Real hyperelliptic curve について

satisfy certain conditions. There are two types of hyperelliptic curves: real hyperelliptic curves and imaginary hyperelliptic curves which differ by the number of points at infinity. In this page, we describe more about real hyperelliptic curves, these are curves having two points at infinity while imaginary hyperelliptic curves have one point at infinity.
==Definition==
A real hyperelliptic curve of genus ''g'' over ''K'' is defined by an equation of the form

C:y^2+h(x)y=f(x)

where

h(x) \in K

has degree not larger than ''g+1'' while

f(x) \in K

must have degree ''2g+1'' or ''2g+2''. This curve is a non singular curve where no point

(x,y)

in the algebraic closure of

K

satisfies the curve equation

y^2+h(x)y=f(x)

and both partial derivative equations:

2y+h(x)=0

and

h'(x)y=f'(x)

.
The set of (finite)

K

–rational points on ''C'' is given by
:

C(K) = \ \cup  S

Where

S

is the set of points at infinity. For real hyperelliptic curves, there are two points at infinity,

\infty_1

and

\infty_2

. For any point

P(a,b)\in C(K)

, the opposite point of

P

is given by

\overline = (a, -b-h)

; it is the other point with ''x''-coordinate ''a'' that also lies on the curve.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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