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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Polynomials, meanders, and paths in the lattice of noncrossing partitions
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by David Savitt PDF
Trans. Amer. Math. Soc. 361 (2009), 3083-3107 Request permission

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
  • 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
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 361 (2009), 3083-3107
  • MSC (2000): Primary 05A18; Secondary 14P25
  • DOI: https://doi.org/10.1090/S0002-9947-08-04579-0
  • MathSciNet review: 2485419