Diophantine approximation on polynomial curves

TL;DR

This paper generalizes previous results on rational approximation of polynomial curves in higher-dimensional Euclidean spaces, focusing on non-degenerate polynomial parametrizations.

Contribution

It extends earlier work by Budarina, Dickinson, and Levesley to higher dimensions and broader polynomial conditions, excluding certain degenerate cases.

Findings

Established new approximation bounds for polynomial curves in dimensions ≥ 3

Generalized previous results to broader classes of polynomial parametrizations

Excluded cases of linear dependence and trivial polynomial cases

Abstract

In a paper from 2010, Budarina, Dickinson and Levesley studied the rational approximation properties of curves parametrized by polynomials with integral coefficients in Euclidean space of arbitrary dimension. Assuming the dimension is at least three and excluding the case of linear dependence of the polynomials together with $P(X)\equiv 1$ over the rational number field, we establish proper generalizations of their main result.