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I am working on a problem which just arrived at the following question related to roots of polynomials.

Let $f(x)=x^7-2x^6+1$ and $g(x)=x^{10}-2x^9+1$. These polynomials have $x=1$ as their unique common root (because $f(x)/(x-1)$ and $g(x)/(x-1)$ are irreducible).

However, I would like to proof the following: If $\alpha\neq 1$ and $\beta\neq 1$ are roots of $f(x)$ and $g(x)$, respectively, then $|\alpha|\neq |\beta|$.

I would be very grateful for any suggestion.

Jean
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  • By brute force, this would mean verifying that this polynomial of degree $54$ has no roots on the unit circle. – dxiv Jul 29 '21 at 05:08
  • @dxiv thanks for your suggestion. However, brute force is not very useful since I want to generalize this problem for higher degrees. I tried some inversion on circles etc, but always arrive at a very big polynomial for which we want information on size of roots. – Jean Jul 29 '21 at 23:05
  • You can work out resultants symbolically, too, though it's not necessarily pretty. As for the magnitude of roots, that's a well studied problem in relation to stable polynomials. – dxiv Jul 29 '21 at 23:55

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