Superelliptic curve

In mathematics, a superelliptic curve is an algebraic curve defined by an equation of the form
where formula_2 is an integer and "f" is a polynomial of degree formula_3 with coefficients in a field formula_4; more precisely, it is the smooth projective curve whose function field defined by this equation.
The case formula_5 and formula_6 is an "elliptic curve", the case formula_5 and formula_8 is a "hyperelliptic curve", and the case formula_9 and formula_10 is an example of a "trigonal curve".
Some authors impose additional restrictions, for example, that the integer formula_11 should not be divisible by the characteristic of formula_4, that the polynomial formula_13 should be square free, that the integers "m" and "d" should be coprime, or some combination of these.
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.
Definition.
More generally, a "superelliptic curve" is a cyclic branched covering
of the projective line of degree formula_2 coprime to the characteristic of the field of definition. The degree formula_11 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.
The fundamental theorem of Kummer theory implies that a superelliptic curve of degree formula_11 defined over a field formula_4 has an affine model given by an equation
for some polynomial formula_20 of degree formula_11 with each root having order formula_22, provided that formula_23 has a point defined over formula_4, that is, if the set formula_25 of formula_4-rational points of formula_23 is not empty. For example, this is always the case when formula_4 is algebraically closed. In particular, function field extension formula_29 is a Kummer extension.
Ramification.
Let formula_30 be a superelliptic curve defined over an algebraically closed field formula_4, and formula_32 denote the set of roots of formula_13 in formula_4. Define set
Then formula_36 is the set of branch points of the covering map formula_14 given by formula_38.
For an affine branch point formula_39, let formula_40 denote the order of formula_41 as a root of formula_13. As before, we assume that formula_43. Then
is the ramification index formula_45 at each of the formula_46 ramification points formula_47 of the curve lying over formula_48 (that is actually true for any formula_49).
For the point at infinity, define integer formula_50 as follows. If 
then formula_52. Note that formula_53. Then analogously to the other ramification points,
is the ramification index formula_55 at the formula_56 points formula_57 that lie over formula_58. In particular, the curve is unramified over infinity if and only if its degree formula_11 divides formula_60.
Curve formula_23 defined as above is connected precisely when formula_11 and formula_40 are relatively prime (not necessarily pairwise), which is assumed to be the case.
Genus.
By the Riemann-Hurwitz formula, the genus of a superelliptic curve is given by