Asymptotic estimates for the number of integer solutions to decomposable form inequalities

[摘要] For homogeneous decomposable forms F(X) in n variables with integer coefficients, we consider the number of integer solutions ${f x}inmathbb{Z}^n$ to the inequality $|F({f x})|le m$ as $mightarrowinfty$. We give asymptotic estimates which improve on those given previously by the author in Ann. of Math. (2) 153 (2001), 767–804. Here our error terms display desirable behaviour as a function of the height whenever the degree of the form and the number of variables are relatively prime.

[学科分类] 数学（综合）