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   Author  Topic: holomorphic polynomials  (Read 5016 times)
Mary I
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holomorphic polynomials  
« on: Dec 2nd, 2005, 5:04am »
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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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Re: holomorphic polynomials  
« Reply #1 on: Dec 2nd, 2005, 5:23am »
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No.  Note that any holomorphic function which has no zeroes on a simply-connected domain like D(0,1) has a holomorphic square root there.
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Re: holomorphic polynomials  
« Reply #2 on: Dec 2nd, 2005, 3:48pm »
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For example, f(x) = sqrt(x2 + 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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