Logarithmic structure of the generalized bifurcation set

S. Janeczko

S. Janeczko

Annales Polonici Mathematici, Tome 63 (1996), p. 187-197 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $G : ℂ^{n} \times ℂ^{r} \to ℂ$ be a holomorphic family of functions. If $Λ \subset ℂ^{n} \times ℂ^{r}$ , $π_{r} : ℂ^{n} \times ℂ^{r} \to ℂ^{r}$ is an analytic variety then $Q_{Λ} (G) = (x, u) \in ℂ^{n} \times ℂ^{r} : G (\cdot, u) h a s a c r i t i c a l p o i n t i n Λ \cap π_{r}^{- 1} (u)$ 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.

Publié le : 1996-01-01

Zbl 0868.58014

EUDML-ID : urn:eudml:doc:262533

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     author = {S. Janeczko},
     title = {Logarithmic structure of the generalized bifurcation set},
     journal = {Annales Polonici Mathematici},
     volume = {63},
     year = {1996},
     pages = {187-197},
     zbl = {0868.58014},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv63z2p187bwm}
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S. Janeczko. Logarithmic structure of the generalized bifurcation set. Annales Polonici Mathematici, Tome 63 (1996) pp. 187-197. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv63z2p187bwm/

Bibliographie

[00000] [1] V. I. Arnold, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps, Vol. 1, Birkhäuser, Boston, 1985. | Zbl 1297.32001

[00001] [2] J. W. Bruce, Functions on discriminants, J. London Math. Soc. (2) 30 (1984), 551-567. | Zbl 0605.58011

[00002] [3] J. W. Bruce and R. M. Roberts, Critical points of functions on analytic varieties, Topology 27 (1988), 57-90. | Zbl 0639.32008

[00003] [4] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge Univ. Press, Cambridge, 1984. | Zbl 0576.58012

[00004] [5] S. Izumiya, Generic bifurcations of varieties, Manuscripta Math. 46 (1984), 137-164. | Zbl 0537.58009

[00005] [6] S. Janeczko, On isotropic submanifolds and evolution of quasicaustics, Pacific J. of Math. 158 (1993), 317-334. | Zbl 0806.58023

[00006] [7] S. Janeczko, On quasicaustics and their logarithmic vector fields, Bull. Austral. Math. Soc. 43 (1991), 365-376. | Zbl 0732.58006

[00007] [8] A. Kas and M. Schlessinger, On the versal deformation of a complex space with an isolated singularity, Math. Ann. 196 (1972), 23-29. | Zbl 0242.32014

[00008] [9] S. Łojasiewicz, Introduction to Complex Analytic Geometry, Birkhäuser, 1991. | Zbl 0747.32001

[00009] [10] O. W. Lyashko, Classification of critical points of functions on a manifold with singular boundary, Funktsional. Anal. i Prilozhen. 17 (3) (1983), 28-36 (in Russian).

[00010] [11] K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 265-291. | Zbl 0496.32007

[00011] [12] H. Terao, The bifurcation set and logarithmic vector fields, Math. Ann. 263 (1983), 313-321. | Zbl 0497.32016

[00012] [13] C. T. Wall, A splitting theorem for maps into ℝ², Math. Ann. 259 (1982), 443-453. | Zbl 0468.58004

[00013] [14] V. M. Zakalyukin, Bifurcations of wavefronts depending on one parameter, Functional Anal. Appl. 10 (1976), 139-140. | Zbl 0345.58008