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Topic: holomorphic polynomials (Read 5260 times) 

Mary I
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Hello! If we know that f is holomorphic on D(0,1) and assume that f^2 is a holomorphic polynomial on D(0,1), does it follow that f is also a polynomial on D(0,1)? I am almost sure that f is a polynomial but how can I show it? I have tried to use the Cauchy product but it didn't work out.


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Eigenray
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Re: holomorphic polynomials
« Reply #1 on: Dec 2^{nd}, 2005, 5:23am » 
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No. Note that any holomorphic function which has no zeroes on a simplyconnected domain like D(0,1) has a holomorphic square root there.


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Icarus
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Re: holomorphic polynomials
« Reply #2 on: Dec 2^{nd}, 2005, 3:48pm » 
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For example, f(x) = sqrt(x^{2} + 4) satisfies all your conditions, but is not a polynomial. (By the way, "holomorphic polynomial" is redundant. All polynomials are holomorphic.)


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