Integration with respect to the Euler characteristic over a function space and the Alexander polynomial of a plane curve singularity (Q2759260)

Let \(C\) be a germ of a reduced plane curve at the origin of \(\mathbb{C}^2\) and let \(\varphi_i:(\widetilde C_i,0)\to(\mathbb{C}^2,0)\), be uniformizations of the branches \(C_i\) of \(C\), \(i=1,\dots,h\). For a germ \(g\in{\mathcal O}_{\mathbb{C}^2,0}\), denote by \(v_i=v_i(g)\) and by \(a_i=a_i (g)\), \(i=1,\dots,h\), the exponent and the coefficient, respectively, of the leading monomial in the expansion of the germ \(g\circ\varphi_i: (\widetilde C_i,0)\to(\mathbb{C}^2,0)\) as a power series ring. For any \(v= (v_1,\dots,v_h)\in\mathbb{Z}^h\) set \(J(v)=\{g\in{\mathcal O}_C:v_i(g)\geq v_i,\;i=1, \dots,h\}\). Then \(J(v)\) is an ideal of \({\mathcal O}_C\), and a linear map \(J(v) \to\mathbb{C}^h\) mapping a germ \(g\) to the vector \((a_1,\dots,a_h)\) is well defined. Let \(C(v)\subset\mathbb{C}^h\) be the image of this map, \(F_v= C(v)\cap (\mathbb{C}^*)^h\), and \(c(v)=\dim C(v)\).NEWLINENEWLINE First the authors consider the following formal power series NEWLINE\[NEWLINE((t_1-1)\cdots(t_h-1)\cdot \sum_{v\in\mathbb{Z}^h} c(v)t^v)/(t_1 \cdots t_h-1),NEWLINE\]NEWLINE where \(t^v= t_1^{v_1} \cdots t_h^{v_h}\). They state that it is in fact a polynomial which is equal to the Alexander polynomial of the link \(C\cap S^3_\varepsilon \subset S^3_\varepsilon\) for sufficiently small \(\varepsilon >0\). The second theorem states that the Alexander polynomial is also equal to \(\sum_{v\in\mathbb{Z}^h_{\geq 0}}\chi (\mathbb{P}(f_v))\cdot t^v\), where \(\chi(\mathbb{P}(F_v))\) is the Euler characteristic of the projectivization of \(F_v\).NEWLINENEWLINE The proofs of both results will be published somewhere later.