| 书目名称 | On Some Applications of Diophantine Approximations | | 副标题 | A translation of C.L | | 编辑 | Umberto Zannier | | 视频video | | | 概述 | Presents for the first time in English a landmark paper by Siegel which introduced most important ideas in the realm of Diophantine approximation and its applications.Presents the ideas of Siegel in o | | 丛书名称 | Publications of the Scuola Normale Superiore | | 图书封面 |  | | 描述 | This book consists mainly of the translation, by C. Fuchs, of the 1929 landmark paper "Über einige Anwendungen diophantischer Approximationen" by C.L. Siegel. The paper contains proofs of most important results in transcendence theory and diophantine analysis, notably Siegel’s celebrated theorem on integral points on algebraic curves. Many modern versions of Siegel’s proof have appeared, but none seem to faithfully reproduce all features of the original one. This translation makes Siegel’s original ideas and proofs available for the first time in English. The volume also contains the original version of the paper (in German) and an article by the translator and U. Zannier, commenting on some aspects of the evolution of this field following Siegel’s paper. To end, it presents three modern proofs of Siegel’s theorem on integral points. | | 出版日期 | Book 2014 | | 关键词 | Siegel‘s theorem; diophantine analysis; transcendence theory | | 版次 | 1 | | doi | https://doi.org/10.1007/978-88-7642-520-2 | | isbn_softcover | 978-88-7642-519-6 | | isbn_ebook | 978-88-7642-520-2Series ISSN 2239-1460 Series E-ISSN 2532-1668 | | issn_series | 2239-1460 | | copyright | The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerl |
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Front Matter |
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Abstract
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,On some applications of Diophantine approximations, |
Clemens Fuchs |
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Abstract
The well-known simple deduction rule according to which for any distribution of more than . objects to . drawers at least one drawer contains at least two objects, gives rise to a generalization of the Euclidean algorithm, which by investigations due to D., H. and M. turned out to be the source of important arithmetic laws. In particular it implies a statement on how precisely the number 0 can be at least approximated by a linear combination
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,Über einige Anwendungen diophantischer Approximationen, |
Carl L. Siegel |
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Abstract
.ie bekannte einfache Schlußweise, daß bei einer Verteilung von mehr als . Dingen auf . Fächer in mindestens einem Fach mindestens zwei Dinge gelegen sind, enthält eine Verallgemeinerung des euklidischen Algorithmus, welche sich durch die Untersuchungen von D., H. und M. als die Quelle wichtiger arithmetischer Gesetze erwiesen hat. Sie liefert speziell eine Aussage darüber, wie genau sich . die Zahl o durch eine lineare Verbindung
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,Integral points on curves: Siegel’s theorem after Siegel’s proof, |
Clemens Fuchs,Umberto Zannier |
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Abstract
In this article, conceived as an . to the present translation of Siegel’s paper, we shall present (in brief form) some fairly modern arguments for Siegel’s theorem on integral points on curves, appearing in the second part of his paper [48] that is translated here; we shall refer to the more modern statement appearing as Theorem 3.2 below. The arguments presented here appeared after the original proof. All of these proofs rely on Diophantine Approximation and use suitable versions of Roth’s theorem (1955) or Schmidt’s Subspace Theorem (about 1970). Siegel had not Roth’s theorem [45], which led to considerable complications in his proof. Some of the arguments below may be considered versions of Siegel’s one, simplified both by the use of Roth’s theorem and also by geometrical results on abelian varieties..
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Back Matter |
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Abstract
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发表于 2025-3-21 17:31:56
