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I have a question, which seems deceptively simple to me, but when I thought about it, I couldn't really come up with a rigorous proof. Here goes,

Are the roots of a polynomial equation unique?

Suppose we have a general monic polynomial equation:

[tex]z^{n} + c_{1}z^{n-1} + c_{2}z^{n-2} + \ldots + c_{n} = 0[/tex]

where [itex]z \in \mathbb{C}[/itex] and [itex]c_{1}, c_{2}, \ldots[/itex] are real coefficients. Now let S be the set of all roots of this equation (counted according to their multiplicities so that if x is a root of multiplicity n, then S contains n occurrences of x. Strictly S is not a set, but rather a list, but you see my point.)

Can we find some [itex]S' \neq S[/itex] such that every member [itex]z^{'} \in S'[/itex] is a root of the above equation (and [itex]z^{'} \notin S[/itex])? Stated differently, does this equation have two different sets of roots?

The picture I have in mind for real roots is as follows: if the roots were not unique, the zero crossings of the function on the left hand side of the equation would not be unique, which is impossible. This reasoning does not work for complex roots :-(

Any ideas?

Thanks in advance.

-Vivek

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# Uniqueness of the roots of a polynomial equation

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