N\oe ther decomposition for birational maps

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