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Nœther decomposition for birational maps

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