### 博碩士論文 942401002 完整後設資料紀錄

 DC 欄位 值 語言 DC.contributor 數學系 zh_TW DC.creator 陳慧錚 zh_TW DC.creator Huei Jeng en_US dc.date.accessioned 2011-7-15T07:39:07Z dc.date.available 2011-7-15T07:39:07Z dc.date.issued 2011 dc.identifier.uri http://ir.lib.ncu.edu.tw:88/thesis/view_etd.asp?URN=942401002 dc.contributor.department 數學系 zh_TW DC.description 國立中央大學 zh_TW DC.description National Central University en_US dc.description.abstract 正特徵值函數體上的丟番圖逼近和有理數體以及零特徵值函數體上的丟番圖逼近不同, Mahler 舉出一個例子指出一個代數數逼近指數可以和它的擴張指數相同. Schmidt 和 Thakur 證明出, 給定任何一個介於 2和q+1的有理數m 我們都可以找到一組代數數使得它們的逼近指數等於m, 並且它的擴張指數比q+1小. 在此論文的第一部分中我們證明出了我們可以找出一組代數數使得它們的逼近指數等於m, 並且它的擴張指數等於q+1. 第二部分我們完整的描述了在 IA(q)的這個集合中的元素在區間 (2,q+1] 的逼近指數的分布. Thakur已經證明出在q小的時候大部分IA(q)的元素的逼近指數很接近2. 第三部分我們給出一些特殊代數數(由Carlitz 模來的) 的連分數公式以及逼近指數的計算. 第四部份我們給出了另一些特殊代數數(也是由Carlitz 模來的) 的逼近指數的上界. zh_TW dc.description.abstract In contrast to Roth’’s (Uchiyama’’s respectively) theorem that algebraic real numbers (algebraic power series in characteristic zero respectively) have Diophantine approximation exponents equal to \$2\$, Mahler had shown that Liouville bound is the best possible in finite characteristic. Schmidt and Thakur proved that given any rational number \$mu\$ between \$2\$ and \$q+1\$, where \$q\$ is a power of a prime \$p\$, there exists (explicitly given) algebraic Laurent series \$alpha\$ in characteristic \$p\$, with their Diophantine approximation exponent equal to \$mu\$ and with degree of \$alpha\$ being at most \$q+1\$. We first refine this result by showing that degree of \$alpha\$ can be prescribed to be equal to \$q+1\$. Next we describe how the exponents of \$alpha\$’’s are asymptotically distributed with respect to their heights in the case of algebraic elements of class IA for function fields over finite fields. A result of Thakur says that for low values of \$q\$ most elements \$alpha\$ have exponents near \$2\$. We refine this result and give more precise descriptions of the distribution of the approximation exponents of such elements \$alpha\$ of Class IA. In the last chapter, we compute the continued fractions and approximation exponents of certain families of elements related to Carlitz torsion. en_US DC.subject 丟番圖 zh_TW DC.subject 逼近指數 zh_TW DC.subject Carlitz torsion en_US DC.subject exponents en_US DC.subject Diophantine approximation en_US DC.title 正特徵值函數體上的逼近指數之研究 zh_TW dc.language.iso zh-TW zh-TW DC.title Distribution of Diophantine approximation exponentsfor algebraic quantities in finite characteristic en_US DC.type 博碩士論文 zh_TW DC.type thesis en_US DC.publisher National Central University en_US