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



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Coste, M. and Ruiz Sancho, Jesús María and Shiota, Masahiro (2001) Uniform bounds on complexity and transfer of global properties of Nash functions. Journal fur die Reine und Angewandte Mathematik, 536 . pp. 209-235. ISSN 0075-4102

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Official URL: http://perso.univ-rennes1.fr/michel.coste/publis/TransNash.pdf


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.

Item Type:Article
Uncontrolled Keywords:Nash functions; real spectrum; manifolds; affine Nash manifold; complexity of Nash functions; Tarski-Seidenberg principle; separation problem
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:19835
Deposited On:07 Feb 2013 11:27
Last Modified:12 Dec 2018 15:13

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