## Abstract

We describe subquadratic algorithms, in the algebraic decision-tree model of computation, for detecting whether there exists a triple of points, belonging to three respective sets A, B, and C of points in the plane, that satisfy a pair of polynomial equations. In particular, this has an application to detect collinearity among three sets A, B, C of n points each, in the complex plane, when each of the sets A, B, C lies on some constant-degree algebraic curve. In another development, we present a subquadratic algorithm, in the algebraic decision-tree model, for the following problem: Given a pair of sets A, B each consisting of n pairwise disjoint line segments in the plane, and a third set C of arbitrary line segments in the plane, determine whether A× B× C contains a triple of concurrent segments. This is one of four 3sum-hard geometric problems recently studied by Chan (2020). The results reported in this extended abstract are based on the recent studies of the author with Aronov and Sharir (2020, 2021).

Original language | English |
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Title of host publication | Connecting with Computability - 17th Conference on Computability in Europe, CiE 2021, Proceedings |

Editors | Liesbeth De Mol, Andreas Weiermann, Florin Manea, David Fernández-Duque |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 178-188 |

Number of pages | 11 |

ISBN (Print) | 9783030800482 |

DOIs | |

State | Published - 2021 |

Event | 17th Conference on Computability in Europe, CiE 2021 - Virtual, Online Duration: 5 Jul 2021 → 9 Jul 2021 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 12813 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 17th Conference on Computability in Europe, CiE 2021 |
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City | Virtual, Online |

Period | 5/07/21 → 9/07/21 |

### Bibliographical note

Publisher Copyright:© 2021, Springer Nature Switzerland AG.

## Keywords

- 3SUM-hard problems
- Algebraic decision tree model
- Collinearity testing
- Segment concurrency