Strictly analytic functions on p-adic analytic open sets.

• Volume: 43, Issue: 1, page 127-162
• ISSN: 0214-1493

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Abstract

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Let K be an algebraically closed complete ultrametric field. M. Krasner and P. Robba defined theories of analytic functions in K, but when K is not spherically complete both theories have the disadvantage of containing functions that may not be expanded in Taylor series in some disks. On other hand, affinoid theories are only defined in a small class of sets (union of affinoid sets) [2], [13] and [17]. Here, we suppose the field K topologically separable (example Cp). Then, we give a new definition of strictly analytic functions over a large class of domains called analoid sets. Our theory uses the notion of T-sequence which caracterizes analytic sets in the sense of Robba. Thereby we obtain analytic functions satisfying the property of analytic continuation and which, however, will admit expansion in power series (resp. Laurent series) in any disk (resp. in any annulus). Moreover, the algebra of analytic functions will be stable by derivation. The process consists of defining a large class of analytic sets D, and a class of admissible sets making a covering of such a D, so that we obtain a sheaf on D. We finally give an example of differential equation whose solutions are strictly analytic functions in an analoid set. Such an example might not be involved in theories based on affinoid sets.

How to cite

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Boussaf, Kamal. "Strictly analytic functions on p-adic analytic open sets.." Publicacions Matemàtiques 43.1 (1999): 127-162. <http://eudml.org/doc/41367>.

@article{Boussaf1999,
abstract = {Let K be an algebraically closed complete ultrametric field. M. Krasner and P. Robba defined theories of analytic functions in K, but when K is not spherically complete both theories have the disadvantage of containing functions that may not be expanded in Taylor series in some disks. On other hand, affinoid theories are only defined in a small class of sets (union of affinoid sets) [2], [13] and [17]. Here, we suppose the field K topologically separable (example Cp). Then, we give a new definition of strictly analytic functions over a large class of domains called analoid sets. Our theory uses the notion of T-sequence which caracterizes analytic sets in the sense of Robba. Thereby we obtain analytic functions satisfying the property of analytic continuation and which, however, will admit expansion in power series (resp. Laurent series) in any disk (resp. in any annulus). Moreover, the algebra of analytic functions will be stable by derivation. The process consists of defining a large class of analytic sets D, and a class of admissible sets making a covering of such a D, so that we obtain a sheaf on D. We finally give an example of differential equation whose solutions are strictly analytic functions in an analoid set. Such an example might not be involved in theories based on affinoid sets.},
author = {Boussaf, Kamal},
journal = {Publicacions Matemàtiques},
keywords = {Funciones analíticas; Ecuaciones diferenciales ordinarias; Topología compacto abierta; -adic; strictly analytic function},
language = {eng},
number = {1},
pages = {127-162},
title = {Strictly analytic functions on p-adic analytic open sets.},
url = {http://eudml.org/doc/41367},
volume = {43},
year = {1999},
}

TY - JOUR
AU - Boussaf, Kamal
TI - Strictly analytic functions on p-adic analytic open sets.
JO - Publicacions Matemàtiques
PY - 1999
VL - 43
IS - 1
SP - 127
EP - 162
AB - Let K be an algebraically closed complete ultrametric field. M. Krasner and P. Robba defined theories of analytic functions in K, but when K is not spherically complete both theories have the disadvantage of containing functions that may not be expanded in Taylor series in some disks. On other hand, affinoid theories are only defined in a small class of sets (union of affinoid sets) [2], [13] and [17]. Here, we suppose the field K topologically separable (example Cp). Then, we give a new definition of strictly analytic functions over a large class of domains called analoid sets. Our theory uses the notion of T-sequence which caracterizes analytic sets in the sense of Robba. Thereby we obtain analytic functions satisfying the property of analytic continuation and which, however, will admit expansion in power series (resp. Laurent series) in any disk (resp. in any annulus). Moreover, the algebra of analytic functions will be stable by derivation. The process consists of defining a large class of analytic sets D, and a class of admissible sets making a covering of such a D, so that we obtain a sheaf on D. We finally give an example of differential equation whose solutions are strictly analytic functions in an analoid set. Such an example might not be involved in theories based on affinoid sets.
LA - eng
KW - Funciones analíticas; Ecuaciones diferenciales ordinarias; Topología compacto abierta; -adic; strictly analytic function
UR - http://eudml.org/doc/41367
ER -

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