(1) Department of Mathematics, University of Illinois, 1409 W. Green St., 61801 Urbana, IL;(2) Center for Nonlinear Studies, Los Alamos National Laboratory, MS B258, 87545 Los Alamos, NM
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.