The tangent cones $\alpha X^m+\bar\alpha Y^m$ and $u^m+v^m$ each vanish at exactly $m$ points of $\mathbb{P}^1$

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The tangent cones $\alpha X^m+\bar\alpha Y^m$ and $u^m+v^m$ each vanish at exactly $m$ points of $\mathbb{P}^1$ · Prove2Me