An improved bound on sums of square roots via the subspace theorem

Abstract

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.