N\oe ther decomposition for birational maps

Let $\phi$ be a birational map of the complex projective plane. We know that $\phi$ can be written as a composition of automorphisms of $\mathbb{P}^2_\mathbb{C}$ and the standard quadratic birational map $\sigma$. This writing, that is non-unique, is minimal if the number $\mathfrak{n}(\phi)$ of $\sigma$ is as small as possible. We prove that if $\phi$ is of degree $d\geq 2$, then $\big[\frac{\ln d}{\ln 2}\big]\leq\mathfrak{n}(\phi)\leq 2(2d-1)$.