On the topological type of a set of plane valuations with symmetriesстатья
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Аннотация:Let $\{C_i : i=1,\ldots,r\}$ be a set of irreducible plane curve singularities. For an action of a finite group $G$, let $\Delta^{L}(\{t_{a i}\})$ be the Alexander polynomial in $r|G|}$ variables of the algebraic link $L=(\bigcup\limits_{i=1}^{r}\bigcup\limits_{a\in G}a C_i )\cap S^3_{\varepsilon}$ and let $\zeta(t_1, \ldots, t_r) = \Delta^L(t_1, \ldots, t_1, t_2, \ldots, t_2, t_r, \ldots, t_r)$ with $|G|$ identical variables in each group. (If $r=1$, $\zeta(t)$ is the monodromy zeta function of the function germ $\prod\limits_{a\in G} a^*f$, where $f=0$ is an equation defining the curve $C_1$.) We prove that $\zeta(t_1, \ldots, t_r)$ determines the topological type of the link $L$. We prove an analogous statement for plane divisorial valuations formulated in terms of the Poincar'e series of a set of valuations.