Given an Nth order complex polynomial $P(z) = \sum\limits_{n=0}^N a_nz^n$ such that $a_n = a^*_{N-n}$ i.e. conjugate reciprocal, Lakatos and Losonczi mention that a necessary and sufficient condition for all roots of $P(z)$ to be on the unit circle is that all zeros of $P'(z)$ lie on $|z|<1$.

Also they showed that if

$|a_N| \geq \frac{1}{2}\sum\limits_{n=1}^{N-1}|a_n|$,

$P(z)$ has all roots on unit circle.

I'm unable to see how this condition holds when all coefficients $a_n = 1$ or $|a_n| = 1$ i.e. uni-modular coefficients and conjugate reciprocal.

Please help me understand what I'm missing here?

  • 1
    $\begingroup$ I think it's just a sufficient condition. There are probably many polynomials which do not satisfy $|a_N| \geq \frac{1}{2} \sum |a_n|$ but which have zeros on the unit circle. $\endgroup$ – Antonio Vargas Jul 13 '14 at 16:58
  • $\begingroup$ @AntonioVargas I think you are correct. The condition mentioned above is merely a sufficient condition. $\endgroup$ – sauravrt Jul 14 '14 at 22:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.