Harmonic algebraic curves and noncrossing partitions

Jeremy L. Martin, David L Savitt, Ted Singer

Research output: Contribution to journalArticle

2 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)267-286
Number of pages20
JournalDiscrete and Computational Geometry
Volume37
Issue number2
DOIs
StatePublished - 2007

Fingerprint

Noncrossing Partitions
Algebraic curve
Harmonic
Topology
Polynomials
Maximum principle
Fundamental theorem of algebra
Algebra
Curve
Complex Polynomials
Maximum Principle
Gauss
Deduce
Polynomial

ASJC Scopus subject areas

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

Cite this

Harmonic algebraic curves and noncrossing partitions. / Martin, Jeremy L.; Savitt, David L; Singer, Ted.

In: Discrete and Computational Geometry, Vol. 37, No. 2, 2007, p. 267-286.

Research output: Contribution to journalArticle

Martin, Jeremy L. ; Savitt, David L ; Singer, Ted. / Harmonic algebraic curves and noncrossing partitions. In: Discrete and Computational Geometry. 2007 ; Vol. 37, No. 2. pp. 267-286.
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