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The geometry of continued fractions and the topology of surface singularities, arxiv:math.GT/0506432
 SingularitiesSapporo 2004, Advanced Studies in Pure Mathematics
, 2006
"... Abstract. We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence ..."
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Abstract. We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing structure on the abstract boundaries (also called links) of normal surface singularities. The duality between supplementary cones gives in particular a geometric interpretation of a duality discovered by Hirzebruch between the continued fraction expansions of two numbers λ> 1 and λ λ−1.
QUASI ORDINARY SINGULARITIES, ESSENTIAL DIVISORS AND POINCARÉ SERIES
, 2008
"... We define Poincaré series associated to a toric or analytically irreducible quasiordinary hypersurface singularity, (S, 0), by a finite sequence of monomial valuations, such that at least one of them is centered at the origin 0. This involves the definition of a multigraded ring associated to the a ..."
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We define Poincaré series associated to a toric or analytically irreducible quasiordinary hypersurface singularity, (S, 0), by a finite sequence of monomial valuations, such that at least one of them is centered at the origin 0. This involves the definition of a multigraded ring associated to the analytic algebra of the singularity by the sequence of valuations. We prove that the Poincaré series is a rational function with integer coefficients, which can be defined also as an integral with respect of the Euler characteristic, over the projectivization of the analytic algebra of the singularity, of a function defined by the valuations. In particular, the Poincaré series associated to the set of divisorial valuations corresponding to the essential divisors, considered both over the singular locus and over the point 0, is an analytic invariant of the singularity. In the quasiordinary hypersurface case we prove that this Poincaré series determines and it is determined by the normalized sequence of characteristic monomials. These monomials in the analytic case define a complete invariant of the embedded topological type of the hypersurface singularity.