---
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.'
address:
- 'Jeremy L. Martin, Department of Mathematics, University of Kansas, 405 Snow Hall, 1460 Jayhawk Blvd., Lawrence, KS 66045'
- 'David Savitt, Department of Mathematics, University of Arizona, 617 N.Santa Rita Ave., Tucson, AZ 85721'
- 'Ted Singer, 38 Greenacres, Scarsdale, NY 10583'
author:
- 'Jeremy Martin, David Savitt, and Ted Singer'
title: Harmonic algebraic curves and noncrossing partitions
---

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Introduction {#introduction .unnumbered}
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The first proof of the Fundamental Theorem of Algebra was published in 1746 by d’Alembert [@dAlembert]. In an attempt to correct the lack of rigor in d’Alembert’s approach, as well as in subsequent attempts, Gauss offered a new proof in 1799 