  Let $\zeta$ be a real transcendental number. We introduce a new method to
find upper bounds for the classical exponent $\widehat{w}_{n}(\zeta)$
concerning uniform polynomial approximation. Our method is based on the
parametric geometry of numbers introduced by Schmidt and Summerer, and
transference of the original approximation problem in dimension $n$ to suitable
higher dimensions. For large $n$, we can provide an unconditional bound of
order $\widehat{w}_{n}(\zeta)\leq 2n-2+o(1)$. While this improves the bound of
order $2n-\frac{3}{2}+o(1)$ due to Bugeaud and the author, it is unfortunately
slightly weaker than what can be obtained when incorporating a recently proved
conjecture of Schmidt and Summerer. However, the method also enables us to
establish significantly stronger conditional bound upon a certain presumably
weak assumption on the structure of the best approximation polynomials. Thereby
we provide serious evidence that the exponent should be reasonably smaller than
the known upper bounds.
