TY - JOUR
T1 - Plane branches with Newton non-degenerate polars
AU - Hefez, A.
AU - Hernandes, M. E.
AU - Hernandez Iglesias, Mauro Fernando
N1 - Publisher Copyright:
© 2018 World Scientific Publishing Company.
Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.
PY - 2018/1/1
Y1 - 2018/1/1
N2 - To an equisingularity class of complex plane branches, described by its multiplicity n and characteristic exponents β1 < ⋯ < βr, 0 < n < β1, there is a naturally associated family K(n,β1,...,βr) of equations containing a complete set of analytic representatives for all branches of the class. We show in this paper that the general polar curve of any member of K(n,β1,...,βr) is Newton degenerate, except when r = 1, in which case the general member of K(n,β1) corresponds to a curve which has a Newton non-degenerate general polar curve with a fixed Newton polygon, or when r = 2, n = 2p, β1 = 2q, β2 = 2q + d, with GCD(p,q) = 1 and d ≥ 1 is odd, in which case K(2p, 2q, 2q + d) has a subset containing a complete set of analytic representatives for all branches of the class whose general member has also a Newton non-degenerate general polar curve with a fixed Newton polygon. In both cases, we give explicit Zariski open sets the points of which represent branches with Newton non-degenerate polars and describe the topology of their general polars.
AB - To an equisingularity class of complex plane branches, described by its multiplicity n and characteristic exponents β1 < ⋯ < βr, 0 < n < β1, there is a naturally associated family K(n,β1,...,βr) of equations containing a complete set of analytic representatives for all branches of the class. We show in this paper that the general polar curve of any member of K(n,β1,...,βr) is Newton degenerate, except when r = 1, in which case the general member of K(n,β1) corresponds to a curve which has a Newton non-degenerate general polar curve with a fixed Newton polygon, or when r = 2, n = 2p, β1 = 2q, β2 = 2q + d, with GCD(p,q) = 1 and d ≥ 1 is odd, in which case K(2p, 2q, 2q + d) has a subset containing a complete set of analytic representatives for all branches of the class whose general member has also a Newton non-degenerate general polar curve with a fixed Newton polygon. In both cases, we give explicit Zariski open sets the points of which represent branches with Newton non-degenerate polars and describe the topology of their general polars.
KW - analytic equivalence
KW - equisingularity
KW - Polar curves
UR - http://www.scopus.com/inward/record.url?scp=85040953836&partnerID=8YFLogxK
U2 - 10.1142/S0129167X18500015
DO - 10.1142/S0129167X18500015
M3 - Artículo
AN - SCOPUS:85040953836
VL - 29
JO - International Journal of Mathematics
JF - International Journal of Mathematics
SN - 0129-167X
IS - 1
M1 - 1850001
ER -