The sum of square roots is as follows: Given x_1,… ,x_n ∈ ℤ and a₁,… ,a_n ∈ ℕ decide whether E = ∑_{i=1}^n x_i √{a_i} ≥ 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| ≥ (n ⋅ max_i (|x_i| ⋅√{a_i})) ^{-2ⁿ}, which is doubly exponentially small. We provide a new bound of the form |E| ≥ γ ⋅ (n ⋅ max_i |x_i|)^{-2n} where γ is a constant depending on a₁,… ,a_n. This is singly exponential in n for fixed a_1,… ,a_n. The constant γ is not explicit and stems from the subspace theorem, a deep result in the geometry of numbers.
@InProceedings{eisenbrand_et_al:LIPIcs.SoCG.2024.54, author = {Eisenbrand, Friedrich and Haeberle, Matthieu and Singer, Neta}, title = {{An Improved Bound on Sums of Square Roots via the Subspace Theorem}}, booktitle = {40th International Symposium on Computational Geometry (SoCG 2024)}, pages = {54:1--54:8}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-316-4}, ISSN = {1868-8969}, year = {2024}, volume = {293}, editor = {Mulzer, Wolfgang and Phillips, Jeff M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.54}, URN = {urn:nbn:de:0030-drops-199993}, doi = {10.4230/LIPIcs.SoCG.2024.54}, annote = {Keywords: Exact computing, Separation Bounds, Computational Geometry, Geometry of Numbers} }
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