Superelliptic curve
In mathematics, a superelliptic curve is a plane curve with an equation of the form
where the exponent m is fixed and f is a polynomial of degree d. The case m = 2 and d > 4 is a hyperelliptic curve: the case m = 3 and d = 4 is a trigonal curve.
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.
More generally, a superelliptic curve is a cyclic branched covering
of the projective line of degree coprime to the characteristic of the field of definition. The degree of the covering map is also referred to as the degree of the curve. By cyclic covering we mean that the Galois group of the covering (i.e., the corresponding function field extension) is cyclic.
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