A sharp bound for the degree of proper monomial mappings between balls
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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.
Math Subject Classifications32B99 32H02 11B309
Key Words and PhrasesProper holomorphic mappings unit ball Lucas polynomials
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