A Geometric Approach to Polynomial and Rational Approximation.

The article focuses on strengthening classical complex approximation theorems—specifically those of Weierstrass, Runge, and Mergelyan—by constructing polynomial and rational approximants with controlled geometric structures. It establishes that for a function \( f \) analytic on a compact set \( K \subset \mathbb{C} \), one can approximate \( f \) uniformly by polynomials or rational functions whose critical points lie within any prescribed domain containing \( K \), and whose critical values lie near the polynomially convex hull of \( f(K) \). The approach involves approximating \( f \) by proper holomorphic maps on analytic domains, extending these to quasiregular mappings on the Riemann sphere, and then applying the Measurable Riemann Mapping Theorem to obtain rational approximants with detailed control over their critical points, critical values, and poles. The results include polynomial approximations when \( K \) is full and rational approximations otherwise, with explicit descriptions of the approximants’ global geometric behavior and critical structures.