The sum of square roots is as follows: Given $x_1,\dots,x_n \in \mathbb{Z}$ and $a_1,\dots,a_n \in \mathbb{N}$ decide whether $ E=\sum_{i=1}^n x_i \sqrt{a_i} \geq 0$. It is a prominent open problem (Problem 33 of the Open Problems Project), whether this can be decided in polynomial time. The state-of-the-art methods rely on separation bounds, which are lower bounds on the minimum nonzero absolute value of $E$. The current best bound shows that $|E| \geq \left(n \cdot \max_i (|x_i| \cdot \sqrt{a_i})\right)^{-2^n} $, which is doubly exponentially small. We provide a new bound of the form $|E| \geq \gamma \cdot (n \cdot \max_i|x_i|)^{-2n}$ where $\gamma $ is a constant depending on $a_1,\dots,a_n$. This is singly exponential in $n$ for fixed $a_1,\dots,a_n$. The constant $\gamma$ is not explicit and stems from the subspace theorem, a deep result in the geometry of numbers.

翻译：平方根和问题如下：给定 $x_1,\dots,x_n \in \mathbb{Z}$ 和 $a_1,\dots,a_n \in \mathbb{N}$，判断 $ E=\sum_{i=1}^n x_i \sqrt{a_i} \geq 0$ 是否成立。这是一个著名的开放问题（开放问题项目中的第33个问题），即能否在多项式时间内判定该问题。当前最先进的方法依赖于分离界，即 $E$ 的最小非零绝对值的下界。目前的最佳界表明 $|E| \geq \left(n \cdot \max_i (|x_i| \cdot \sqrt{a_i})\right)^{-2^n} $，这是一个双指数小量。我们提供了一个新下界，形式为 $|E| \geq \gamma \cdot (n \cdot \max_i|x_i|)^{-2n}$，其中 $\gamma $ 是依赖于 $a_1,\dots,a_n$ 的常数。对于固定的 $a_1,\dots,a_n$，该下界是 $n$ 的单指数。常数 $\gamma$ 并非显式给出，而是源于子空间定理，这是数的几何中的一个深刻结果。