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Harmonic algebraic curves and noncrossing partitions
Martin, Jeremy L. Savitt, David Singer, Ted
Martin, Jeremy L.
Savitt, David
Singer, Ted
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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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This is the author's accepted manuscript.
Date
2007-02
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Springer Verlag
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Harmonic algebraic curves and noncrossing partitions (with David Savitt and Ted Singer), Discrete and Computational Geometry 37, no. 2 (2007), 267--286.