臺灣博碩士論文加值系統

摘要
這篇論文是在討論擬固定多項式問題。共分成五個章節。
第一章: 這份論文的介紹。
第二章: 在實數空間上的擬固定多項式問題, 討論當解的個數為無限時的充要條件及當
解的個數為有限時, 解的個數的上界。
第三章: 推廣在第二章中, 當解的個數為有限時, 我們可找出解個數之上限是個緊的上界,
同時找出當解個數之最佳上界的充要條件。
第四章: 在k-維度空間中, 討論當解的個數為無限時的充要條件及當解的個數為有限時,
解的個數之上界。
第五章: 在第二章中, 當解的個數為有限時, 我們提供一個有效率的演算法計算出所有的
解。

Abstract
In this dissertation, we study some quasi-fixed polynomial problem. It
includes five chapters in this dissertation.
Chapter 1 : Introduction of this dissertation.
Chapter 2 : We focus a real-valued polynomial function F : Rn R ! R
and consider the number of all quasi-fixed (polynomial) solutions. We prove the
necessary and sufficient conditions when the number of all quasi-fixed polyno-
mial solutions is infinitely many and find the upper bound of the number of all
solutions when the number is finitely many.
Chapter 3 : We prove that the upper bound s + 2 in chapter 2 is necessary.
Moreover, we find the necessary or sufficient conditions when the number of all
solutions is s + 2 and s + 1.
Chapter 4 : We focus a vector-valued polynomial function F : Rn R ! Rk
and consider the number of all quasi-fixed (polynomial) solutions. We prove the
necessary and sufficient conditions when the number of all quasi-fixed (poly-
nomial) solutions is infinitely many and we also find the upper bound of the
number of all solutions when the number is finitely many.
Chapter 5 : If the number of all quasi-fixed polynomial solutions in Chapter
2 is finitely many. We provide an efficient algorithm to solve all quasi-fixed
polynomial solutions.

Contents
Abstract (in Chinese) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I
Abstract (in English) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .V
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .VII
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 A Quasi-fixed Polynomial Problem for Polynomial Function . . . . 1
1.2 Quasi-fixed polynomial solutions of real-valued polynomial equa-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Quasi-fixed Point for vector-valued Polynomial Functions on Rn
R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2
1.4 Approximate Solutions For Quasi-Fixed Polynomial Problems of
Real-Valued Polynomial Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2. A Quasi-fixed Point Problem for Polynomial Function . . . . . . . .5
2.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5
2.2 Some Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
3. Quasi-fixed polynomial solutions of real-valued polynomial equa-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
3.3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4. Quasi-fixed Point for vector-valued Polynomial Functions on
Rn R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Preliminary and some Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Characterization for vector polynomial function . . . . . . . . . . . . . . . 47
4.4 Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52
5. Approximate Solutions For Quasi-Fixed Polynomial Problems
of Real-Valued Polynomial Equation . . . . . . . . . . . . . . . . . . . . . . . . . . .57
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Main Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

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