Inequalities for decomposable forms of degree $n+1$ in $n$ variables

Abstract:

We consider the number of integral solutions to the inequality $|F(\mathbf {x}) |\le m$, where $F(\mathbf {X} )\in \mathbb {Z} [\mathbf {X} ]$ is a decomposable form of degree $n+1$ in $n$ variables. We show that the number of such solutions is finite for all $m$ only if the discriminant of $F$ is not zero. We get estimates for the number of such solutions that display appropriate behavior in terms of the discriminant. These estimates sharpen recent results of the author for the general case of arbitrary degree.