Let $d\in\mathbb{N}$. We consider the vector space $V=\mathbb{C}^2\otimes\mathbb{C}[x_0,x_1]_d$ where $\mathbb{C}[x_0,x_1]_d$ is the space of homogeneous binary forms of degree $d$. We have a natural action of $G=\textrm{SL}_2(\mathbb{C})\times \textrm{SL}_2(\mathbb{C})$ on $V$ where the second copy acts by a linear change of coordinates on $x_0,x_1$. Is it possible to write down explicit normal forms for this group action?
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