Orderings On Real Surfaces

Abstract

After describing explicitly all total orderings in the ring R[[x, y]], we prove that each ordering in the quotient field of the ring of germs of real analytic functions at an irreducible point O of a real analytic surface X is defined by a half-branch of the germ at O of some curve on X, which is analytic off the origin. Then follows an analogous result for real algebraic surfaces.