有理擬似解的一些特性

Some Properties of Rational Near Solutions

甘李揚(Li-Yang Gan)

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/cycu200600367

摘要

令F(x,y)為係數布於有理數體的一個多項式，我們會好奇有沒有一個有理函數r(x)可以使得F(x,r(x))是某一個有理數b。如果有，我們就稱呼這樣的r(x)是F(x,y)=0的一個有理擬似解。在此篇論文，我們主要是探究當給定一個係數布於有理數體的多項式F(x,y)時，有理擬似解個數的上界。
　　對於給定一個係數布於有理數體的多項式F(x,y)，F(x,y)=0可能會有無限多個有理擬似解。在這種情況下，F(x,y)必定為Σ_c_i^n(p(x)y-q(x))^i的型式，其中c_iεQ，p(x)、q(x)εQ[x]。同時，只要α是一個有理數，(q(x)+α)/p(x)很明顯就會是F(x,y)=0的一個有理擬似解。而且如果是F(x,y)=0的一個有理擬似解的話，一定會是(q(x)+α)/p(x)的型式，其中α是某個有理數。
　　另一方面，當F(x,y)=0只有有限多個的有理擬似解時，我們更加想要知道有沒有一個簡易的方法來判斷F(x,y)=0究竟最多有多少有理擬似解。因為有一些現象顯示有理擬似解個數不會離deg_y(F(x,y))太遠，所以雖然F(x,y)=0的有理擬似解個數很可能不只和deg_y(F(x,y))有關，我們仍然期盼F(x,y)=0的有理擬似解個數僅與deg_y(F(x,y))有關，就像是一般解的個數上界只被deg_y(F(x,y))所界定住那樣。例如，我們觀察出任意兩個F(x,y)=0的有理擬似解q_1(x)/p_1(x)、q_2(x)/p_2(x)，如果F(x,q_1(x)/p_1(x))≠F(x,q_2(x)/p_2(x))，則p_1(x)q_2(x)-p_2(x)q_1(x)必定整除p_1(x)和p_2(x)的最大公因式的某個冪次。而要存在許多那樣滿足一系列相對應的整除關係的多項式配對實屬困難。
　　在此，我們企圖找尋一些關於F(x,y)=0的有理擬似解上界和deg_y(F(x,y))之間的關係，但是最後還是沒有發現理想的結果。

關鍵字

擬似解 ； 有理擬似解

並列摘要

Let F(x,y) be a polynomial over the field Q of all rational numbers. We may wonder whether there is a rational function r(x) in {Q}(x) such that F(x,r(x))=b, for some rational number b. Here we call such r(x) a rational near solution of F(x,y)=0. In this paper, we mainly explore the upper bound of the number of rational near solutions of
a given polynomial F(x,y) in Q[x,y]. Given a polynomial F(x,y) in Q[x,y], it may happen that F(x,y)=0 has infinitely many rational near solutions. In this situation, F(x,y) must be of the form Σ_c_i^n(p(x)y-q(x))^ic_i, where c_i is in Q, p(x), q(x) are in Q[x]. And, for any rational number α, (q(x)+α)x)/p(x) is clearly a rational near solution of F(x,y)=0, and all rational near solutions of F(x,y)=0 are of the form (q(x)+α)x)/p(x) also, for some rational number α. On the other hand, when F(x,y)=0 has only finitely many rational near solutions,
we indeed wonder whether there is an easy way to know how many rational near solutions F(x,y)=0 has. With respect to variable y, we know that the number of usual solutions is bounded by the degree of (F(x,y)) in y. Here we hope that the number of rational near solutions is simply dependent on the degree of (F(x,y)) in y also.
Even if it is quite possible that the upper bound of the number of all rational near solutions of
F(x,y)=0 is dependent not merely on the degree of F(x,y) in y, there are some evidences that the number of rational near solutions of F(x, y)=0 should not be much larger than the degree of F(x, y) in y. For instance, we observe that if
q_1(x)/p_1(x), q_2(x)}/p_2(x) are two rational near solutions of F(x, y)=0 with F(x,q_1(x)/p_1(x))≠F(x,q_2(x)/p_2(x)), then p_1(x)q_2(x)-p_2(x)q_1(x) must divide a certain power of the greatest common divisor of p_1(x) and p_2(x).
And it seems difficult that there exist a lot of such pairs satisfying these division relations. In brief, for a given polynomial F(x,y) in Q[x, y] we intend to find some relations between
the upper bound of the number of rational near solutions of F(x,y)=0 and the degree of F(x,y) in y. Here we cannot find an ideal result.

並列關鍵字

rational near solutions ； near solutions

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.