${\mathrm{SL}}_2$-equivariant polynomial automorphisms of the binary forms

Kurth, Alexandre

Annales de l'Institut Fourier, Tome 47 (1997) no. 2, pp. 585-597.

Soit $R_n:=ℂ[x,y{]}_n$ l’espace des formes binaires de degré $n\ge 1$. Nous montrons que chaque automorphisme polynomial de $R_n$ qui commute avec l’action linéaire de ${\mathrm{SL}}_2\left(ℂ\right)$ et qui conserve la variété des formes avec racines deux à deux distinctes, est un multiple scalaire de l’identité sur $R_n$.

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 $R_n$.

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