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Suppose I have a polynomial $$ p(x)=\sum_{i=0}^n p_ix^i. $$ For simplicity furthermore assume $p_n=1$.

As it is well known we may use Gershgorin circles to give an upper bound for the absolute values of the roots of $p(x)$. The theorem states that all roots are contained within a circle with radius $$ r=\max\{|p_0|, 1+|p_1|,\ldots, 1+|p_{n-1}|\}. $$


Now I wonder if there is something like an inverse of this theorem. Suppose I know that all roots are contained within a circle of radius $r$. Is there anything that can be said about the maximum coefficient, i.e. $$ \max |p_i|\leq \text{some function of }r? $$

I would also be graceful for a counter example.

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There are actually stronger estimates that deal with all of the roots. Let $r_1,\ldots,r_n$ be the roots of your polynomial (with multiplicities as appropriate). Then $$ \max_{0\le i\le n} |p_i| \ge \frac{1}{4^n} \prod_{j=1}^n \max\bigl\{|r_j|,1\bigr\} $$ and $$ \max_{0\le i\le n} |p_i| \le 4^n \prod_{j=1}^n \max\bigl\{|r_j|,1\bigr\}. $$ For references to much more general results, see the answers to Bounds on coefficients of factors of a multivariate polynomial and Getting a bound on the coefficients of the factor polynomial . (The $4^{\pm n}$ constants are not best possible.)

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Since the coefficients are $\pm 1$ times sums of products of the roots, this is obvious: for a polynomial of degree $d$ with all roots of absolute value $\le r$,

$$ |p_i| \le {d \choose i} r^{d-i} $$

The maximum of the right side is at $i = \left\lfloor \dfrac{d-r}{1+r} \right\rfloor$ or $\left\lceil \dfrac{d-r}{1+r} \right\rceil$ if $r < d$, or at $i=0$ if $r \ge d$.

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  • $\begingroup$ Okay, that was in fact quite obvious. Thanks. $\endgroup$ Jul 23, 2014 at 16:31

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