2026-07-23T16:35:25Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:17872 2024-03-06T11:00:35Z ddc:004 open_access

Improved Algebraic Degeneracy Testing Cardinal, Jean Sharir, Micha Degeneracy testing k-SUM problem incidence bounds Hocroft’s problem polynomial method algebraic decision trees In the classical linear degeneracy testing problem, we are given n real numbers and a k-variate linear polynomial F, for some constant k, and have to determine whether there exist k numbers a_1,…,a_k from the set such that F(a_1,…,a_k) = 0. We consider a generalization of this problem in which F is an arbitrary constant-degree polynomial, we are given k sets of n real numbers, and have to determine whether there exists a k-tuple of numbers, one in each set, on which F vanishes. We give the first improvement over the naïve O^*(n^{k-1}) algorithm for this problem (where the O^*(⋅) notation omits subpolynomial factors). We show that the problem can be solved in time O^*(n^{k - 2 + 4/(k+2)}) for even k and in time O^*(n^{k - 2 + (4k-8)/(k²-5)}) for odd k in the real RAM model of computation. We also prove that for k = 4, the problem can be solved in time O^*(n^2.625) in the algebraic decision tree model, and for k = 5 it can be solved in time O^*(n^3.56) in the same model, both improving on the above uniform bounds. All our results rely on an algebraic generalization of the standard meet-in-the-middle algorithm for k-SUM, powered by recent algorithmic advances in the polynomial method for semi-algebraic range searching. In fact, our main technical result is much more broadly applicable, as it provides a general tool for detecting incidences and other interactions between points and algebraic surfaces in any dimension. In particular, it yields an efficient algorithm for a general, algebraic version of Hopcroft’s point-line incidence detection problem in any dimension. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Jean Cardinal and Micha Sharir 2023 Is Part Of LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2023.22 urn:nbn:de:0030-drops-178723 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.22 eng https://creativecommons.org/licenses/by/4.0/legalcode