Newton process and semigroups of irreducible quasi-ordinary power series

Original Paper


The Newton process were introduced by Artal-Bartolo, Cassou-Noguès, Luengo and Melle-Hernández as a generalization of the Newton algorithm associated to plane curve singularities. Newton process is useful to study \(\nu \)-quasi-ordinary and quasi-ordinary polynomials in any number of variables. We describe numerically the Newton process associated to a quasi-ordinary branch of an irreducible quasi-ordinary polynomial in terms of its characteristic exponents. We show the relation between these numerical data and the semigroup of the singularity, give a criterium for irreducibility of quasi-ordinary polynomials and describe the normalization of irreducible quasi-ordinary surfaces in terms of the numerical data. We also study why and when irreducibility fails to be preserved by the Newton process.


Quasi-ordinary power series 

Mathematics Subject Classification (2000)

14B05 32S05 32S10 



I would like to thank very much E. Artal-Bartolo, Pi. Cassou-Noguès, P. D. González-Pérez, I. Luengo and A. Melle Hernández for several useful discussions and suggestions during the preparation of this work. I also the referees for interesting reports which sensibly reshaped this work.


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Copyright information

© Springer-Verlag Italia 2013

Authors and Affiliations

  1. 1.Dpto. de Algebra. Facultad de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain
  2. 2.MATCH, Univ. HeidelbergHeidelbergGermany

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