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Sufficiency of jets with line singularities
Author(s) -
Brodersen Hans
Publication year - 2009
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/bdp016
Subject(s) - mathematics , gravitational singularity , invariant (physics) , jet (fluid) , combinatorics , homeomorphism (graph theory) , manifold (fluid mechanics) , line (geometry) , pure mathematics , mathematical analysis , geometry , mathematical physics , physics , mechanical engineering , engineering , thermodynamics
Let z : (ℝ n + 1 , 0) → (ℝ, 0) be an r ‐jet with a singular set containing a 1‐dimensional manifold L . Letℛ 0 Lbe the set of homeomorphism germs h : (ℝ n + 1 , 0) → (ℝ n + 1 , 0) leaving L invariant. Letε [ r ] Lbe the set of C r germs, f : (ℝ n + 1 , 0) → (ℝ, 0), with singular sets containing L . We say that z is sufficient inε [ r ] Lif any two f and g inε [ r ] Lwithj r f ( 0 ) = j r g ( 0 ) = z areℛ 0 L ‐equivalent. In this paper we give necessary and sufficient conditions in terms of Łojasiewicz inequalities for such a jet z to be sufficient inε [ r ] L .