Some Properties of Rational Near Solutions

• Main Authors: Li-Yang Gan, 甘李揚
• Other Authors: Shih-Ping Tung
• Format: Others
• Language: en_US
• Published: 2006
• Online Access: http://ndltd.ncl.edu.tw/handle/25598893131228824128

碩士 === 中原大學 === 應用數學研究所 === 94 === 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.