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

Published online by Cambridge University Press:  10 February 2005

Jeffrey Lin Thunder
Jeffrey Lin Thunder
Affiliation:
Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, USAjthunder@math.niu.edu
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Abstract

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For homogeneous decomposable forms F(X) in n variables with integer coefficients, we consider the number of integer solutions ${\bf x}\in\mathbb{Z}^n$ to the inequality $|F({\bf x})|\le m$ as $m\rightarrow\infty$. 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.

Keywords

Diophantine inequalitiesdecomposable formssubspace theorem11D7511D45 (primary)11D72 (secondary)
Type
Research Article
Information
Compositio Mathematica , Volume 141 , Issue 2 , March 2005 , pp. 271 - 292
Copyright
Foundation Compositio Mathematica 2005