Upper bounds for the number of facets

of a simplicial complex

Authors:
Jürgen Herzog and Takayuki Hibi

Journal:
Proc. Amer. Math. Soc. **125** (1997), 1579-1583

MSC (1991):
Primary 05D05; Secondary 13D40

DOI:
https://doi.org/10.1090/S0002-9939-97-03704-0

MathSciNet review:
1363424

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Abstract | References | Similar Articles | Additional Information

Abstract: Here we study the maximal dimension of the annihilator ideals

of artinian graded rings with a given Hilbert function, where is the polynomial ring in the variables over a field with each , is a graded ideal of , and is the graded maximal ideal of . As an application to combinatorics, we introduce the notion of -facets and obtain some informations on the number of -facets of simplicial complexes with a given -vector.

**[A-H--H]**A. Aramova, J. Herzog and T. Hibi,*Squarefree lexsegment ideals*, Math. Z., to appear.**[B-H]**Winfried Bruns and Jürgen Herzog,*Cohen-Macaulay rings*, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR**1251956****[B-H--V]**W. Bruns, J. Herzog and U. Vetter,*Syzygies and Walks*, in ``Commutative Algebra'' (G. Valla, et al., Eds.), World Scientific, 1994, pp. 36 - 57.**[C-L]**G. F. Clements and B. Lindström,*A generalization of a combinatorial theorem of Macaulay*, J. Combinatorial Theory**7**(1969), 230–238. MR**0246781****[G-K]**Curtis Greene and Daniel J. Kleitman,*Proof techniques in the theory of finite sets*, Studies in combinatorics, MAA Stud. Math., vol. 17, Math. Assoc. America, Washington, D.C., 1978, pp. 22–79. MR**513002****[H]**T. Hibi, ``Algebraic Combinatorics on Convex Polytopes,'' Carslaw Publications, Glebe, N.S.W., Australia, 1992.**[Hoc]**Ja. Ja. Dambit,*An algorithm for superimposing a nonplanar graph onto the plane with a nearly minimal number of crossings*, Latviĭsk. Mat. Ežegodnik**Vyp. 21**(1977), 152–163, 237 (Russian). MR**0480193****[M-M]**H. Michael Möller and Ferdinando Mora,*New constructive methods in classical ideal theory*, J. Algebra**100**(1986), no. 1, 138–178. MR**839576**, https://doi.org/10.1016/0021-8693(86)90071-2**[Sta]**Richard P. Stanley,*Combinatorics and commutative algebra*, Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1983. MR**725505**

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Additional Information

**Jürgen Herzog**

Affiliation:
FB 6 Mathematik und Informatik, Universität–GHS–Essen, 45117 Essen, Germany

Email:
mat300@uni-essen.de

**Takayuki Hibi**

Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560, Japan

Email:
hibi@math.sci.osaka-u.ac.jp

DOI:
https://doi.org/10.1090/S0002-9939-97-03704-0

Received by editor(s):
August 28, 1995

Received by editor(s) in revised form:
October 26, 1995

Additional Notes:
This paper was written while the authors were staying at the Mathematische Forschungsinstitut Oberwolfach in the frame of the RiP program which is financed by Volkswagen–Stiftung

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 1997
American Mathematical Society