On Zeros of Reciprocal Polynomials of Odd Degree

Volume 4, Issue 3 (GI8), 2003

Article 60

ON ZEROS OF RECIPROCAL POLYNOMIALS OF ODD DEGREE

PIROSKA LAKATOS AND LÁSZLÓ LOSONCZI

INSTITUTE OF MATHEMATICS,
DEBRECEN UNIVERSITY,
4010 DEBRECEN, PF.12, HUNGARY.
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

are near to the

st roots of unity (except the root

). 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 $A_k\in\mathbb{C}, A_m\ne 0$ 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

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

are on the unit circle if

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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