The algebraic structure of the set of solutions to the Thue equation

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• 作者：Aleks ; er Grytczuk ; Marek W&#xf3 ; jtowicz
• 关键词：Thue equation
• 刊名：Journal of Number Theory
• 出版年：2010
• 出版时间：July 2010
• 年：2010
• 卷：130
• 期：7
• 页码：1480-1487
• 全文大小：168 K

文摘

Let F_n be a binary form with integral coefficients of degree n2, let d denote the greatest common divisor of all non-zero coefficients of F_n, and let h2 be an integer. We prove that if d=1 then the Thue equation (T) F_n(x,y)=h has relatively few solutions: if is a subset of the set of all solutions to (T), with , then

(#) h divides the number , where , 1kr, and δ(ξ_k,ξ_l)=x_ky_l−x_ly_k. As a corollary we obtain that if h is a prime number then, under weak assumptions on F_n, there is a partition of into at most n subsets maximal with respect to condition (#).