Uniform dual approximation to Veronese curves in small dimension

TL;DR

This paper improves upper bounds for uniform approximation exponents on Veronese curves in dimensions 3 to 9, using geometric methods, and compares these bounds with recent results by Poels.

Contribution

It refines bounds for approximation exponents on Veronese curves and demonstrates that previous bounds in dimension three are not sharp, using parametric geometry of numbers.

Findings

Improved upper bounds for dimensions 3 to 9

Previous bounds in dimension three are not sharp

Same bounds as Poels obtained conditionally

Abstract

We refine upper bounds for the classical exponents of uniform approximation for a linear form on the Veronese curve in dimension from $3$ to $9$. For dimension three, this in particular shows that a bound previously obtained by two different methods is not sharp. Our proof involves parametric geometry of numbers and investigation of geometric properties of best approximation polynomials. Slightly stronger bounds have been obtained by Poels with a different method contemporarily. In fact, we obtain the same bounds as a conditional result.