Uniform dual approximation to Veronese curves in small dimension

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.