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]**W. Bruns and J. Herzog, ``Cohen-Macaulay Rings,'' Cambridge University Press, Cambridge / New York / Sydney, 1993. MR**95h:13020****[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. Combin. Theory**7**(1969), 230 - 238 MR**40:50****[G-K]**C. Greene and D. Kleitman,*Proof techniques in the theory of finite sets*, in ``Studies in Combinatorics'' (G.-C. Rota, ed.), Mathematical Association of America, Washington, D.C., 1978, pp. 22 - 79. MR**80a:05006****[H]**T. Hibi, ``Algebraic Combinatorics on Convex Polytopes,'' Carslaw Publications, Glebe, N.S.W., Australia, 1992.**[Hoc]**M. Hochster,*Cohen-Macaulay rings, combinatorics, and simplicial complexes*, in ``Ring Theory II'' (B. R. McDonald and R. Morris, eds.), Lect. Notes in Pure and Appl Math., No. 26, Dekker, New York, 1977, pp. 171 - 223. MR**58:376****[M-M]**M. Möller and F. Mora,*New constructive methods in classical ideal theory*, J. of Algebra**100**(1986), 138 - 178. MR**88c:13012****[Sta]**R. P. Stanley, ``Combinatorics and Commutative Algebra,'' Birkhäuser, Boston / Basel / Stuttgart, 1983. MR**85b:05002**

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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