正特徵值函數體上的逼近指數之研究

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陳慧錚(Huei Jeng) 查詢紙本館藏

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論文名稱

正特徵值函數體上的逼近指數之研究
(Distribution of Diophantine approximation exponentsfor algebraic quantities in finite characteristic)

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摘要(中)

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.

摘要(英)

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.

關鍵字(中)

★ 丟番圖
★ 逼近指數

關鍵字(英)

★ Carlitz torsion
★ exponents
★ Diophantine approximation

論文目次

1 Introduction 1
1.1 Diophantine Approximation for Number Fields . . . . . . . . 1
1.2 Diophantine Approximation for Function Fields . . . . . . . . 3
1.3 Analogue of Thue's Theorem and Distribution of Approximation
Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Distribution of Approximation Exponents of Elements in Class
IA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Outline of This Thesis . . . . . . . . . . . . . . . . . . . . . . 9
2 Analogue of Thue's Theorem and The Class I, Class IA 11
2.1 Thue's Method . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Di erential Method . . . . . . . . . . . . . . . . . . . 11
2.2 The Class I and Class IA . . . . . . . . . . . . . . . . . . . . 13
2.3 Results from W. Schmidt and D.Thakur . . . . . . . . . . . . 15
2.3.1 Continued Fractions and Exponents . . . . . . . . . . 15
2.3.2 Continued Fractions and Approximation Exponents of
Elements in Class IA . . . . . . . . . . . . . . . . . . . 16
2.4 Distribution of Exponents for Fixed Special Degrees . . . . . 17
3 Asymptotic Distribution of Exponents for Algebraic Ele-
ments in Class IA 2
3.1 Heights on IA(q) . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Distribution of Exponents of Elements in IA(q) in (2; q + 1] . 32
4 Approximation exponents of Carlitz torsion 57
4.1 Drinfeld Modules and Their Torsion . . . . . . . . . . . . . . 57
4.2 Continued Fractions and Linear Fractional
Transformations . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Continued Fractions and Approximation
Exponents of (q

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指導教授

夏良忠(Liang-Chung Hsia)

審核日期

2011-7-15

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