# Polynomial convexity.

1. Nov 25, 2008

### lark

I've been reading Gunning and Rossi's book on complex analysis in several variables (good book!).
They define something called "polynomial convexity" for a domain D. "domain" = connected open set, I think.
The point of polynomial convexity is that if D $$\subset C^n$$ is polynomially convex, then an analytic function on D can be approximated by polynomials, and the approximation is uniform on compact subsets of D.
Is the converse true? i.e. if D isn't polynomially convex, is there a function that's analytic on D that can't be approximated by polynomials uniformly on compact subsets?
If D isn't polynomially convex, that means there's a compact subset K of D and a point w $$\notin$$ D such that
|p(w)| $$\leq$$ max(|p(z)|, z $$\in$$ K), for all polynomials p.
Laura

Last edited: Nov 25, 2008
2. Nov 27, 2008

### lark

I got an answer to this. The converse isn't true in $$C^n, n >1$$, but it's true in C.