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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Imaginary hyperelliptic curve ： ウィキペディア英語版 Imaginary hyperelliptic curve A hyperelliptic curve is a particular kind of algebraic curve. There exist hyperelliptic curves of every genus $g\; \backslash geq\; 1$. If the genus of a hyperelliptic curve equals 1, we simply call the curve an elliptic curve. Hence we can see hyperelliptic curves as generalizations of elliptic curves. There is a well-known group structure on the set of points lying on an elliptic curve over some field $K$, which we can describe geometrically with chords and tangents. Generalizing this group structure to the hyperelliptic case is not straightforward. We cannot define the same group law on the set of points lying on a hyperelliptic curve, instead a group structure can be defined on the so-called Jacobian of a hyperelliptic curve. The computations differ depending on the number of points at infinity. This article is about imaginary hyperelliptic curves, these are hyperelliptic curves with exactly 1 point at infinity. Real hyperelliptic curves have two points at infinity. == Formal definition == Hyperelliptic curves can be defined over fields of any characteristic. Hence we consider an arbitrary field $K$ and its algebraic closure $\backslash overline$. An (imaginary) hyperelliptic curve of genus $g$ over $K$ is given by an equation of the form $$ C : y^2 + h(x) y = f(x) \in K() where $h(x)\; \backslash in\; K()$ is a polynomial of degree not larger than $g$ and $f(x)\; \backslash in\; K()$ is a monic polynomial of degree $2g\; +\; 1$. Furthermore we require the curve to have no singular points. In our setting, this entails that no point $(x,y)\; \backslash in\; \backslash overline\; \backslash times\; \backslash overline$ satisfies both $y^2\; +\; h(x)\; y\; =\; f(x)$ and the equations $2y\; +\; h(x)\; =\; 0$ and $h\text{'}(x)y\; =\; f\text{'}(x)$. This definition differs from the definition of a general hyperelliptic curve in the fact that $f$ can also have degree $2g+2$ in the general case. From now on we drop the adjective imaginary and simply talk about hyperelliptic curves, as is often done in literature. Note that the case $g\; =\; 1$ corresponds to $f$ being a cubic polynomial, agreeing with the definition of an elliptic curve. If we view the curve as lying in the projective plane $\backslash mathbb^2(K)$ with coordinates $(X:Y:Z)$, we see that there is a particular point lying on the curve, namely the point at infinity $(0:1:0)$ denoted by $O$. So we could write $C\; =\; \backslash \; \backslash cup\; \backslash$. Suppose the point $P\; =\; (a,b)$ not equal to $O$ lies on the curve and consider $\backslash overline\; =\; (a,\; -b-h(a))$. As $(-b-h(a))^2\; +\; h(a)(-b-h(a))$ can be simplified to $b^2\; +h(a)\; b$, we see that $\backslash overline$ is also a point on the curve. $\backslash overline$ is called the opposite of $P$ and $P$ is called a Weierstrass point if $P\; =\; \backslash overline$, i.e. $h(a)\; =\; -2b$. Furthermore, the opposite of $O$ is simply defined as $\backslash overline\; =\; O$. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Imaginary hyperelliptic curve」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.