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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 Weierstrass point ： ウィキペディア英語版 Weierstrass point In mathematics, a Weierstrass point $P$ on a nonsingular algebraic curve $C$ defined over the complex numbers is a point such that there are more functions on $C$, with their poles restricted to $P$ only, than would be predicted by the Riemann–Roch theorem. That is, looking at the vector spaces :$L(0),\; \backslash \; L(P),\; \backslash \; L(2P),\; \backslash \; L(3P),\; \backslash ldots$ where $L(kP)$ is the space of meromorphic functions on $C$ whose order at $P$ is at least − $k$ and with no other poles. The concept is named after Karl Weierstrass. We know three things: the dimension is at least 1, because of the constant functions on $C$, it is non-decreasing, and from the Riemann–Roch theorem the dimension eventually increments by exactly 1 as we move to the right. In fact if $g$ is the genus of $C$, the dimension from the $k$-th term is known to be :$l(kP)\; =\; k\; -g\; +\; 1$, for $k\; \backslash geq;\; 2g-\; 1$. Our knowledge of the sequence is therefore :1, ?, ?, ..., ?, ''g'', ''g'' + 1, ''g'' + 2, ... . What we know about the ? entries is that they can increment by at most 1 each time (this is a simple argument: if $f$ and $g$ have the same order of pole at $P$, then $f+cg$ will have a pole of lower order if the constant $c$ is chosen to cancel the leading term). There are :$2g-2$ question marks here, so the cases $g=0$ or $1$ need no further discussion and do not give rise to Weierstrass points. Assume therefore $g\; \backslash geq\; 2$. There will be $g-1$ steps up, and $g-1$ steps where there is no increment. A non-Weierstrass point of $C$ occurs whenever the increments are all as far to the right as possible: i.e. the sequence looks like :1, 1, ..., 1, 2, 3, 4, ..., ''g'' − 1, ''g'', ''g'' + 1, ... . Any other case is a Weierstrass point. A Weierstrass gap for $P$ is a value of $k$ such that no function on $C$ has exactly a $k$-fold pole at $P$ only. The gap sequence is :$1,\; \backslash \; 2,\; \backslash \; \backslash ldots,\; \backslash \; g$ for a non-Weierstrass point. For a Weierstrass point it contains at least one higher number. (The Weierstrass gap theorem or Lückensatz is the statement that there must be $g$ gaps.) For hyperelliptic curves, for example, we may have a function $F$ with a double pole at $P$ only. Its powers have poles of order $4,\; \backslash \; 6$ and so on. Therefore, such a $P$ has the gap sequence :1, 3, 5, ..., 2''g'' − 1. In general if the gap sequence is :$a,\; \backslash \; b,\; \backslash \; c,\; \backslash \; \backslash ldots$ the weight of the Weierstrass point is :$(a-1)+(b-2)+(c-3)+\backslash ldots$ This is introduced because of a counting theorem: on a Riemann surface the sum of the weights of the Weierstrass points is :$g(g^2-1)$. For example, a hyperelliptic Weierstrass point, as above, has weight ''g''(''g'' − 1)/2. Therefore, there are (at most) 2(''g'' + 1) of them; as those can be found (for example, the six points of ramification when ''g'' = 2 and $C$ is presented as a ramified covering of the projective line) this exhausts all the Weierstrass points on $C$. Further information on the gaps comes from applying Clifford's theorem. Multiplication of functions gives the non-gaps a semigroup structure, and an old question of Adolf Hurwitz asked for a characterization of the semigroups occurring. A new necessary condition was found by Buchweitz in 1980, and he gave an example of a subsemigroup of the nonnegative integers with 16 gaps that does not occur as the semigroup of non-gaps at a point on a curve of genus 16. A definition of Weierstrass point for a nonsingular curve over a field of positive characteristic was given by F. K. Schmidt in 1939. ==References== * * 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「Weierstrass point」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.