Vol. 92, No. 1, 1981

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The Riemann mapping theorem for planar Nash rings

Gustave Adam Efroymson

Vol. 92 (1981), No. 1, 73–78

Recall f : D R2 is a Nash function if f is analytic and if there exists a polynomial pf(x,y,z)0 with pf(x,y,f(x,y)) = 0 for all (x,y) in D. We wish to show Theorem 1: If D is a semi-algebraic simply connected, open domain in R2, then D is Nash isomorphic to R2. This means that there exists a map (x,y) (f1(x,y),f2(x,y)) where f1 and f2 are Nash functions on D and the map is a Nash isomorphism of D with R2.

As a corollary, we obviously get NDNR2 where ND = {f : D R|f is a Nash function on D}. Moreover, if D is any connected semi-algebraic domain in R2, it follows that D is Nash isomorphic to R2 minus n points where n = the number of holes in D. Here a domain is always considered to be open. The problem of classification of nonopen regions even in the plane is much more complicated and not settled as far as I know.

Mathematical Subject Classification 2000
Primary: 30F20
Secondary: 14E20, 14G30
Received: 26 June 1978
Published: 1 January 1981
Gustave Adam Efroymson