Displaying similar documents to “On real algebraic links in S 3 ”

Approximation of C ∞ -functions without changing their zero-set

F. Broglia, A. Tognoli (1989)

Annales de l'institut Fourier

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For a C ∞ function ϕ : M → ℝ (where M is a real algebraic manifold) the following problem is studied. If ϕ - 1 ( 0 ) is an algebraic subvariety of M , can ϕ be approximated by rational regular functions f such that f - 1 ( 0 ) = ϕ - 1 ( 0 ) ? We find that this is possible if and only if there exists a rational regular function g : M → ℝ such that g - 1 ( 0 ) = ϕ - 1 ( 0 ) and g(x) · ϕ ( x ) ≥ 0 for any x in ℝ n . Similar results are obtained also in the analytic and in the Nash cases. For non approximable functions the minimal flatness locus...

The trivial locus of an analytic map germ

H. Hauser, G. Muller (1989)

Annales de l'institut Fourier

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We prove: For a local analytic family { X s } s ∈ S of analytic space germs there is a largest subspace T in S such that the family is trivial over T . Moreover the reduction of T equals the germ of those points s in S for which X s is isomorphic to the special fibre X 0 .