  Let $n$ be a positive integer and $\xi$ a transcendental real number. We are
interested in bounding from above the uniform exponent of polynomial
approximation $\widehat{\omega}_n(\xi)$. Davenport and Schmidt's original 1969
inequality $\widehat{\omega}_n(\xi)\leq 2n-1$ was improved recently, and the
best upper bound known to date is $2n-2$ for each $n\geq 10$. In this paper, we
develop new techniques leading us to the improved upper bound
$2n-\frac{1}{3}n^{1/3}+\mathcal{O}(1)$.
