Palindrome-Polynomials with Roots on the Unit Circle

Given a polynomial f(x) of degree n, let f (x) denote its reciprocal, i.e., f (x) = xf(1/x). If a polynomial is equal to its reciprocal, we call it a palindrome since the coefficients are the same when read backwards or forwards. In this mathematical note we show that palindromes whose coefficients satisfy a certain magnitude-condition must have a root on the unit circle. More exactly our main result is the following. If a palindrome f(x) of even degree n with real coefficients 20, 21, . . . , 2n satisfies the condition |2k| ≥ |2n/2| cos(π/([ n/2 n/2−k ] + 2)), for some k ∈ {0, 1, . . . n/2 − 1}, then f(x) has unimodular roots. In particular, palindromes with coefficients 0 and 1 always have a root on the unit circle.