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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 Superelliptic curve ： ウィキペディア英語版 Superelliptic curve In mathematics, a superelliptic curve is a plane curve with an equation of the form :$y^m\; =\; f(x),$ where the exponent ''m'' is fixed and ''f'' is a polynomial of degree ''d''. The case ''m'' = 2 and ''d'' > 4 is a ''hyperelliptic curve'': the case ''m'' = 3 and ''d'' = 4 is a ''trigonal curve''. The Diophantine problem of finding integer points on a superelliptic curve can be solved by a method similar to one used for the resolution of hyperelliptic equations: a Siegel identity is used to reduce to a Thue equation. == Definition == More generally, a ''superelliptic curve'' is a cyclic branched covering :$C\; \backslash to\; \backslash mathbb^1$ of the projective line of degree $m\; \backslash geq\; 2$ coprime to the characteristic of the field of definition. The degree $m$ of the covering map is also referred to as the degree of the curve. By ''cyclic covering'' we mean that the Galois group of the covering (i.e., the corresponding function field extension) is cyclic. The fundamental theorem of Kummer theory implies that a superelliptic curve of degree $m$ defined over a field $k$ has an affine model given by an equation :$y^m\; =\; f(x)$ for some polynomial $f\; \backslash in\; k()$ of degree $m$ with each root having order , provided that $C$ has a point defined over $k$, that is, if the set $C(k)$ of $k$-rational points of $C$ is not empty. For example, this is always the case when $k$ is algebraically closed. In particular, function field extension $k(C)/k(x)$ is a Kummer extension. 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「Superelliptic 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.