On Zeros of Reciprocal Polynomials of Odd Degree

The first author [1] proved that all zeros of the reciprocal polynomial Pm(z) = m ∑ k=0 Akz k (z ∈ C), of degree m ≥ 2 with real coefficients Ak ∈ R (i.e. Am 6= 0 and Ak = Am−k for all k = 0, . . . , [ m 2 ] ) are on the unit circle, provided that |Am| ≥ m ∑ k=0 |Ak −Am| = m−1 ∑ k=1 |Ak −Am|. Moreover, the zeros of Pm are near to the m + 1st roots of unity (except the root 1). A. Schinzel [3] generalized the first part of Lakatos’ result for self–inversive polynomials i.e. polynomials Pm(z) = m ∑ k=0 Akz k for which Ak ∈ C, Am 6= 0 and Āk = Am−k for all k = 0, . . . ,m with a fixed ∈ C, | | = 1. He proved that all zeros of Pm are on the unit circle, provided that |Am| ≥ inf c,d∈C, |d|=1 m ∑ k=0 |cAk − dAm|. 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 |Am| ≥ cos π 2(m+ 1) m−1 ∑ k=1 |Ak −Am|. We conjecture that Schinzel’s result can also be extended similarly: all zeros of Pm are on the unit circle if Pm is self-inversive and |Am| ≥ cos π 2(m+ 1) inf c,d∈C, |d|=1 m ∑