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Let $k$ be any number field (for simplicity we can take $k = \mathbb{Q}$, it doesn't really matter), and suppose we want to study the $k$-rational points on $$y^2 x = f(x),$$ where $f$ is a polynomial of degree greater or equal than 53. In other words, $y^2 x = f(x)$ is a sort of twisted elliptic or hyperelliptic curve. The only way I can think of dealing with this equation is to forget for a moment that the $x$ on the LHS is a variable, and deal instead with the genuine twist $y^2 d = f(x)$. After somehow solving this new equation, we can then check whether any of the finitely many solutions $\{x_i, y_i\}$ has $x_i = d$. For obvious reasons, however, this approach is useless (we'd have to check infinitely many values of $d$). This brings me to the questionQuestion: What are the techniques available (if any) to tackle equations like $y^2 x = f(x)$?

Let $k$ be any number field (for simplicity we can take $k = \mathbb{Q}$, it doesn't really matter), and suppose we want to study the $k$-rational points on $$y^2 x = f(x),$$ where $f$ is a polynomial of degree greater or equal than 5. In other words, $y^2 x = f(x)$ is a sort of twisted hyperelliptic curve. The only way I can think of dealing with this equation is to forget for a moment that the $x$ on the LHS is a variable, and deal instead with the genuine twist $y^2 d = f(x)$. After somehow solving this new equation, we can then check whether any of the finitely many solutions $\{x_i, y_i\}$ has $x_i = d$. For obvious reasons, however, this approach is useless (we'd have to check infinitely many values of $d$). This brings me to the question: What are the techniques available (if any) to tackle equations like $y^2 x = f(x)$?

Let $k$ be any number field, and suppose we want to study the $k$-rational points on $$y^2 x = f(x),$$ where $f$ is a polynomial of degree greater or equal than 3. In other words, $y^2 x = f(x)$ is a sort of twisted elliptic or hyperelliptic curve. Question: What are the techniques available to tackle equations like $y^2 x = f(x)$?

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user40810
user40810

Rational solutions of equations of the form $y^2 x = f(x)$

Let $k$ be any number field (for simplicity we can take $k = \mathbb{Q}$, it doesn't really matter), and suppose we want to study the $k$-rational points on $$y^2 x = f(x),$$ where $f$ is a polynomial of degree greater or equal than 5. In other words, $y^2 x = f(x)$ is a sort of twisted hyperelliptic curve. The only way I can think of dealing with this equation is to forget for a moment that the $x$ on the LHS is a variable, and deal instead with the genuine twist $y^2 d = f(x)$. After somehow solving this new equation, we can then check whether any of the finitely many solutions $\{x_i, y_i\}$ has $x_i = d$. For obvious reasons, however, this approach is useless (we'd have to check infinitely many values of $d$). This brings me to the question: What are the techniques available (if any) to tackle equations like $y^2 x = f(x)$?