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Abstract

We will give new upper bounds for the number of solutions to the inequalities
of the shape $|F(x , y)| \leq h$, where $F(x , y)$ is a sparse binary form,
with integer coefficients, and $h$ is a sufficiently small integer in terms of
the absolute value of the discriminant of the binary form $F$. Our bounds
depend on the number of non-vanishing coefficients of $F(x , y)$. When $F$ is
really sparse, we establish a sharp upper bound for the number of solutions
that is linear in terms of the number of non-vanishing coefficients. This work
will provide affirmative answers to a number of conjectures posed by Mueller
and Schmidt in 1988, for special but important cases.