Let D be a domain in Rn. In
this paper D is assumed to be defined by a finite number of strict polynomial
inequalities. A Nash function on D is a real valued analytic function f(x) such that
there exists a polynomial p(z,x1,⋯,xn) in R[z,x1,⋯,xn] such that p(f(x),x) = 0 for
all x in D. Let AD be the ring of such functions on D. For any real closed field L
containing R, use the Tarski-Seidenberg theorem to extend f to a function
from a domain DL (defined by the same inequalities as D), DL⊆ L(n),
to L. Now let φ : AD→ L be a homomorphism. Since R[x1,⋯,xn] ⊂ AD,
φx = (φx1,⋯,φxn) is a well defined point in L(n) and is in DL. So f(φx) is
defined for any f in AD. In this paper it is shown that f(φx) = φf. From this
result one can deduce Mostowski’s version of the Hilbert Nullstellensatz for
AD.