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Mary I
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 holomorphic polynomials   « on: Dec 2nd, 2005, 5:04am » Quote Modify Remove

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 2nd, 2005, 5:23am » Quote Modify

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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Icarus
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 Re: holomorphic polynomials   « Reply #2 on: Dec 2nd, 2005, 3:48pm » Quote Modify

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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