Vol. 63, No. 1, 1976

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Substitution in Nash functions

Gustave Adam Efroymson

Vol. 63 (1976), No. 1, 137–145

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.

Mathematical Subject Classification 2000
Primary: 14A99
Secondary: 32C05
Received: 12 April 1975
Published: 1 March 1976
Gustave Adam Efroymson