### Abstract

The 3SUM problem asks if an input n-set of real numbers contains a triple whose sum is zero. We consider the 3POL problem, a natural generalization of 3SUM where we replace the sum function by a constant-degree polynomial in three variables. The motivations are threefold. Raz, Sharir, and de Zeeuw gave an O(n^{11}/^{6}) upper bound on the number of solutions of trivariate polynomial equations when the solutions are taken from the cartesian product of three n-sets of real numbers. We give algorithms for the corresponding problem of counting such solutions. Grønlund and Pettie recently designed subquadratic algorithms for 3SUM. We generalize their results to 3POL. Finally, we shed light on the General Position Testing (GPT) problem: "Given n points in the plane, do three of them lie on a line?", a key problem in computational geometry. We prove that there exist bounded-degree algebraic decision trees of depth O(n^{12/7+ϵ}) that solve 3POL, and that 3POL can be solved in O(n^{2}(log log n)3/2/(log n)1/2) time in the real-RAM model. Among the possible applications of those results, we show how to solve GPT in sub-quadratic time when the input points lie on o((log n)1/6/(log log n)1/2) constant-degree polynomial curves. This constitutes the first step towards closing the major open question of whether GPT can be solved in subquadratic time. To obtain these results, we generalize important tools - such as batch range searching and dominance reporting - to a polynomial setting. We expect these new tools to be useful in other applications.

Original language | English (US) |
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Title of host publication | 33rd International Symposium on Computational Geometry, SoCG 2017 |

Editors | Matthew J. Katz, Boris Aronov |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 131-1315 |

Number of pages | 1185 |

ISBN (Electronic) | 9783959770385 |

DOIs | |

State | Published - Jun 1 2017 |

Event | 33rd International Symposium on Computational Geometry, SoCG 2017 - Brisbane, Australia Duration: Jul 4 2017 → Jul 7 2017 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 77 |

ISSN (Print) | 1868-8969 |

### Other

Other | 33rd International Symposium on Computational Geometry, SoCG 2017 |
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Country | Australia |

City | Brisbane |

Period | 7/4/17 → 7/7/17 |

### Keywords

- 3SUM
- Dominance reporting
- General position testing
- Polynomial curves
- Range searching
- Subquadratic algorithms

### ASJC Scopus subject areas

- Software

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## Cite this

*33rd International Symposium on Computational Geometry, SoCG 2017*(pp. 131-1315). (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 77). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2017.13