# Divisors in global analytic sets

• Volume: 013, Issue: 2, page 297-307
• ISSN: 1435-9855

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## Abstract

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We prove that any divisor $Y$ of a global analytic set $X\subset {ℝ}^{n}$ 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 ${H}_{q-1}^{\infty }\left(X,{ℤ}_{2}\right)$ then the divisor $Y$ is globally principal.

## How to cite

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Acquistapace, Francesca, and Díaz-Cano, A.. "Divisors in global analytic sets." Journal of the European Mathematical Society 013.2 (2011): 297-307. <http://eudml.org/doc/277592>.

@article{Acquistapace2011,
abstract = {We prove that any divisor $Y$ of a global analytic set $X \subset \mathbb \{R\}^n$ 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 $H^\infty _\{q-1\}(X,\mathbb \{Z\}_2)$ then the divisor $Y$ is globally principal.},
author = {Acquistapace, Francesca, Díaz-Cano, A.},
journal = {Journal of the European Mathematical Society},
keywords = {real analytic sets; divisors; real analytic set; divisor},
language = {eng},
number = {2},
pages = {297-307},
publisher = {European Mathematical Society Publishing House},
title = {Divisors in global analytic sets},
url = {http://eudml.org/doc/277592},
volume = {013},
year = {2011},
}

TY - JOUR
AU - Acquistapace, Francesca
AU - Díaz-Cano, A.
TI - Divisors in global analytic sets
JO - Journal of the European Mathematical Society
PY - 2011
PB - European Mathematical Society Publishing House
VL - 013
IS - 2
SP - 297
EP - 307
AB - We prove that any divisor $Y$ of a global analytic set $X \subset \mathbb {R}^n$ 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 $H^\infty _{q-1}(X,\mathbb {Z}_2)$ then the divisor $Y$ is globally principal.
LA - eng
KW - real analytic sets; divisors; real analytic set; divisor
UR - http://eudml.org/doc/277592
ER -

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