Critical points of functions on analytic varieties

Bruce, J. W.; Roberts, R. Michael

doi:10.1016/0040-9383(88)90007-9

Cited by 103 publications

(135 citation statements)

References 18 publications

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“…When the point x = 0 is a stratum in the logarithmic stratification of the analytic variety, this is the case when V has an isolated singularity at the origin (see [1] for details), both spaces Θ V and Θ 0 V coincide.…”

Section: §1 Introductionmentioning

confidence: 99%

“…For other results related to the subject discussed in this paper, see for instance [1], [6], [13]. §2.…”

Section: §1 Introductionmentioning

confidence: 99%

“…We denote by Θ V = {η ∈ θ n : η(I(V )) ⊆ I(V )}, the submodule of germs of vector fields tangent to V (see [1] for more details).…”

Section: §1 Introductionmentioning

confidence: 99%

“…The group R V is a geometric subgroup of the contact group, as defined by J. Damon [3], [4], hence the infinitesimal criterion for R V -determinacy holds (see [1] for a proof). …”

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confidence: 99%

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Topological triviality of families of functions on analytic varieties

Ruas¹,

Tomazella²

2004

Nagoya Mathematical Journal

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Abstract. We present in this paper sufficient conditions for the topological triviality of families of germs of functions defined on an analytic variety V . The main result is an infinitesimal criterion based on a convenient weighted inequality, similar to that introduced by T. Fukui and L. Paunescu in [8]. When V is a weighted homogeneous variety, we obtain as a corollary, the topological triviality of deformations by terms of non negative weights of a weighted homogeneous germ consistent with V . Application of the results to deformations of Newton non-degenerate germs with respect to a given variety is also given. §1. Introduction Let V, 0 be the germ of an analytic subvariety of k n , k = R, or C and let R V (respectively C 0 -R V ) be the group of germs of diffeomorphisms (respectively homeomorphisms) preserving V, 0, acting on germs h 0 : k n , 0 → k, 0. The aim of this paper is to study topologically trivial deformations of R Vfinitely determined germs h 0 . The main result is Theorem 3.4 in which we introduce a sufficient condition for the C 0 -R V -triviality of families of map germs h : k n × k, 0 → k, 0, h(x, 0) = h 0 (x), based on a convenient weighted inequality, similar to that introduced by T. Fukui and L. Paunescu in [8]. A non weighted version of this result first appeared in [13]. There, the sufficient condition for topological triviality is formulated in terms of the integral closure of the tangent space to the R V -orbit of h t .As an application of the results, when V is a weighted homogeneous analytic variety, we prove that any deformation by non negative weights of an R V -finitely determined weighted homogeneous germ (consistent with V ) is topologically trivial. This result was previously proved by J. Damon

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Section: §1 Introductionmentioning

confidence: 99%

“…For other results related to the subject discussed in this paper, see for instance [1], [6], [13]. §2.…”

Section: §1 Introductionmentioning

confidence: 99%

“…We denote by Θ V = {η ∈ θ n : η(I(V )) ⊆ I(V )}, the submodule of germs of vector fields tangent to V (see [1] for more details).…”

Section: §1 Introductionmentioning

confidence: 99%

“…The group R V is a geometric subgroup of the contact group, as defined by J. Damon [3], [4], hence the infinitesimal criterion for R V -determinacy holds (see [1] for a proof). …”

mentioning

confidence: 99%

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Topological triviality of families of functions on analytic varieties

Ruas¹,

Tomazella²

2004

Nagoya Mathematical Journal

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“…In this article we answer the question by proving a similar formula for function germs with line singularities on a weighted homogeneous space X with isolated complete intersection singularity (see Proposition 7). Note that for a function germ / with isolated singularity on a weighted homogeneous complete intersection with isolated singularity, Bruce and Robert [1] have proved an algebraic formula for the Milnor number of / . .…”

Section: Introductionmentioning

confidence: 99%