Rational approximation to surfaces defined by polynomials in one variable

TL;DR

This paper investigates how well certain polynomial-defined surfaces can be approximated by rational functions, focusing on manifolds formed by polynomial curves with potential implications for more complex polynomial surfaces.

Contribution

It extends recent results on polynomial curves to manifolds formed by polynomials in one variable, offering a new approach to rational approximation of such surfaces.

Findings

Analysis of rational approximation properties for polynomial manifolds

Generalization of results from polynomial curves to higher-dimensional manifolds

Potential methods for approximating more complex polynomial-defined surfaces

Abstract

We study the rational approximation properties of special manifolds defined by a set of polynomials with rational coefficients. Mostly we will assume the case of all polynomials to depend on only one variable. In this case the manifold can be viewed as a Cartesian product of polynomial curves and it is possible to generalize recent results concerning such curves with similar concepts. There is hope that the method leads to insights on how to treat more general manifolds defined by arbitrary polynomials with rational coefficients.