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Question

Suppose $z$ is some complex number satisfying $|z|=1$. Is there a way to simplify the expression $$\min_{z}\left|\sum_{k=0}^{n}a_k z^k\right|$$ to something that can be easily computed given real numbers $\{a_k\}_{k=0}^n$?


My attempt

Write $z=\cos(\theta)+i\sin(\theta)$ and use De Moivre's formula to get $z^k=\cos(k\theta)+i\sin(k\theta)$. Then, we get $$\left|\sum_{k=0}^{n}a_k z^k\right|=\left|\sum_{k=0}^{n}a_k (\cos(k\theta)+i \sin(k\theta))\right|.$$ Euler's formula, then allows us to write $$\left|\sum_{k=0}^{n}a_k z^k\right|=\left|\sum_{k=0}^{n}a_k e^{k\theta i}\right|.$$ So the problem would reduce to that of solving $$\min_{\theta}\left|\sum_{k=0}^{n}a_k e^{k\theta i}\right|.$$ This seems like it could help but I don't know how to proceed from here.

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    $\begingroup$ There is no need to encumber the development with the constant term $1$. Absorb it in $a_0$. $\endgroup$ – Yves Daoust May 4 at 9:44
  • $\begingroup$ Are the $a_i$ real ? $\endgroup$ – Yves Daoust May 4 at 9:53
  • $\begingroup$ You are right, so I edited the question. Yes, the $a_i$'s are supposed to be real. $\endgroup$ – mzp May 4 at 9:55
  • $\begingroup$ I tried with $az^2+bz+c$ and $az^3+bz^2+cz+d$ but found no nice simplification. $\endgroup$ – Yves Daoust May 4 at 10:13
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    $\begingroup$ In general there is no way to find this minimum. For example, if the polynomial under consideration has a root on the circle then the minimum zero. However, one can find upper bound by integrating the polynomial square, i.e.computing its $L^2$ norm. Clearly, this on is $\sqrt{\sum |a_k|^2}$, hence the minimum is bounded by this value. $\endgroup$ – Salcio May 4 at 10:43

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