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.
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Martin, J., Savitt, D. & Singer, T. Harmonic Algebraic Curves and Noncrossing Partitions. Discrete Comput Geom 37, 267–286 (2007). https://doi.org/10.1007/s00454-006-1283-6
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DOI: https://doi.org/10.1007/s00454-006-1283-6