Logarithmic structure of the generalized bifurcation set

S. Janeczko

Annales Polonici Mathematici (1996)

  • Volume: 63, Issue: 2, page 187-197
  • ISSN: 0066-2216

Abstract

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Let G : ℂ n × ℂ r → ℂ be a holomorphic family of functions. If Λ ⊂ ℂ n × ℂ r , π r : ℂ n × ℂ r → ℂ r is an analytic variety then    Q Λ ( G ) = ( x , u ) ∈ ℂ n × ℂ r : G ( · , u ) h a s a c r i t i c a l p o i n t i n Λ ∩ π r - 1 ( u ) is a natural generalization of the bifurcation variety of G. We investigate the local structure of Q Λ ( G ) for locally trivial deformations of Λ ₀ = π r - 1 ( 0 ) . In particular, we construct an algorithm for determining logarithmic stratifications provided G is versal.

How to cite

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S. Janeczko. "Logarithmic structure of the generalized bifurcation set." Annales Polonici Mathematici 63.2 (1996): 187-197. <http://eudml.org/doc/262533>.

@article{S1996,
abstract = {Let $G: ℂ^\{n\} × ℂ^\{r\} → ℂ$ be a holomorphic family of functions. If $Λ ⊂ ℂ^\{n\} × ℂ^\{r\}$, $π_r: ℂ^\{n\} × ℂ^\{r\} → ℂ^\{r\}$ is an analytic variety then   $Q_\{Λ\}(G) = \{(x,u) ∈ ℂ^\{n\} × ℂ^\{r\}: G(·,u) has a critical point in Λ ∩ π_\{r\}^\{-1\}(u)\} $is a natural generalization of the bifurcation variety of G. We investigate the local structure of $Q_\{Λ\}(G)$ for locally trivial deformations of $Λ₀ = π_\{r\}^\{-1\}(0)$. In particular, we construct an algorithm for determining logarithmic stratifications provided G is versal.},
author = {S. Janeczko},
journal = {Annales Polonici Mathematici},
keywords = {bifurcations; singularities; logarithmic stratifications; generalized bifurcation sets},
language = {eng},
number = {2},
pages = {187-197},
title = {Logarithmic structure of the generalized bifurcation set},
url = {http://eudml.org/doc/262533},
volume = {63},
year = {1996},
}

TY - JOUR
AU - S. Janeczko
TI - Logarithmic structure of the generalized bifurcation set
JO - Annales Polonici Mathematici
PY - 1996
VL - 63
IS - 2
SP - 187
EP - 197
AB - Let $G: ℂ^{n} × ℂ^{r} → ℂ$ be a holomorphic family of functions. If $Λ ⊂ ℂ^{n} × ℂ^{r}$, $π_r: ℂ^{n} × ℂ^{r} → ℂ^{r}$ is an analytic variety then   $Q_{Λ}(G) = {(x,u) ∈ ℂ^{n} × ℂ^{r}: G(·,u) has a critical point in Λ ∩ π_{r}^{-1}(u)} $is a natural generalization of the bifurcation variety of G. We investigate the local structure of $Q_{Λ}(G)$ for locally trivial deformations of $Λ₀ = π_{r}^{-1}(0)$. In particular, we construct an algorithm for determining logarithmic stratifications provided G is versal.
LA - eng
KW - bifurcations; singularities; logarithmic stratifications; generalized bifurcation sets
UR - http://eudml.org/doc/262533
ER -

References

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