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Please just give Hint of the following problem:

$f$ is a polynomial in $\mathbb{C}$, $\alpha,\beta$ be complex number with $\beta\neq 0$, given that $f(z)=\alpha$ when $z^5=\beta$, what can we say about the degree of $f$?

what I have?If I take it is of degree $n$, $f$ is onto, entire,open map, $f(z)=\sum_{k=1}^{n}a_nz^n$

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It is not clear whether "when" is supposed to mean "only when". –  Eckhard Jan 21 '13 at 7:47
    
I agree with Eckhard, although the only difference it makes to the final answer as to possible degree is whether or not the constant polynomial case is allowed. –  Jonas Meyer Jan 21 '13 at 7:50
    
Is the answer unique? If I take $\alpha=\beta=1$ then any $f(z)=z^{5n}$ works. –  PAD Jan 21 '13 at 8:06
    
@Pentelis: It depends on what you mean by unique; there is not a unique possible degree, but there is a unique set of possible degrees. –  Jonas Meyer Jan 21 '13 at 17:05

1 Answer 1

up vote 3 down vote accepted

Hints:

  • How many $z$ are there such that $z^5=\beta$?

  • How many times can a degree $n$ polynomial take the same value?

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okay can I use this result: $f$ is analytic at $z_0$, $f(z_0)=w_0$ and that $f(z)-w_0$ has a zero of order $n$ at $z_0$. If $\epsilon>0$ is small enough, there exist $\delta>0$ such that $\forall a,|a-w_0|<\delta$ the equation $f(z)=w_0$ has exactly $n$ roots in the disc $|z-z_0|<\epsilon$, here $f(z)=\alpha$ has $5$ roots namely all $5$ th roots of $\beta$, so degree $5$? –  Une Femme Douce Jan 21 '13 at 7:22
    
@Panu: That theorem is not needed, and I don't see how it is relevant. We don't know from what's given what the orders of the zeros are. One way to proceed is to note that if $f(z_0)=\alpha$, then $f(z)-\alpha=g(z)(z-z_0)$ for some polynomial $g$. We can apply this result to $g$ if there are more zeros. You will not be able to determine the exact degree, but you can give a bound (and be careful about one special case). You are correct that there are $5$ given solutions to $f(z)=\alpha$. –  Jonas Meyer Jan 21 '13 at 7:35
    
got your point,and then I can say degree $\le 5$, but what disaster will happen if degree $>5$? –  Une Femme Douce Jan 21 '13 at 7:43
    
@Panu: That is incorrect. For example, can a degree one polynomial, $f(z)=a_0+a_1z$ with $a_1\neq 0$, take the same value more than once? A degree two polymonial can: For example, if $f(z)-\alpha = (z-z_1)(z-z_2)$, then $f$ is a degree two polynomial that take the value $\alpha$ twice. –  Jonas Meyer Jan 21 '13 at 7:45
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@Panu: Right; the possible exception is the constant polynomial, $f(z)\equiv \alpha$. A degree $5$ example will occur with $f(z)=\alpha + (z^5 - \beta)$, but any higher degree is also possible. –  Jonas Meyer Jan 21 '13 at 18:56

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