Uniform bounds on complexity and transfer of global properties of Nash functions (Q2729305)

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 semialgebraic graph. Several important global results (solutions to separation, global equations, extension and factorization problems) concerning Nash functions on affine Nash manifolds were obtained by \textit{M. Coste}, \textit{J. M. Ruiz} and \textit{M. Shiota} [Am. J. Math. 117, 905-927 (1995; Zbl 0873.32007) and Compos. Math. 103, 31-62 (1996; Zbl 0885.14029)] and by \textit{M. Coste} and \textit{M. Shiota} [Ann. Sci. Éc. Norm. Super., IV. Ser. 33, 139-149 (2000; see the preceding review Zbl 0981.14027)]. In the present article the authors prove uniform bounds on the complexity of Nash functions and (using the Tarski-Seidenberg principle) obtain a solution to extension and global equations problems over arbitrary real closed fields. This allows 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 problems in this setting. The authors also prove the idempotency of the real spectrum and an abstract version of the separation problem. Conditions for the rings of abstract Nash functions to be noetherian are discussed.