Harmonic algebraic curves and noncrossing partitions

Jeremy L. Martin, David L Savitt, Ted Singer

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2 Scopus citations

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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ASJC Scopus subject areas

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

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