Harmonic algebraic curves and noncrossing partitions
Combinatorics
2011-10-05 v1 Algebraic Geometry
Complex Variables
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.
Cite
@article{arxiv.math/0511248,
title = {Harmonic algebraic curves and noncrossing partitions},
author = {Jeremy Martin and David Savitt and Ted Singer},
journal= {arXiv preprint arXiv:math/0511248},
year = {2011}
}
Comments
18 pages, 14 color figures; uses epsfig and psfrag