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Approximation of C -functions without changing their zero-set

F. BrogliaA. Tognoli — 1989

Annales de l'institut Fourier

For a C function ϕ : M (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 such that g - 1 ( 0 ) = ϕ - 1 ( 0 ) and g(x) · ϕ ( x ) 0 for any x in n . Similar results are obtained also in the analytic and in the Nash cases. For non approximable functions the minimal flatness locus is also studied....

Formal relations between quasianalytic functions of some fixed class

F. BrogliaA. ElkhadiriF. Pieroni — 2004

Annales Polonici Mathematici

In [Ga] Gabrielov has given conditions under which the completion of the kernel of a morphism φ: A → B between analytic rings coincides with the kernel of the induced morphism φ̂: Â → B̂ between the completions. If B is a domain, a sufficient condition is that rk φ = dim(Â/ker φ̂), where rk φ is the rank of the jacobian matrix of φ considered as a matrix over the quotient field of B. We prove that the above property holds in a fixed quasianalytic Denjoy-Carleman class if and only if the class coincides...

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