Monograph

# Polynomial Density Of Compact Smooth Surfaces

## Degree

Doctor of Philosophy

Mathematics

Shafikov, Rasul

## Abstract

We show that any smooth closed surface has polynomial density 3 and that any connected compact smooth surface with boundary has polynomial density 2.

## Summary for Lay Audience

We assume any surfaces we consider are all in one piece. The distance between a pair of functions is defined to be the maximum over all points of the distance between their values. (When choosing functions of a particular form to approximate a given function, this distance can be thought of as an error tolerance.) We say that a specific function f can be approximated by a set of functions F if we can always find a function from F within the distance specified, no matter how small we chose the distance to be.

We say a surface has polynomial density n if there exist smooth functions g_1, g_2, ..., g_n such that the set of polynomials in them can approximate any continuous function on the surface. We confirm that for a surface without a boundary component, a closed surface, the polynomial density is three. On the other hand, for a surface with a boundary component, we show that the polynomial density must be two.