Signatures et composantes connexes

Mahé, Louis‐Pascal

doi:10.1007/bf01457236

Cited by 27 publications

(13 citation statements)

References 8 publications

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“…But this is relative separation of disjoint closed constructible sets, which can be proved by change of base (see [Mh,4.3]). Let U ⊂ Spec r (A) be open constructible and S, F ⊂ U disjoint constructible sets that are closed in U .…”

Section: Proofmentioning

confidence: 99%

Uniform bounds on complexity and transfer of global properties of Nash functions

Coste¹,

Sancho²,

Shiota³

2001

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We show that the complexity of semialgebraic sets and mappings can be used to parametrize Nash sets and mappings by Nash families. From this we deduce uniform bounds on the complexity of Nash functions that lead to first-order descriptions of many properties of Nash functions and a good behaviour under real closed field extension (e.g. primary decomposition). As a distinguished application, we derive the solution of the extension and global equations problems over arbitrary real closed fields, in particular over the field of real algebraic numbers. This last fact and a technique of change of base are used to prove that the Artin-Mazur description holds for abstract Nash functions on the real spectrum of any commutative ring, and solve extension and global equations in that abstract setting. To complete the view, we prove the idempotency of the real spectrum and an abstract version of the separation problem. We also discuss the conditions for the rings of abstract Nash functions to be noetherian .

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Section: Proofmentioning

confidence: 99%

Uniform bounds on complexity and transfer of global properties of Nash functions

Coste¹,

Sancho²,

Shiota³

2001

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…We next use the abstract version of Mostowski's separation theorem [Mah,Th 4.3]: as F 1 and F 2 are closed disjoint constructibles in Spec r C , there exist…”

Section: Proposition 12 Suppose That the Artin-mazur Description Holmentioning

confidence: 99%

The idempotency of the real spectrum implies the extension theorem for Nash functions

Quarez

1998

Math Z

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“…La d6monstration du th6or6me de [21] cit6 ci-dessus fournissait m~me un plan pour aboutir au r6sultat. I1 ne manquait que deux ingr6dients de belle taille: -savoir borner en fonction de la dimension le nombre d'in6galit~s n6cessaires/t la description d'un ouvert semi-alg6brique, -savoir borner en fonction de la dimension le nombre n de carr6s tel que tout ~16ment totalement positif f divise un 616ment de la forme 1 + ~ x 2.…”

Section: ) Si Lest Un Corps De Fonctions Non Rdel De Degrk De Tranunclassified

“…Si q~ est une forme bilin6aire non d6g6n6r6e sur un anneau A, on notera ~:Spec, A~Z la fonction continue qui fi eeSpecrA associe ~b(e)eW(k(e))=Z (~ correspond fi un morphisme A~k(c 0 off k(c 0 est un corps r6el clos [21]). …”

Section: Lore Ktapeunclassified

Th�or�me de Pfister pour les vari�t�s et anneaux de Witt r�duits

Mahé

1986

Invent Math

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Signatures et composantes connexes

Cited by 27 publications

References 8 publications

Uniform bounds on complexity and transfer of global properties of Nash functions

Uniform bounds on complexity and transfer of global properties of Nash functions

The idempotency of the real spectrum implies the extension theorem for Nash functions

Th�or�me de Pfister pour les vari�t�s et anneaux de Witt r�duits

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