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Title: holomorphic polynomials Post by Mary I on Dec 2nd, 2005, 5:04am 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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Title: Re: holomorphic polynomials Post by Eigenray on Dec 2nd, 2005, 5:23am 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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Title: Re: holomorphic polynomials Post by Icarus on Dec 2nd, 2005, 3:48pm 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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