Uniform Dual Approximation to Veronese Curves in Small Dimension

We refine upper bounds for the classical exponents of uniform ap- proximation 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 poly- nomials. Slightly stronger bounds have been obtained by Poels with a different method contemporarily. In fact, we obtain his bounds as a conditional result.