An example concerning rational convexity

Abstract

A rat ionally convex domain, G, in C ~, is a domain of ho lomorphy in C n such tha t every analyt ic funct ion on G can be uniformly approximated, on compac t subsets, by rat ional functions of the coordinates. I f these rational functipns can always be taken to be polynomials, then G is a Runge domain. I n 1941, in Domaines d'holomorphie et domaines rationellement eonvexes (Japan. J . Math. 17, 517--521), K. 0KA presented an example of a bounded domain of ho lomorphy in C ~ which is no t rat ionally convex. This domain is no t simply-connected. I n 1960, in Addendum to " A n Example Concerning Polynomial Convexity" (Math. Ann. 140, 322--323), J. WERMER constructed a bounded domain of ho lomorphy in C a which is not a Runge domain, bu t which is analyt ical ly homeomorphic with a polycylinder. I n this note, we show tha t Wermer ' s domain is also no t ra t ional ly convex. Theorem. There is a bounded domain o/ holomorphy in C a which is analytically homeomorphic with the polycylinder Izil max(IR(y) l : y ~ K}. Express R as g/h, where 9 and h are relatively prime polynomials, and set / = g h. Then /(x) = 0, bu t / does no t vanish a t any point of K. Since K is contractible, there is a continuous, single-valued, branch of log (/) defined on K. I f we recall t h a t the circle, T, lies in K, and now apply the argument principle to / on D, we find tha t