## 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 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 (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^{∗}(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.

Original language | English |
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Title of host publication | 39th International Symposium on Computational Geometry, SoCG 2023 |

Editors | Erin W. Chambers, Joachim Gudmundsson |

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

ISBN (Electronic) | 9783959772730 |

DOIs | |

State | Published - 1 Jun 2023 |

Event | 39th International Symposium on Computational Geometry, SoCG 2023 - Dallas, United States Duration: 12 Jun 2023 → 15 Jun 2023 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 258 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 39th International Symposium on Computational Geometry, SoCG 2023 |
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Country/Territory | United States |

City | Dallas |

Period | 12/06/23 → 15/06/23 |

### Funding

Funders | Funder number |
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Israel Science Foundation | 260/18 |

## Keywords

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