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On Zeros of Reciprocal Polynomials of Odd Degree  
 
  Authors: Piroska Lakatos, Laszlo Losonczi,  
  Keywords: Reciprocal, Semi-reciprocal polynomials, Chebyshev transform, Zeros on the unit circle.  
  Date Received: 12/12/02  
  Date Accepted: 30/07/03  
  Subject Codes:

30C15, 12D10, 42C05.

  
  Editors: Anthony Sofo,  
 
  Abstract:

The first author [1] proved that all zeros of the reciprocal polynomial

$displaystyle P_m(z)=sum_{k=0}^m A_k z^kquad(zin mathbb{C}),$

of degree $ mge 2$ with real coefficients $ A_kin mathbb{R}$ (i.e. $ A_mne 0 $ and $ A_k=A_{m-k}$ for all $ k=0,dots,left[frac{m}{2}right]$) are on the unit circle, provided that

$displaystyle vert A_mvertge sum_{k=0}^{m} vert A_k-A_mvert=sum_{k=1}^{m-1} vert A_k-A_mvert.$

Moreover, the zeros of $ P_m$ are near to the $ m+1$st roots of unity (except the root $ 1$). A. Schinzel [3] generalized the first part of Lakatos' result for self-inversive polynomials i.e. polynomials

$displaystyle P_m(z)=sum_{k=0}^m A_k z^k$

for which $ A_kinmathbb{C}, A_mne 0$ and $ epsilon bar{A}_k=A_{m-k}$ for all $ k=0,dots,m$ with a fixed $ epsilonin mathbb{C}, vertepsilonvert=1.$ He proved that all zeros of $ P_m$ are on the unit circle, provided that

$displaystyle vert A_mvertge inflimits_{c,dinmathbb{C}, vert dvert=1}sum_{k=0}^{m} vert cA_k-d^{m-k}A_mvert.$

If the inequality is strict the zeros are single. The aim of this paper is to show that for real reciprocal polynomials of odd degree Lakatos' result remains valid even if

$displaystyle vert A_mvertgecos^2frac{pi}{2(m+1)} sum_{k=1}^{m-1} vert A_k-A_mvert.$

We conjecture that Schinzel's result can also be extended similarly: all zeros of $ P_m$ are on the unit circle if $ P_m$ is self-inversive and

$displaystyle vert A_mvertge cosfrac{pi}{2(m+1)} inflimits_{c,din mathbb{C}% , vert dvert=1}sum_{k=0}^{m} vert cA_k-d^{m-k}A_mvert . $
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