A Geometric Approach to Polynomial and Rational Approximation

We strengthen the classical approximation theorems of Weierstrass, Runge, and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function f on a compact set K, the critical points of our approximants may be taken to lie in any given domain containing K, and all the critical values in any given neighborhood of the polynomially convex hull of f(K).