On the topological type of a set of plane valuations with symmetries

2016/04/27 by Antonio Campillo, Campillo, A., F. Delgado +3 · 1 citation
Mathematics · #13A18 #14B05 #14R20 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1604.08152

openalex publication_date 2016/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \C_i : i=1,…,r\ be a set of irreducible plane curve singularities. For an action of a finite group G, let Δ^L(\t_{a i}\) be the Alexander polynomial in r\vert G\vert variables of the algebraic link (\bigcup_i=1^r\bigcup_{a∈ G}a C_i )∩ S³_ε and let ζ(t₁,…, t_r) = Δ^L(t₁,…,t₁,t₂,…,t₂, …,t_r,…,t_r) with \vert G\vert identical variables in each group. (If r=1, ζ(t) is the monodromy zeta function of the function germ ∏_{a∈ G} a^*f, where f=0 is an equation defining the curve C₁.) We prove that ζ(t₁,…, t_r) determines the topological type of the link L. We prove an analogous statement for plane divisorial valuations formulated in terms of the Poincaré series of a set of valuations.

Cited by

• Poincaré series of valuations on functions over a field

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