Flag Hilbert-Poincaré series and Igusa zeta functions of hyperplane arrangements (Q6660131)

The authors introduce multivariate rational functions associated with hyperplane arrangement called the Hilbert-Poincaré series. Let \(\mathcal{A}\) be an affine hyperplane arrangement in \(\mathbb{K}^{d}\) with \(\mathbb{K}\) being a field, and let \(d = \dim(\mathcal{A})\). For an arrangement \(\mathcal{A}\) we define the intersection poset \(L(\mathcal{A})\) ordered by the reverse-inclusion. Denote by \(0\) and \(1\) the bottom and the top element, respectively, in the poset \(L(\mathcal{A})\) and observe that \(1 \in L(\mathcal{A})\) if and only if \(\mathcal{A}\) is central. For \(x \in L(\mathcal{A})\) we write \(rk(x)\) for the rank of \(x\) which is the supremum over the lengths of all chains from the bottom element \(0\) to \(x\). For \(x \in L(\mathcal{A})\) we define two hyperplane arrangements \N\[\N\mathcal{A}_{x} = \{H \in \mathcal{A} \, : \, x \subseteq H\},\N\]\N\[\N\mathcal{A}^{x} = \{x \cap H \, : \, H \in \mathcal{A} \setminus \mathcal{A}_{x}, \, x \cap H \neq \emptyset\}.\N\]\NThe Poincaré polynomial of \(\mathcal{A}\) is defined as \N\[\N\pi_{\mathcal{A}}(Y) = \sum_{x \in L(\mathcal{A})}\mu(x) (-Y)^{rk(x)} \in \mathbb{Z}[Y],\N\]\Nwhere \(\mu\) is the Möbius function defined on \(L(\mathcal{A})\). Then the Poincaré polynomial adapted to flag can be defined as follows. For a poset \(P\) we define the order complex \(\triangle(P)\) associated with \(P\) which is the simplicial complex with the vertex set \(P\) whose simplices are the flags of \(P\). For \(F = (x_{1} < \ldots < x_{l}) \in \triangle(L(\mathcal{A}))\), set \(x_{0}=0\), \(x_{l+1} = \emptyset\) and \N\[\N\pi_{F}(Y)= \prod_{k=0}^{\ell}\pi_{\mathcal{A}_{x_{k+1}}^{x_{k}}}(Y) \in \mathbb{Z}[Y].\N\]\NDenote \(\widetilde{L(\mathcal{A})} = L(\mathcal{A})\setminus \{0\}\). Let \(\textbf{T} : = (T_{x})_{x \in \widetilde{L(\mathcal{A})}}\) be indeterminantes. The flag Hilbert-Poincaré series associated with \(\mathcal{A}\) is defined as \N\[\N\verb}fHP}_{\mathcal{A}}(Y,\textbf{T}) = \sum_{F \in \triangle(\widetilde{L(\mathcal{A})})}\pi_{F}(T)\prod_{x \in F}\frac{T_{x}}{1-T_{x}} \in \mathbb{Q}(\textbf{T})[Y].\N\]\NThe main result of the paper can be formulated as follows.\N\NTheorem. Let \(\mathcal{A}\) be a central hyperplane arrangement over a field of characteristic zero and denote by \(rk(\mathcal{A})\) the rank of \(\mathcal{A}\), i.e., the rank of a maximal element of \(L(\mathcal{A})\). Then \N\[\N\verb}fHP}_{\mathcal{A}}(Y^{-1},(T_{x}^{-1})_{x \in \widetilde{L(\mathcal{A})}}) = (-Y)^{-rk(\mathcal{A})}T_{1}\cdot \verb}fHP}_{\mathcal{A}}(Y,\textbf{T}).\N\]\NIt turns out that by various substitutions of the variables of the flag Hilbert-Poincaré series one gets some connections with different enumeration problems. For example, one can find a connection with Ingusa local zeta functions associated with products of linear polynomials and topological zeta functions.