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Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.) We are ...

According to several sources, it is conjectured (or at least believed) that the rational points of curves over the rationals of genus $g > 1$ are uniformly bounded by $g$. E.g. here p. 1. Assuming ...

Let $f(x)=x^4+b_3 x^3+ b_2 x^2+b_1 x + b_0$. Let $g(x)=x^4 f(1/x)$. Let $C : g(x)=y^2$. $C$ is birationally equivalent to $f(x)=y^2$. The constant coefficient of $g(x)$ is $1$ since $f$ is monic and $(...