Editing Hyperelliptic curve (section)

{{Short description|Algebraic curve}}
[[File:Example of a hyperelliptic curve.svg|right|thumb|Fig. 1: The graph of the hyperelliptic curve <math>C : y^2 = f(x)</math> where
<math display="block">f(x) = x^5 - 2x^4 - 7x^3 + 8x^2 + 12x = x (x + 1) (x - 3) (x + 2) (x - 2). </math>
]]
In [[algebraic geometry]], a '''hyperelliptic curve''' is an [[algebraic curve]] of [[Genus (mathematics)|genus]] ''g'' > 1, given by an equation of the form
<math display="block">y^2 + h(x)y = f(x)</math>
where ''f''(''x'') is a [[polynomial]] of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' distinct roots, and ''h''(''x'') is a polynomial of degree < ''g'' + 2 (if the characteristic of the ground field is not 2, one can take ''h''(''x'') = 0).

A '''hyperelliptic function''' is an element of the [[function field of an algebraic variety|function field]] of such a curve, or of the [[Jacobian variety]] on the curve; these two concepts are identical for [[elliptic function]]s, but different for hyperelliptic functions.