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• W2994852306 abstract "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 $e_iin {1,0, -1}$ for all $1leq ileq k$. 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 $2^{2k}$." @default.
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• W2994852306 date "2009-09-30" @default.
• W2994852306 modified "2023-09-27" @default.
• W2994852306 title "Bounding the sum of square roots via lattice reduction" @default.
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• W2994852306 doi "https://doi.org/10.1090/s0025-5718-09-02304-7" @default.
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