Along with Nash functions one defines Nash manifolds, which are semialgebraic analytic submanifolds of some Rn. A Nash mapping between Nash manifolds is then an analytic mapping with semialgebraic graph. Nash functions and manifolds are named after John Forbes Nash, Jr., who proved that any compact smooth manifold admits a Nash manifold structure, i.e., is diffeomorphic to some Nash manifold. More generally, a smooth manifold admits a Nash manifold structure if and only if it is diffeomorphic to the interior of some compact smooth manifold possibly with boundary. Nash's result was later completed by Alberto Tognoli who proved that any compact smooth manifold is diffeomorphic to some affine real algebraic manifold; actually, any Nash manifold is Nash diffeomorphic to an affine real algebraic manifold. These results exemplify the fact that the Nash category is somewhat intermediate between the smooth and the algebraic categories.
Local properties
The local properties of Nash functions are well understood. The ring of germs of Nash functions at a point of a Nash manifold of dimension n is isomorphic to the ring of algebraic power series in n variables, which is the henselization of the ring of germs of rational functions. In particular, it is a regular local ring of dimension n.
Global properties
The global properties are more difficult to obtain. The fact that the ring of Nash functions on a Nash manifold is noetherian was proved independently by Jean-Jacques Risler and Gustave Efroymson. Nash manifolds have properties similar to but weaker than Cartan's theorems A and B on Stein manifolds. Let denote the sheaf of Nash function germs on a Nash manifold M, and be a coherent sheaf of -ideals. Assume is finite, i.e., there exists a finite open semialgebraic covering of M such that, for each i, is generated by Nash functions on. Then is globally generated by Nash functions on M, and the natural map is surjective. However contrarily to the case of Stein manifolds.
Generalizations
Nash functions and manifolds can be defined over any real closed field instead of the field of real numbers, and the above statements still hold. Abstract Nash functions can also be defined on the real spectrum of any commutative ring.
