The Riemann mapping theorem for planar Nash rings

Gustave Adam Efroymson

Vol. 92 (1981), No. 1, 73–78

DOI: 10.2140/pjm.1981.92.73

Abstract

Recall f : D → R² is a Nash function if f is analytic and if there exists a polynomial p_f(x,y,z)≢0 with p_f(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 N_D≅N_R² where N_D = {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

Milestones

Received: 26 June 1978

Published: 1 January 1981