= Polynomial pencil with four square members
Let $u,v\in\mathbb C[t]$ be <coprime polynomials>. If $\alpha u+\beta v$ is a square for four distinct $(\alpha:\beta)$, then $u,v$ are constant. For independent $u,v$, the four square members $L_j=s_j^2$ are pairwise <coprime polynomials>. Let $d$ and $e$ be their maximal and minimal degrees; at least three have degree $d$. For independent members of degrees $e,d$, the nonzero <polynomial> $W=L_k'L_l-L_kL_l'$ has degree at most $d+e-1$. Each $s_j$ divides $W$, giving $\deg W\geq(3d+e)/2>d+e-1$, a contradiction. The dependent case follows directly from <coprimality of polynomials>.
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