Hyperelliptic curve

In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus g > 1, given by an equation of the form $${\displaystyle y^{2}+h(x)y=f(x)}$$ where f(x) is a polynomial of degree n = 2g + 1 > 4 or n = 2g + 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 such a curve, or of the Jacobian variety on the curve; these two concepts are identical for elliptic functions, but different for hyperelliptic functions.