### Abstract

Motivated by Gauss's first proof of the Fundamental Theorem of Algebra, we study the topology of harmonic algebraic curves. By the maximum principle, a harmonic curve has no bounded components; its topology is determined by the combinatorial data of a noncrossing matching. Similarly, every complex polynomial gives rise to a related combinatorial object that we call a basketball, consisting of a pair of noncrossing matchings satisfying one additional constraint. We prove that every noncrossing matching arises from some harmonic curve, and deduce from this that every basketball arises from some polynomial.

Original language | English (US) |
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Pages (from-to) | 267-286 |

Number of pages | 20 |

Journal | Discrete and Computational Geometry |

Volume | 37 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2007 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

Martin, J. L., Savitt, D., & Singer, T. (2007). Harmonic algebraic curves and noncrossing partitions.

*Discrete and Computational Geometry*,*37*(2), 267-286. https://doi.org/10.1007/s00454-006-1283-6