Rational approximation and Lagrangian inclusions

TL;DR

This paper proves that certain Lagrangian inclusions in complex 2-space are rationally convex and demonstrates that most compact surfaces can be approximated uniformly by rational functions composed with two smooth complex functions.

Contribution

It establishes the rational convexity of specific Lagrangian inclusions and shows that most compact surfaces admit uniform rational approximation via two smooth functions.

Findings

Lagrangian inclusions with specific singularities are rationally convex

Most compact surfaces (except S^2 and RP^2) admit rational approximation

Any continuous function on these surfaces can be uniformly approximated by rational functions in two variables

Abstract

We show that a Lagrangian inclusion in $\mathbb C^2$ with double transverse self-intersection points and standard open Whitney umbrellas is rationally convex. As an application we show that any compact surface $S$, except $S^2$ and $\mathbb RP_2$, admits a pair of smooth complex-valued functions $f_1$, $f_2$ with the property that any continuous complex valued function on $S$ is a uniform limit of a sequence of $R_j(f_1,f_2)$, where $R_j(z_1,z_2)$ are rational functions in $\mathbb C^2$.