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Let $n\in\mathbb N^*$, $P(x)=a_0+\dotsb+a_{n-1}x^{n-1}+x^n$ and $r_1,\dotsc,r_n\in\mathbb C$ the roots of $P$.

Is it true $\lim\limits_{\max(\lvert a_i\rvert,i=0\dotsc n-1)\rightarrow 0} \max(\lvert r_i\rvert,i=1\dotsc n)=0$?

If there is an inequality between the two maxes, can you give it?

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    $\begingroup$ Yes. This follows from Rouché’s theorem. $\endgroup$
    – Anthony Quas
    Commented Apr 2, 2022 at 15:22
  • $\begingroup$ Isn't there a standard inequality between the two maxes $\endgroup$
    – markvs
    Commented Apr 2, 2022 at 15:23
  • $\begingroup$ @markvs can you give it ? $\endgroup$
    – Dattier
    Commented Apr 2, 2022 at 15:27

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See an application of Rouché's theorem to polynomials. As I remember, it is used in one of the proofs of The Fundamental Theorem of Algebra.

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This has nothing to do with complex numbers, or Rouche's theorem. In any normed field, denote $r=\max|r_j|$ and $A=\max|a_j|$. Then $$r^n\leq An\max\{1,r^{n-1}\}.$$ So if $r\leq 1$ then $r\leq (An)^{1/n}$ and if $r>1$, then $r\leq An$.

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