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For any complex polynomial $P(X)$ we denote by $Z_0(P)$ the set of zeroes of $P$ and by $Z_1(P)$ the set of zeroes of $P(X)-1$. Prove that if $Z_0(P)=Z_0(Q)$ and $Z_1(P)=Z_1(Q)$ then $P=Q$. Assume that P(X) and Q(X) are nonconstant.

I am trying to limit the degrees and number of roots, but multiplicity always gets in the way.

Hint: Use the theorem that a root is a double root if and only if it is also the root of its derivative.

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    This is wrong as stated, with e.g. $P(X)=5, Q(X)=42$. You have to assume that at least one polynomial is non-constant or one of the sets is non-empty. – gammatester Mar 31 '15 at 08:10

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Since zeros of both polynomials are same, implies that their factors are same. Next, if possible,assume some factor is repeating m times in one and n times in other say it is (x-a). Now the values for which a polynomial becomes 1 is also same for both . This gives (x-a)^(n-m) = 1 which has more than one solution. So, a contradiction. Thus both polynomials are same.

Sry
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