Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 46, No. 1, pp. 1-18 (2005)

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The Apery algorithm for a plane singularity with two branches

Valentina Barucci, Marco d'Anna and Ralf Fröberg

Dipartimento di matematica, Università di Roma 1, Piazzale Aldo Moro 2, 00185 Roma, Italy, e-mail:; Dipartimento di matematica, Università di Catania, Viale Andrea Doria 6, 95125 Catania, Italy, e-mail:; Matematiska institutionen, Stockholms Universitet, 10691 Stockholm, Sweden e-mail:

Abstract: The equisingularity class of a plane irreducible curve is determined by the semigroup of the curve or, equivalently, by its multiplicity sequence. For a curve with two branches, the semigroup (now a subsemigroup of ${\mathbb N}^2$) still determines the equisingularity class. We introduce the ``multiplicity tree'' for the curve, which also determines the equisingularity class, and construct an algorithm to go back and forth between the semigroup and the multiplicity tree. Moreover we characterize the multiplicity trees of plane curve singularities with two branches.

Classification (MSC2000): 13H15, 13A18, 14H50

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Electronic version published on: 11 Mar 2005. This page was last modified: 4 May 2006.

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