# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] J. BOCHNAK , M. COSTE and M.-F. ROY , Real Algebraic Geometry , Springer, Berlin, 1998 . MR 2000a:14067 | Zbl 0912.14023 · Zbl 0912.14023 [2] M. COSTE , J. RUIZ and M. SHIOTA , Approximation in compact Nash manifolds , Amer. J. Math. 117 ( 1995 ) 905-927. MR 96f:32015 | Zbl 0873.32007 · Zbl 0873.32007 · doi:10.2307/2374953 [3] M. COSTE , J. RUIZ and M. SHIOTA , Separation, factorization and finite sheaves on Nash manifolds , Compositio Math. 103 ( 1996 ) 31-62. Numdam | MR 97g:14039 | Zbl 0885.14029 · Zbl 0885.14029 · numdam:CM_1996__103_1_31_0 · eudml:90460 [4] M. COSTE and M. SHIOTA , Thom’s first isotopy lemma : semialgebraic version, with uniform bound , in : F. Broglia, ed., Real Analytic and Algebraic Geometry, De Gruyter, 1995 , pp. 83-101. MR 96i:14047 | Zbl 0844.14025 · Zbl 0844.14025 [5] M. SHIOTA , Nash Manifolds , Lecture Notes in Math., Vol. 1269, Springer, Berlin, 1987 . MR 89b:58011 | Zbl 0629.58002 · Zbl 0629.58002 [6] J.-C. TOUGERON , Idéaux de Fonctions Différentiables , Springer, Berlin, 1972 . MR 55 #13472 | Zbl 0251.58001 · Zbl 0251.58001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.