Arc spaces and Alexander invariants
Let \(X\) be a complex algebraic variety of dimension \(d\) and \(f= (f_1,\dots, f_p)\) be \(p\) regular functions on \(X\). We denote by \(Z_f(T)\) the motivic Igusa local zeta function associated to \(f\) (the definition of motivic Igusa local zeta function is after Denef and Loeser). We put \(X_0= \bigcap_{1\leq\nu\leq p} (f_\nu=0)\) and \(K\) its compact subset, then we can define the Alexander polynomial \({^A\zeta_{K,f}}\), in which \(A\) is the ring \(\mathbb{C}[\mathbb{Z}^p]\). Also, we can define the realization of Alexander \({^A\zeta_K}\). One of the main results of this paper is the relation \[ {^A\zeta_K}\Biggl( \lim_{T\to\infty}\,Z_f(T^\alpha)\Biggr)= -{^A\zeta_{K,f}} \] in which we put \(T^\alpha= T^{\alpha_1}\cdots T^{\alpha_p}\) \((\alpha= (\alpha_1,\dots, \alpha_p)\in (\mathbb{N}^\times)^p)\). From this relation, the author shows explicit formulae of Alexander polynomials associated to plane curves.
scientific article; zbMATH DE number 1883363