In real algebraic geometry, a Nash function on an open semialgebraic subset U ⊂ Rn is an analytic function f: U → R satisfying a nontrivial polynomial equation P(x,f(x)) = 0 for all x in U (A semialgebraic subset of Rn is a subset obtained from subsets of the form {x in Rn : P(x)=0} or {x in Rn : P(x) > 0}, where P is a polynomial, by taking finite unions, finite intersections and complements).
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 (i.e., those series satisfying a nontrivial polynomial equation), which is the henselization of the ring of germs of rational functions.
In particular, it is a regular local ring of dimension n. The global properties are more difficult to obtain.
The fact that the ring of Nash functions on a Nash manifold (even noncompact) is noetherian was proved independently (1973) by Jean-Jacques Risler and Gustave Efroymson.
is globally generated by Nash functions on M, and the natural map is surjective.
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.