We deal with the following question of Dolgachev : is the Cremona group generated by involutions ? Answer is yes in dimension $2$ (Cerveau-Deserti). We give an upper bound of the minimal number $\mathfrak{ n}_\varphi$ of involutions we need to write a birational self map $\varphi$ of $\mathbb{ P}^2_\mathbb{ C}$. We prove that de Jonqui\`eres maps of $\mathbb{ P}^3_\mathbb{ C}$ and maps of small bidegree of $\mathbb{ P}^3_\mathbb{ C}$ can be written as a composition of involutions of $\mathbb{ P}^3_\mathbb{ C}$ and give an upper bound of $\mathfrak{ n}_\varphi$ for such maps $\varphi$. We get similar results in particular for automorphisms of $(\mathbb{ P}^1_\mathbb{ C})^n$, automorphisms of $\mathbb{ P}^n_\mathbb{ C}$, tame automorphisms of $\mathbb{ C}^n$, monomial maps of $\mathbb{ P}^n_\mathbb{ C}$, and elements of the subgroup generated by the standard involution of $\mathbb{ P}^n_\mathbb{ C}$ and $\mathrm{ PGL}(n+1,\mathbb{ C})$.
Categories: math.AG