We consider the space of binary forms of degree $n\ge 1$ denoted by $R_n:=ℂ[x,y{]}_n$. We will show that every polynomial automorphism of $R_n$ which commutes with the linear ${\mathrm{SL}}_2\left(ℂ\right)$-action and which maps the variety of forms with pairwise distinct zeroes into itself, is a multiple of the identity on $R_n$.

On s’intéresse aux difféomorphismes birationnels des surfaces algébriques réelles qui possèdent une dynamique réelle simple et une dynamique complexe riche. On donne un exemple d’une telle transformation sur ${\mathbb{P}}^1\times {\mathbb{P}}^1$, mais on montre qu’une telle situation est exceptionnelle et impose des conditions fortes à la fois sur la topologie du lieu réel et sur la dynamique réelle.

We describe the structure of the group of algebraic automorphisms of the following surfaces 1) P1,k x P1,k minus a diagonal; 2) P1,k x P1,k minus a fiber. The motivation is to get a new proof of two theorems proven respectively by L. Makar-Limanov and H. Nagao. We also discuss the structure of the semi-group of polynomial proper maps from C2 to C2.