擬固定多項式問題研究

A Quasi-Fixed Polynomial Problem

陳怡州(Yi-Chou Chen)

指導教授 ： 賴漢卿

中原大學/理學院/應用數學研究所/博士(2010年)

https://doi.org/10.6840/cycu201000243

摘要

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。

關鍵字

擬固定多項式解 ； 擬固定值

並列摘要

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.

並列關鍵字

quasi-fixed value ； quasi-fixed (polynomial) solution

參考文獻

[1] LAI HANG-CHIN AND CHEN YI-CHOU, A Quasi-fixed Polynomial Prob-

Analysis, Volume 11, Number 1, (2010), 101-114..

mania, to be published in 2011, pp 1-14.

Conference on Fixed Point Theory and Applications, on July 16-22, 2009

[6] LENSTRA, A.K., Factoring multivariate polynomials over algebraic number