analytic classification of typical degenerate elementary singular
Russian Mathematics (Iz. VUZ)
Vol. 46, No. 1, pp.11{14, 2002
Izvestiya VUZ. Matematika
UDC 517.9
ANALYTIC CLASSIFICATION OF TYPICAL DEGENERATE
ELEMENTARY SINGULAR POINTS OF THE GERMS
OF HOLOMORPHIC VECTOR FIELDS ON THE COMPLEX PLANE
S.M. Voronin and Yu.I. Meshcheryakova
Introduction
Let be the class of germs of holomorphic vector elds in (C 2 0) with degenerate elementary
singular point 0 (i. e., such that the linear part of a germ in zero is degenerate, but not all its
eigenvalues are equal to zero).
The germs v and ve from are said to be analytically (formally) equivalent if a local holomorphic
(formal) change of coordinates H in (C 2 0) exists which translates one germ into other, H v = ve H .
The germs v and ve from are said to be orbital analytically (formally) equivalent if a local
holomorphic change of coordinates exists which translates the phase portrait of one germ into the
phase portrait of other germ (if a formal change of coordinates H and a formal power series k with
nonzero free term exist such that H v = k ve H ).
The orbital analytic and formal classications of germs from are well-known (see 1] 2], 3,
pp. 29, 33). In this article we will obtain an analytic classication of the typical germs from . We
prove that this classication has two functional modules and four numerical ones, which exceeds
twice the quantity of modules of the orbital analytic classication. We prove also the theorem on
sectorial normalization.
V
V
0
V
0
V
x
V
1. Formal classication
As is known (see 1] 2], 3, p. 29 4]), the typical germ from
to one of the germs
x
Let
@ + y2 @
v = x @x
1 + y @y
V
2
V
is formally orbital equivalent
C:
be the class of germs which are formally orbital equivalent to v .
Theorem 1 (on formal classication, see 6]). Every germ from
of the germs
the germs v
v ab = v (a + by) a b
ab
2
C
V
is formally equivalent to one
a = 0
6
with distinguished indices are pairwise formally non-equivalent.
Partially supported by the Russian Foundation for Basic Research (grant no. 98-01-00821) and Ministry
of Education (grant no. 97-0-1.8-57).
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