  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.
