Superelliptic curve information

In mathematics, a superelliptic curve is an algebraic curve defined by an equation of the form

${\displaystyle y^{m}=f(x),}$

where ${\displaystyle m\geq 2}$ is an integer and f is a polynomial of degree ${\displaystyle d\geq 3}$ with coefficients in a field ${\displaystyle k}$; more precisely, it is the smooth projective curve whose function field defined by this equation. The case ${\displaystyle m=2}$ and ${\displaystyle d=3}$ is an elliptic curve, the case ${\displaystyle m=2}$ and ${\displaystyle d\geq 5}$ is a hyperelliptic curve, and the case ${\displaystyle m=3}$ and ${\displaystyle d\geq 4}$ is an example of a trigonal curve.

Some authors impose additional restrictions, for example, that the integer ${\displaystyle m}$ should not be divisible by the characteristic of ${\displaystyle k}$, that the polynomial ${\displaystyle f}$ should be square free, that the integers m and d should be coprime, or some combination of these.^[1]

The Diophantine problem of finding integer points on a superelliptic curve can be solved by a method similar to one used for the resolution of hyperelliptic equations: a Siegel identity is used to reduce to a Thue equation.