近解的推廣

Generalization of Approximate Solutions

李玥瑩(Yueh-Ying Lee)

指導教授 ： 董世平

中原大學/理學院/應用數學系/碩士(2006年)

https://doi.org/10.6840/cycu200600356

摘要

令 $F(x,y)$ 為佈於有理數體的多項式，若 $F(x,g(x))in Q$，則我們可稱佈於有理數體的多項式 $g(x)$ 是 $F(x,y)$ 的近解(approximate solution) $[1]$。在此我們將推廣近解，得到擬似解(near solutions)與有理擬似解(rational near solutions)。若存在 $cin Q$ 使得 $F(x,h(x))equiv cx^{m}$，則我們稱佈於有理數體的多項式 $h(x)$ 是 $m-$擬似解($m-$near solution)及 $c$ 為 $F(x,y)$ 對應 $h(x)$ 的 $m-$值($m-$value) $[2]$。我們將延伸近解的定義，利用兩種近解來尋找擬似解及得到一些性質，以及將近解由多項式推廣到有理函數。 若存在 $cin Q$ 使得 $F(x,r(x))equiv c$，則我們稱佈於有理數體的有理函數 $r(x)$ 為有理擬似解，且若有理擬似解的分母為一個固定的多項式，則可得到分子次數的上界值。

關鍵字

近解 ； 有理擬似解 ； 擬似解

並列摘要

Let $F(x,y)$ be a polynomial over $Q$, we will call a polynomial $g(x)$ over $Q$ an approximate solution of $F(x,y)$ if $F(x,g(x))in Q$ $[1]$. Here we generalize approximate solutions
to obtain near solutions and rational near solutions. If there exists a $cin Q$ such that $F(x,h(x))equiv cx^{m}$, then we call a polynomial $h(x)$ over $Q$ a $m-$near solution and the number $c$ the $m-$value of $F(x,y)$ corresponding to $h(x)$ $[2]$. Now we extend the definition of approximate solutions, making use of two kinds of approximate solutions to look for near solutions, and getting some properties. We will be able to extend the approximate solution from polynomials to rational functions. If there exists a $cin Q$ such that $F(x,r(x))equiv c$, then we call a rational function $r(x)$ over $Q$ a rational near solution. And if the denominator of approximate solutions is a fixed polynomial, then an upper bound on the degrees of the numerators of approximate solutions is obtained.

並列關鍵字

rational near solution ； approximate solution ； near solution

參考文獻

[1] S. P. Tung, Approximate solutions of polynomial equations, J. Symbolic Comput., 33 (2002), 239-254.

[2] S. P. Tung, Near solutions of polynomial equations, to appear, Acta Arith.