On some problem in Arithmetic Dynamical System and Diophantine Approximation in Positive Characteristic

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王季豪(Chi-Hao Wang) 查詢紙本館藏

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論文名稱

(On some problem in Arithmetic Dynamical System and Diophantine Approximation in Positive Characteristic)

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摘要(中)

這份博士論文研究的內容包含了三個部份，前二部分研究簇上函數疊代所引發的丟番圖問題，
第三個部份是有關於正特徵值函數體上丟番圖逼近的研究。
令 X 是一個 K-簇，並且在 X 上面給定一個半有限函數，
我們經由推廣動態多項式定義了在 X 上的推廣動態循環，
然後我們證明了在 X 上的推廣動態循環對於這個半有限函數是有效的。
第二部分是研究單項函數上的標準高度函數。
我們證明了在某些條件下，
單項函數上的標準高度函數不滿足 Northcott 有限性質。
針對這個問題，
我們定義了總和標準高度函數來修正它，
而且建立了幾個總和標準高度函數的基本性質，
特別來說，總和標準高度函數在適當條件下滿足 Northcott 有限性質。
最後一個部份是有關於正特徵值函數體上丟番圖逼近的研究。
古典丟番圖逼近是研究有理數可以多麼有效地逼近一個給定的無理實數，
在函數體上的逼近問題通常需要集合 I 的幫助，而 Lasjaunias 的研
究給出了集合 I 的定義。依此定義，本論文在這部份的研究給出了分
辨一個元素在正特徵值函數體上是否屬於集合 I 的準則。

摘要(英)

There are three parts in this thesis.
The first two parts study Diophantine problems arising from iterations of maps on varieties.
The third part concentrates on a question in Diophantine approximations over function fields of positive characteristic.
Let X be a K-variety equipped with a quasi-finite morphism over K.
First we generalize the definition of dynatomic polynomials to define generalized dynatomic cycles on X.
And we show that the generalized dynatomic cycles are effective for quasi-finite morphisms on X.
The second part is a study of canonical height associated to monomial maps.
We show that, under certain conditions, the canonical height function associated to monomial maps does not satisfy the Northcott finiteness property.
To remedy such defect, we modify the definition of canonical height by introducing the total canonical height.
Then we establish several basic properties of the total canonical height.
In particular, we show that under certain mild conditions, the total canonical heights satisfy the Northcott finiteness property.
In the last part, we study a question about Diophantine approximation in positive characteristic.
Let $K = FF_q( heta)$ be the rational function field in variable $ heta$ over the finite field $FF_q$.
Where q is a power of the prime number p.
Let $alpha in K_infty := FF_q((frac{1}{ heta}))$ be an element algebraic over K.
We give an effective criterion for $alpha$ being in Class I under certain conditions satisfied by $alpha$.

關鍵字(中)

★ 動態系統
★ 標準高度函數
★ 丟翻圖逼近

關鍵字(英)

★ dynamical system
★ canonical height
★ diophantine approximation

論文目次

中文摘要 i
英文摘要 iii
謝誌 v
目錄 vii
一、 Introduction 1
1.1 Generalized Dynatomic Polynomial and Cycles 6
1.2 Canonical Height Functions For Monomial Maps 8
1.3 Diophantine Approximation in Positive Character-
istic - A Criterion for Elements in Class I 11
二、 Arithmetic Dynamical System 17
2.1 Notation 17
2.2 Generalized Dynatomic Cycles 17
2.3 Serre’s Local Algebra 24
2.4 Computing a P (m, n) 27
2.5 Deformation and Flatness 32
2.6 Proof of Theorem 2.2.4 33
三、 Height and Canonical Height 37
3.1 Height 37
3.2 Canonical Height 44
3.3 Dynamical Degree and Canonical Height 46
3.4 Properties of Canonical Height Functions 48
3.5 The Total Canonical Height Function 52
3.6 Monomial Maps associated to Non-diagonalizable Ma-trices 61
3.6.1 Two-dimensional non-diagonalizable matrices 61
3.6.2 Higher dimension 63
3.7 Points with Small Total Canonical Height 65
四、 Diophantine Approximation in Positive Characteristic 67
4.1 Number Field 67
4.2 Function Fields 68
4.3 First Condition 72
4.4 Criterion 77
4.5 Dierentiation of algebraic power series 82
4.6 Application 85
索引 89
參考文獻 89

參考文獻

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指導教授

夏良忠、呂明光
(Liang-Chung Hsia、Ming-Guang Leu)

審核日期

2013-11-27

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