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.

中文翻译：

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用稀疏二进制形式表示整数

我们将为形状 $|F(x , y)| 的不等式的解数给出新的上限。\leq h$，其中 $F(x , y)$ 是稀疏二进制形式，具有整数系数，而 $h$ 是一个足够小的整数，就二进制形式 $F$ 的判别式的绝对值而言。我们的界限取决于 $F(x, y)$ 的非消失系数的数量。当 $F$ 真的很稀疏时，我们为解的数量建立一个尖锐的上限，该上限与非消失系数的数量呈线性关系。这项工作将为穆勒和施密特在 1988 年针对特殊但重要的情况提出的许多猜想提供肯定的答案。