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A sharp bound for the degree of proper monomial mappings between balls

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Abstract

The authors prove that a proper monomial holomorphic mapping from the two-ball to the N-ball has degree at most 2N-3, and that this result is sharp. The authors first show that certain group-invariant polynomials (related to Lucas polynomials) achieve the bound. To establish the bound the authors introduce a graph-theoretic approach that requires determining the number of sinks in a directed graph associated with the quotient polynomial. The proof also relies on a result of the first author that expresses all proper polynomial holomorphic mappings between balls in terms of tensor products.

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Authors and Affiliations

  1. Department of Mathematics, University of Illinois, 1409 W. Green St., 61801, Urbana, IL

    John P. D’Angelo & Emily Riehl

  2. Center for Nonlinear Studies, Los Alamos National Laboratory, MS B258, 87545, Los Alamos, NM

    Šimon Kos

Authors
  1. John P. D’Angelo
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  2. Šimon Kos
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  3. Emily Riehl
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Communicated by Steven Krantz

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D’Angelo, J.P., Kos, Š. & Riehl, E. A sharp bound for the degree of proper monomial mappings between balls. J Geom Anal 13, 581–593 (2003). https://doi.org/10.1007/BF02921879

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  • DOI: https://doi.org/10.1007/BF02921879

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