  Consider an irreducible bilinear form $f(x_1,x_2;y_1,y_2)$ with integer
coefficients. We derive an upper bound for the number of integer points
$(\mathbf{x},\mathbf{y})\in\mathbb{P}^1\times\mathbb{P}^1$ inside a box
satisfying the equation $f=0$. Our bound seems to be the best possible bound
and the main term decreases with a larger determinant of the form $f$. We
further discuss the case when $f(x_1,x_2;y_1,y_2;z_1,z_2)$ is an irreducible
non-singular trilinear form defined on $\mathbb{P}^1\times
\mathbb{P}^1\times\mathbb{P}^1$, with integer coefficients. In this case, we
examine the singularity and reducibility conditions of $f$. To do this, we
employ the Cayley hyperdeterminant $D$ associated to $f$. We then derive an
upper bound for the number of integer points in boxes on such trilinear forms.
The main term of the estimate improves with larger $D$. Our methods are based
on elementary lattice results.
