The zeta function of a simplicial complex
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.
scientific article; zbMATH DE number 1250638
scientific article