Harmonic algebraic curves and noncrossing partitions

Details Harmonic algebraic curves and noncrossing partitions Harmonic algebraic curves and noncrossing partitions

2005-11-10

arxiv.org/abs/math/0511248v1

Discrete Comput. Geom. 37, no. 2 (2007), 267-286

Mathematics

Combinatorics

18 pages, 14 color figures; uses epsfig and psfrag

Scientific paper

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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