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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.

MSC:
14P20 Nash functions and manifolds
58A07 Real-analytic and Nash manifolds
32C07 Real-analytic sets, complex Nash functions
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References:
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