Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bounding the Sum of Square Roots via Lattice Reduction   Let $k$ and $n$ be positive integers. Define $R(n,k)$ to be the minimum
positive value of $$ | e_i \sqrt{s_1} + e_2 \sqrt{s_2} + ... + e_k \sqrt{s_k}
-t | $$ where $ s_1, s_2, ..., 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 $2^{2k}$.

20
The categories of this paper are: acc-phys
24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
5
The categories of this paper are: gr-qc
6
The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
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The categories of this paper are: chem-ph
14
The categories of this paper are: math-ph
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg
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The categories of this paper are: quant-ph
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The categories of this paper are: stat