Hyperelliptic Curve

A hyperelliptic curve is an algebraic curve given by an equation of the form y^2=f(x), where f(x) is a polynomial of degree n>4 with n distinct roots.

If f(x) is a cubic or quartic polynomial, then the curve is called an elliptic curve.

The genus of a hyperelliptic curve is related to the degree of the polynomial. A polynomial of degree 2g+1 or 2g+2 gives a curve of genus g. All curves of genus 2 are hyperelliptic, but for larger genus there are curves that are not hyperelliptic.

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This entry contributed by Alan Simpson

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Simpson, Alan. "Hyperelliptic Curve." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HyperellipticCurve.html

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