### 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) |
---|---|

Pages (from-to) | 267-286 |

Number of pages | 20 |

Journal | Discrete and Computational Geometry |

Volume | 37 |

Issue number | 2 |

DOIs | |

State | Published - 2007 |

### Fingerprint

### ASJC Scopus subject areas

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

### Cite this

*Discrete and Computational Geometry*,

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

**Harmonic algebraic curves and noncrossing partitions.** / Martin, Jeremy L.; Savitt, David L; Singer, Ted.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 37, no. 2, pp. 267-286. https://doi.org/10.1007/s00454-006-1283-6

}

TY - JOUR

T1 - Harmonic algebraic curves and noncrossing partitions

AU - Martin, Jeremy L.

AU - Savitt, David L

AU - Singer, Ted

PY - 2007

Y1 - 2007

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=33847312328&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33847312328&partnerID=8YFLogxK

U2 - 10.1007/s00454-006-1283-6

DO - 10.1007/s00454-006-1283-6

M3 - Article

AN - SCOPUS:33847312328

VL - 37

SP - 267

EP - 286

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 2

ER -