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Superelliptic curve information


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

where is an integer and f is a polynomial of degree with coefficients in a field ; more precisely, it is the smooth projective curve whose function field defined by this equation. The case and is an elliptic curve, the case and is a hyperelliptic curve, and the case and is an example of a trigonal curve.

Some authors impose additional restrictions, for example, that the integer should not be divisible by the characteristic of , that the polynomial 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.

  1. ^ Galbraith, S.D.; Paulhus, S.M.; Smart, N.P. (2002). "Arithmetic on superelliptic curves". Mathematics of Computation. 71: 394–405. doi:10.1090/S0025-5718-00-01297-7. MR 1863009.

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