On Orderings In Real Surfaces

Abstract

It is well-known that if C is an algebraic curve over the real closed field R and is a total ordering of the function field R(C) of C then there is a semi-algebraic embedding w : (0, 1) ! C such that f 2 R(C) is positive with respect to if and only if there is some t 2 R, 0 < t such that fw is defined and positive on (0,t). In the present paper it is shown that the total orderings of the function field of an algebraic surface over the field R of real numbers admits a similar geometric description. Let V be an irreducible algebraic surface over R embedded in some Rn. Using a discussion of the orderings of the meromorphic function germs of an irreducible analytic surface germ the following is proved: If is a total ordering of R(V ) then there is an analytic map c : (0, 1) ! V such that f 2 R(V ) is positive with respect to if and only if fc is defined and positive on (0,t) for some 0 < t 2 R.