No. 4, Mathematiska Institutionen, Stockholms Universitet, Sweden, 1982.
R. Stanley, Linear diophantine equations and local cohomology, Inv. Math. 68 (1982), 175–193.
R. Stanley, An introduction to combinatorial algebra, Proceedings of the Silver Jubilee Conference on Combinatorics, University of Waterloo (to appear).
B. Tessier, Variétés toriques et polytopes, Séminaire Bourbaki, 1980/81, no. 565, Lecture Notes in Mathematics, Vol. 901, Springer Verlag, Berlin, Heidelberg, 1981.
D. W. Walkup, The lower bound conjecture for 3- and 4-manifolds, Acta Math. 125 (1970), 75–107.
O. Zariski and P. Samuel, Commutative Algebra, Vol. II, van Nostrand, Princeton, N.J., 1960.
K. Baclawski and A. M. Garsia, Combinatorial decompositions of a class of rings, Advances in Math. 39 (1981), 155–184.
R. Stanley, Balanced Cohen-Macaulay complexes, Trans. Amer. Math. Soc. 249 (1979), 139–157.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1983 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Billera, L.J. (1983). Polyhedral Theory and Commutative Algebra. In: Bachem, A., Korte, B., Grötschel, M. (eds) Mathematical Programming The State of the Art. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-68874-4_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-68874-4_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-68876-8
Online ISBN: 978-3-642-68874-4
eBook Packages: Springer Book Archive