We investigate the rationality of Weil sums of binomials of the form $W^{K,s}_u=\sum_{x \in K} \psi(x^s - u x)$, where $K$ is a finite field whose canonical additive character is $\psi$, and where $u$ is an element of $K^{\times}$ and $s$ is a positive integer relatively prime to $|K^\times|$, so that $x \mapsto x^s$ is a permutation of $K$. The Weil spectrum for $K$ and $s$, which is the family of values $W^{K,s}_u$ as $u$ runs through $K^\times$, is of interest in arithmetic geometry and in several information-theoretic applications. The Weil spectrum always contains at least three distinct values if $s$ is nondegenerate (i.e., if $s$ is not a power of $p$ modulo $|K^\times|$, where $p$ is the characteristic of $K$). It is already known that if the Weil spectrum contains precisely three distinct values, then they must all be rational integers. We show that if the Weil spectrum contains precisely four distinct values, then they must all be rational integers, with the sole exception of the case where $|K|=5$ and $s \equiv 3 \pmod{4}$.

翻译：我们研究形如 $W^{K,s}_u=\sum_{x \in K} \psi(x^s - u x)$ 的二项式韦尔和的有理性，其中 $K$ 是有限域，其典范加法特征为 $\psi$，$u$ 是 $K^{\times}$ 中的元素，$s$ 是与 $|K^\times|$ 互质的正整数，因此映射 $x \mapsto x^s$ 是 $K$ 上的置换。对于给定 $K$ 和 $s$，韦尔谱（即当 $u$ 遍历 $K^{\times}$ 时 $W^{K,s}_u$ 的取值族）在算术几何及若干信息论应用中具有重要意义。若 $s$ 非退化（即 $s$ 不是 $p$ 模 $|K^\times|$ 的幂，其中 $p$ 为 $K$ 的特征），则韦尔谱至少包含三个不同的值。已知当韦尔谱恰好包含三个不同值时，这些值必均为有理整数。本文证明，除 $|K|=5$ 且 $s \equiv 3 \pmod{4}$ 这一特例情形外，若韦尔谱恰好包含四个不同值，则这些值必均为有理整数。