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On Tangent Cones of Analytic Sets and Łojasiewicz Exponents
For an analytic set V in K n , K = R or C , which contains the origin 0 ∈ K n , the geometric tangent cone of V at 0 is the set of vectors in K n which are the limits of secant lines passing through the origin and non-zero sequences in V that converge to the origin; the algebraic tangent cone of V a...
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| Published in: | Bulletin of the Iranian Mathematical Society 2020-04, Vol.46 (2), p.355-380 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites |
| Online Access: | Get full text |
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| Summary: | For an analytic set
V
in
K
n
,
K
=
R
or
C
, which contains the origin
0
∈
K
n
, the
geometric tangent cone
of
V
at 0 is the set of vectors in
K
n
which are the limits of secant lines passing through the origin and non-zero sequences in
V
that converge to the origin; the
algebraic tangent cone
of
V
at 0 is the algebraic set defined by the ideal generated by the initial forms of all analytic functions
f
whose germs at 0 are in the ring of germs of analytic functions in
K
n
about 0 which vanish on the germ of
V
at 0. In this paper, we give some characterization for the geometric tangent cone and compare these two tangent cones of
V
at 0. In particular, we characterize the geometric tangent cone of
V
at 0 via the so-called
Ł
ojasiewicz exponent of an analytic map germ along a line
and compute this number in some special cases. |
|---|---|
| ISSN: | 1017-060X 1735-8515 |
| DOI: | 10.1007/s41980-019-00261-z |