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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Polynomials, meanders, and paths in the lattice of noncrossing partitions


Author: David Savitt
Journal: Trans. Amer. Math. Soc. 361 (2009), 3083-3107
MSC (2000): Primary 05A18; Secondary 14P25
Published electronically: December 30, 2008
MathSciNet review: 2485419
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Abstract: For every polynomial $ f$ of degree $ n$ with no double roots, there is an associated family $ \mathcal{C}(f)$ of harmonic algebraic curves, fibred over the circle, with at most $ n-1$ singular fibres. We study the combinatorial topology of  $ \mathcal{C}(f)$ in the generic case when there are exactly $ n-1$ singular fibres. In this case, the topology of  $ \mathcal{C}(f)$ is determined by the data of an $ n$-tuple of noncrossing matchings on the set $ \{0,1,\ldots,2n-1\}$ with certain extra properties. We prove that there are $ 2(2n)^{n-2}$ such $ n$-tuples, and that all of them arise from the topology of $ \mathcal{C}(f)$ for some polynomial $ f$.


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Additional Information

David Savitt
Affiliation: Department of Mathematics, University of Arizona, 617 N. Santa Rita Avenue, Tucson, Arizona 85721
Email: savitt@math.arizona.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04579-0
PII: S 0002-9947(08)04579-0
Received by editor(s): June 8, 2006
Received by editor(s) in revised form: May 1, 2007, and June 18, 2007
Published electronically: December 30, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.