Abstract
We consider Diophantine inequalities of the kind $|F(\mathrm{x})| \le m$, where $F(\mathbf{X})\in \mathbb{Z}[\mathbf{X}]$ is a homogeneous polynomial which can be expressed as a product of $d$ homogeneous linear forms in $n$ variables with complex coefficients and $m\ge 1$. We say suc 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 $S\subseteq \mathbb{R}^n$ defined over $\mathbb{Q}$, the corresponding $n’$-dimensional volume for $F$ restricted to $S$ is also finite.
We show that the number of integral solutions $\mathbf{x} \in \mathbb{Z}^n$ 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 $m\to \infty$ 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 $d$.
