Decomposable form inequalities

• 成果类型: Article
• 署名作者: Thunder, JL
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/2661368
• 发表日期: 2001
• 页码: 767-804
• 关键词: equations

摘要

We consider Diophantine inequalities of the kind /F(x)/ less than or equal to m, where F(X) is an element of Z[X] is a homogeneous polynomial which can be expressed as a product of d homogeneous linear forms in n variables with complex coefficients and m greater than or equal to 1. We say such a form is of finite type if the total volume of all real solutions to this inequality is finite and if, for every n ' -dimensional subspace S subset of or equal to R-n defined over Q, the corresponding n ' -dimensional volume for F restricted to S is also finite. We show that the number of integral solutions x is an element of Z(n) to our inequality above is finite for all m if and only if the form F is of finite type. When F is of finite type, we show that the number of integral solutions is estimated asymptotically as m --> infinity by the total volume of all real solutions. This generalizes a previous result due to Mahler for the case n = 2. Further, we prove a conjecture of W. M. Schmidt, showing that for F of finite type the number of integral solutions is bounded above by c(n, d)m(n/d), where c(n, d) is an effectively computable constant depending only on n and d.