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Nash functions on noncompact Nash manifolds. (English) Zbl 0981.14027
An affine Nash manifold in \(\mathbb{R}^n\) is a semialgebraic analytic submanifold of \(\mathbb{R}^n\). A Nash mapping between affine Nash manifolds is an analytic mapping with semi-algebraic graph. It was shown by M. Coste, J. M. Ruiz and M. Shiota [Compos. Math. 103, 31-62 (1996; Zbl 0885.14029)] that three important conjectures concerning Nash functions on affine Nash manifolds (separation, global equations and extension conjectures; last two of these conjectures are Nash versions of Cartan’s theorems A and B) are equivalent and that these three conjectures imply the factorization conjecture. An earlier article by M. Coste, J. M. Ruiz and M. Shiota [Am. J. Math. 117, 905-927 (1995; Zbl 0873.32007)] gave a positive answer to the conjectures in the case of compact affine Nash manifolds. In the present article, the authors prove the conjectures for all affine Nash manifolds.

14P20 Nash functions and manifolds
58A07 Real-analytic and Nash manifolds
32C07 Real-analytic sets, complex Nash functions
Full Text: DOI Numdam EuDML
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