The zeta function of a simplicial complex
type: Item
instance of: scholarly article
title: The zeta function of a simplicial complex
zbMATH Open document ID: 0932.57023
DOI: 10.1007/BF02762266
published in: Israel Journal of Mathematics
publication date: 2000-03-14T00:00:00Z
review text: For a given field \(F\), a simplicial complex \(\Delta\) with \(n\) vertices defines a (singular) projective variety \(V(\Delta,F)\) in projective \((n-1)\)-space. The algebro-geometric zeta function \(Z_\Delta(q,t)\), in the sense of A. Weil, is then called the zeta function of the given complex \(\Delta\). In the present paper, the authors derive a simple rational expression for this particular zeta function, in a purely combinatorial-topological way, and then they show how this formula can be used to compute the singular homology groups \(H_i(V(\Delta, \mathbb{C}),\mathbb{Z})\) of the complex projective variety \(V(\Delta, \mathbb{C})\). This result reveals the highly interesting fact that both the Betti numbers and the zeta function of the singular projective variety \(V(\Delta,\mathbb{C})\) are completely determined by the combinatorial topology of the underlying complex \(\Delta\). Then, after a brief discussion of the possible geometric representations of the variety \(V(\Delta,\mathbb{C})\), the case of Cohen-Macaulay complexes \(\Delta\) is studied in greater detail. In this situation, where the homogeneous coordinate ring of the variety \(V(\Delta, \mathbb{C})\) is a Cohen-Macaulay ring, the formula for the zeta function \(Z_\Delta (t)\) becomes particularly simple. At the end of the paper, the authors point out that their results are also intimately related to the so-called ``\(f\)-vector theory'' in combinatorial topology and commutative algebra.
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 57Q05
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 55N10
Mathematics Subject Classification ID: 14F25
zbMATH DE Number: 1250638
zbMATH Keywords: \(f\)-vector theory
zbMATH Keywords: Stanley-Reisner rings
zbMATH Keywords: simplicial complex
zbMATH Keywords: singular homology
zbMATH Keywords: Betti numbers
zbMATH Keywords: zeta function
zbMATH Keywords: singular projective variety
zbMATH Keywords: Cohen-Macaulay complexes
author: Anders Bjoerner
author: Karanbir Singh Sarkaria
MaRDI profile type: Publication
cites work: Q4324358
cites work: La conjecture de Weil. I
cites work: On the Rationality of the Zeta Function of an Algebraic Variety
cites work: Q4039844
cites work: Q3827224
cites work: A "Riemann Hypothesis" for Triangulable Manifolds
cites work: Q3748279
cites work: Homotopy types of subspace arrangements via diagrams of spaces
full work available at URL: https://doi.org/10.1007/bf02762266
OpenAlex ID: W1987876025
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description: scientific article; zbMATH DE number 1250638