Bounding the sum of square roots via lattice reduction

Abstract:

Let $k$ and $n$ be positive integers. Define $R(n,k)$ to be the minimum positive value of \[ \left | e_i \sqrt {s_1} + e_2 \sqrt {s_2} + \cdots + e_k \sqrt {s_k} -t \right |, \] where $s_1, s_2, \cdots , s_k$ are positive integers no larger than $n$, $t$ is an integer and $e_i\in \{1,0, -1\}$ for all $1\leq i\leq k$. It is important in computational geometry to determine a good lower and upper bound of $R(n,k)$. In this paper we show that this problem is closely related to the shortest vector problem in certain integral lattices and present an algorithm to find lower bounds based on lattice reduction algorithms. Although we can only prove an exponential time upper bound for the algorithm, it is efficient for large $k$ when an exhaustive search for the minimum value is clearly infeasible. It produces lower bounds much better than the root separation technique does. Based on numerical data, we formulate a conjecture on the length of the shortest nonzero vector in the lattice, whose validation implies that our algorithm runs in polynomial time and the problem of comparing two sums of square roots of small integers can be solved in polynomial time. As a side result, we obtain constructive upper bounds for $R(n,k)$ when $n$ is much smaller than $2^{2k}$.