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This is a follow-up to this question: Quadratic squares

Consider now a polynomial in two variables $f(x, y).$ Are there bounds (upper or lower) for how many $x_0$ of height less than $N$ such that $f(x_0, y)$ is a square (of a polynomial)?

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I assume you are talking about integer coefficients but here is something you can say with coefficients in an arbitrary field $k$ (of characteristic zero, or perhaps $\ne 2$). Look at $X: z^2 = f(x,y)$ as defining an hyperelliptic curve (in the coordinates $x,z$) over $k(y)$, so if $f$ as a polynomial in $x$ over $k(y)$ has distinct roots and degree at least $3$, this is a hyperelliptic curve of positive genus. If this curve is non-isotrivial (i.e. it really depends on $y$) then by the function field analogue of Siegel's theorem, it has finitely many points with coordinates in $k[y]$ so, in particular, there are only finitely many $x_0 \in k$ such that $f(x_0,y)$ is the square of a polynomial in $y$, for if $f(x_0,y)=g(y)^2$, then $(x_0,g(y))$ is an integral point of $X$.

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