关于Thue方程的解数

On the Number of Solutions of Thue's Equation

设ｍ是正整数，ｆ（Ｘ，Ｙ）＝ａ０Ｘｎ＋ａ１Ｘ（ｎ－１）Ｙ＋．．．＋ａｎＹｎ∈Ｚ［Ｘ，Ｙ］是Ｑ上不可约化的叫ｎ（ｎ≥３）次齐次多项式。本文证明了：当ｇｃｄ（ｍ，ａ０）＝１，ｎ≥４００且ｍ≥１０（３５）时，方程｜ｆ（ｘ，ｙ）｜＝ｍ，ｘ，ｙ∈ｚ，ｇｃｄ（ｘ，ｙ）＝１，至多有６ｎｖ（ｍ）组解（ｘ，ｙ），其中ｖ（ｍ）是同余式Ｆ（ｚ）＝ｆ（ｚ，１）≡０（ｍｏｄｍ）的解数。特别是当ｇｃｄ（ｍ，ＤＦ）＝１时，该方程至多有６ｎ（ω（ｍ）＋１）组解（ｘ，ｙ），其中ＤＦ是多项式Ｆ的判别式，ω（ｍ）是ｍ的不同素因数的个数．

Let m be a positive integer, and let f(X, Y) = aoXn + a1Xn(-1)Y + ... + anYn ∈z[X,Y] be an irreducible binary form of degree n witn n≥3.In this paper we prove that if gcd(m, a0)=1,n≥400 and m≥10(35),then the equation |f(x,y)|= m has at most 6nv(m)integer solutions(x, y) with gcd (x, y) =1,where v(m)is the number of solutions of the congruence F(z) = f(z, 1) ≡ 0(modm). Moreover, if gcd(m,DF) = 1, Where DF is the discriminant of F, then the equation has at most 6n(ω(m)+1) solutions(x,y), where ω(m)is the number of distinct prime factors of m.