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Relations among analytic functions. I

Edward Bierstone, P. D. Milman (1987)

Annales de l'institut Fourier

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 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 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 category)....

Relations among analytic functions. II

Edward Bierstone, P. D. Milman (1987)

Annales de l'institut Fourier

This is a sequel to “Relations among analytic functions I”, Ann. Inst. Fourier, 37, fasc. 1, [pp. 187-239]. We reduce to semicontinuity of local invariants the problem of finding 𝒞 solutions to systems of equations involving division and composition by analytic functions. We prove semicontinuity in several general cases : in the algebraic category, for “regular” mappings, and for module homomorphisms over a finite mapping.

Résidu de Grothendieck et forme de Chow.

Mohamed Elkadi (1994)

Publicacions Matemàtiques

We show an explicit relation between the Chow form and the Grothendieck residue; and we clarify the role that the residue can play in the intersection theory besides its role in the division problem.

Solutions d'un système d'équations analytiques réelles et applications

Jean-Claude Tougeron (1976)

Annales de l'institut Fourier

On démontre que toute solution formelle y ( x ) d’un système d’équations analytiques réelles (resp. polynomiales réelles) f ( x , y ) = 0 , se relève en une solution C homotope à une solution analytique (resp. à une solution de Nash) aussi proche que l’on veut de y ( x ) pour la topologie de Krull. On utilise ce théorème pour démontrer l’algébricité (ou l’analyticité) de certains idéaux de R { x } (ou R [ [ x ] ] ), et aussi pour construire des déformations analytiques de germes d’ensembles analytiques en germes d’ensembles de Nash.

Solving power series equations. II. Change of ground field

Joseph Becker (1979)

Annales de l'institut Fourier

We study the effect of changing the residue field, on the topological properties of local algebra homomorphisms of analytic algebras (quotients of convergent power series rings). Although injectivity is not preserved, openness and closedness in the Krull topology, simple topology, and inductive topology is preserved.

Spaces of polynomial functions of bounded degrees on an embedded manifold and their duals

Shuzo Izumi (2015)

Annales Polonici Mathematici

Let (U) denote the algebra of holomorphic functions on an open subset U ⊂ ℂⁿ and Z ⊂ (U) its finite-dimensional vector subspace. By the theory of least spaces of de Boor and Ron, there exists a projection b from the local ring n , b onto the space Z b of germs of elements of Z at b. At a general point b ∈ U its kernel is an ideal and b induces the structure of an Artinian algebra on Z b . In particular, this holds at points where the kth jets of elements of Z form a vector bundle for each k ∈ ℕ. For an embedded...

Sur la dualité locale

Marcos Sebastiani (1980)

Annales de l'institut Fourier

Une construction explicite et élémentaire de l’homomorphisme trace pour les applications analytiques locales de type fini entre des espaces normaux est donnée. On généralise le théorème de dualité locale dans le cas où l’anneau local à la source est un anneau de factorisation unique. Des exemples et des applications sont donnés.

Sur le théorème de division de Weierstrass

Jacques Chaumat, Anne-Marie Chollet (1995)

Studia Mathematica

We prove a Weierstrass division formula for C Whitney jets ∂̅-flat on arbitrary compact subsets of the complex plane. We also give results for Carleman classes.

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