Volume 4,  Issue 3 (GI8), 2003

Article 60



E-Mail: lapi@math.klte.hu

E-Mail: losi@math.klte.hu

Received 12 December, 2002; Accepted 30 July, 2003.
Communicated by: A. Sofo

ABSTRACT.    The first author [1] proved that all zeros of the reciprocal polynomial
$\displaystyle P_m(z)=\sum_{k=0}^m A_k z^k\quad(z\in \mathbb{C}),$
of degree $ m\ge 2$ with real coefficients $ A_k\in \mathbb{R}$ (i.e. $ A_m\ne 0 $ and $ A_k=A_{m-k}$ for all $ k=0,\dots,\left[\frac{m}{2}\right]$) are on the unit circle, provided that
$\displaystyle \vert A_m\vert\ge \sum_{k=0}^{m} \vert A_k-A_m\vert=\sum_{k=1}^{m-1} \vert A_k-A_m\vert.$
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
for which $ A_k\in\mathbb{C}, A_m\ne 0$ and $ \epsilon
\bar{A}_k=A_{m-k}$ for all $ k=0,\dots,m$ with a fixed $ \epsilon\in
\mathbb{C}, \vert\epsilon\vert=1.$ He proved that all zeros of $ P_m$ are on the unit circle, provided that
$\displaystyle \vert A_m\vert\ge \inf\limits_{c,d\in\mathbb{C}, \vert d\vert=1}\sum_{k=0}^{m}
\vert cA_k-d^{m-k}A_m\vert.$
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
$\displaystyle \vert A_m\vert\ge\cos^2\frac{\pi}{2(m+1)} \sum_{k=1}^{m-1} \vert A_k-A_m\vert.$
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_m\vert\ge \cos\frac{\pi}{2(m+1)} \inf\limits_{c,d\in \mathbb{C}%
, \vert d\vert=1}\sum_{k=0}^{m} \vert cA_k-d^{m-k}A_m\vert .


[1] P. LAKATOS, On zeros of reciprocal polynomials, Publ. Math. (Debrecen) 61 (2002), 645-661.
[3] A. SCHINZEL, Self-inversive polynomials with all zeros on the unit circle, Ramanujan Journal, to appear.

Key words:
Reciprocal, Semi-reciprocal polynomials, Chebyshev transform, Zeros on the unit circle.

2000 Mathematics Subject Classification:
Primary 30C15, Secondary 12D10, 42C05.

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