Displaying similar documents to “The trivial locus of an analytic map germ”

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...

On the Pythagoras numbers of real analytic set germs

José F. Fernando, Jesús M. Ruiz (2005)

Bulletin de la Société Mathématique de France

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We show that (i) the Pythagoras number of a real analytic set germ is the supremum of the Pythagoras numbers of the curve germs it contains, and (ii) every real analytic curve germ is contained in a real analytic surface germ with the same Pythagoras number (or Pythagoras number 2 if the curve is Pythagorean). This gives new examples and counterexamples concerning sums of squares and positive semidefinite analytic function germs.

Overstability and resonance

Augustin Fruchard, Reinhard Schäfke (2003)

Annales de l’institut Fourier

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We consider a singularity perturbed nonlinear differential equation ε u ' = f ( x ) u + + ε P ( x , u , ε ) which we suppose real analytic for x near some interval [ a , b ] and small | u | , | ε | . We furthermore suppose that 0 is a turning point, namely that x f ( x ) is positive if x ≠ 0 . We prove that the existence of nicely behaved (as ϵ → 0 ) local (at x = 0 ) or global, real analytic or C ∞ solutions is equivalent to the existence of a formal series solution ∑ u n ( x ) ε n with u n analytic at x = 0 . The main tool of a proof is a new “principle of analytic continuation” for...

Relations among analytic functions. I

Edward Bierstone, P. D. Milman (1987)

Annales de l'institut Fourier

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Neither real analytic sets nor the images of real or complex analytic mappings are, in general, coherent. Let Φ : X → Y be a morphism of real analytic spaces, and let Ψ : 𝒢 → ℱ be a homomorphism of coherent modules over the induced ring homomorphism Φ * : 𝒪 Y → 𝒪 X . We conjecture that, despite the failure of coherence, certain natural discrete invariants of the modules of formal relations ℛ a = Ker Ψ ^ a , a ∈ X , are upper semi-continuous in the analytic Zariski topology of X . We prove semicontinuity in many cases (e.g. in the algebraic...

Topological invariants of analytic sets associated with Noetherian families

Aleksandra Nowel (2005)

Annales de l’institut Fourier

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Let Ω ⊂ ℝ n be a compact semianalytic set and let ℱ be a collection of real analytic functions defined in some neighbourhood of Ω . Let Y ω be the germ at ω of the set ⋂ f ∈ ℱ f - 1 ( 0 ) . Then there exist analytic functions v 1 , v 2 , ... , v s defined in a neighbourhood of Ω such that 1 2 χ ( lk ( ω , Y ω ) ) = ∑ i = 1 s sgn v i ( ω ) , for all ω ∈ Ω .