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Find the number of roots $f(z)=z^{10}+10z+9$ in $D(0,1)$. I want to find $g(z)$ s.t. $|f(z)-g(z)|<|g(z)|$, but I cannot. Any hint is appreciated.

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up vote 3 down vote accepted

First, we factor by $z+1$ to get $f(z)=(z+1)(z^9-z^8+z^7+\dots-z^2+z+9)$. Let $F(z):=z^9-z^8+z^7+\dots-z^2+z+9$ and $G(z)=9$. Then for $F$ of modulus $1$, $|F(z)-G(z)|\leqslant 9=|G(z)|$ (the equality is actually strict as $|z-z^2|=|1-z|<2$).

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@Sanchez You can post your alternative approach as an answer. –  Davide Giraudo Nov 30 '12 at 23:13
    
actually let me delete the last comment, as I hope that OP can figure it out himself. –  user27126 Nov 30 '12 at 23:14
    
Thank you very much, @Davide. I think you are right and there is no root inside the region. Only when we remove $(x+1)$, things get clearer. Also, thank Sanchez's comments. –  Sam Nov 30 '12 at 23:20
    
How about $z=-1$ again? –  Ma Ming May 3 '13 at 13:56

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