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Divisors in global analytic sets

Francesca AcquistapaceA. Díaz-Cano — 2011

Journal of the European Mathematical Society

We prove that any divisor Y of a global analytic set X n has a generic equation, that is, there is an analytic function vanishing on Y with multiplicity one along each irreducible component of Y . We also prove that there are functions with arbitrary multiplicities along Y . The main result states that if X is pure dimensional, Y is locally principal, X / Y is not connected and Y represents the zero class in H q - 1 ( X , 2 ) then the divisor Y is globally principal.

Sulla non validità di un teorema di approssimazione

Francesca AcquistapaceFabrizio BrogliaAlberto Tognoli — 1973

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

The following theorem is true: if U is open in 𝐑 n , V U is a coherent real analytic set, and f : U 𝐑 is a C function such that f V is analytic, then it is possible to approximate f (together with its derivatives) by analytic functions { f n } such that f n V = f V . In this paper we prove that this result is not true unless V is coherent (with the reduced structure).

On the finiteness of Pythagoras numbers of real meromorphic functions

Francesca AcquistapaceFabrizio BrogliaJosé F. FernandoJesús M. Ruiz — 2010

Bulletin de la Société Mathématique de France

We consider the 17th Hilbert Problem for global real analytic functions in a modified form that involves infinite sums of squares. Then we prove a local-global principle for a real global analytic function to be a sum of squares of global real meromorphic functions. We deduce that an affirmative solution to the 17th Hilbert Problem for global real analytic functions implies the finiteness of the Pythagoras number of the field of global real meromorphic functions, hence that of the field of real...

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