Improved Algebraic Degeneracy Testing

Jean Cardinal*, Micha Sharir*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

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 a1, ..., ak from the set such that F(a1, ..., ak) = 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 (Equation presented) algorithm for this problem (where the O(·) notation omits subpolynomial factors). We show that the problem can be solved in time (Equation presented) for even k and in time (Equation presented) 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(n2.625) in the algebraic decision tree model, and for k = 5 it can be solved in time O(n3.56) 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.

Original languageEnglish
Title of host publication39th International Symposium on Computational Geometry, SoCG 2023
EditorsErin W. Chambers, Joachim Gudmundsson
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772730
DOIs
StatePublished - 1 Jun 2023
Event39th International Symposium on Computational Geometry, SoCG 2023 - Dallas, United States
Duration: 12 Jun 202315 Jun 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume258
ISSN (Print)1868-8969

Conference

Conference39th International Symposium on Computational Geometry, SoCG 2023
Country/TerritoryUnited States
CityDallas
Period12/06/2315/06/23

Funding

FundersFunder number
Israel Science Foundation260/18

    Keywords

    • Degeneracy testing
    • Hocroft's problem
    • algebraic decision trees
    • incidence bounds
    • k-SUM problem
    • polynomial method

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