Newton process and semigroups of irreducible quasi-ordinary power series

The Newton process were introduced by Artal-Bartolo, Cassou-Nogues, Luengo and Melle-Hernandez as a generalization of the Newton algorithm associated to plane curve singularities. Newton process is useful to study v-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.