Let \(P(X)=a_nX^n+a_{n-1} X^{n-1}+\dots+a_1X+a_0\) be a polynomial with coefficients in \(\mathbb C\). \(P(X)\) is called a Littlewood polynomial if the coefficients \(a_i\in\{1,-1\}, \;i=0,\dots,n\). The polynomial \(P(X)\) is said to be reciprocal if it satisfies \(P(X)=X^n P(X^{-1})\). A root of \(P(X)\) is called unimodular if it has an absolute value 1.
K. Mukunda [“Littlewood Pisot numbers”, J. Number Theory 117, No. 1, 106–121 (2006; Zbl 1095.11049)] proved that every reciprocal Littlewood polynomial of odd degree at least 3 has at least three unimodular roots. The main theorem of this paper is a generalization of Mukunda’s result to:
Theorem. Every reciprocal Littlewood polynomial of odd degree \(d\geq 7\) has at least five unimodular roots. Every reciprocal Littlewood polynomial of even degree \(d\geq 14\) has at least four unimodular roots.
A Pisot number is a real algebraic integer \(\alpha>1\), all of whose conjugates lie inside the open unit disc. Let \(N\) and \(d\) be a positive integers. Denote \[ A_N= \{X^d+\sum_{k=0}^{d-1} a_k X^k\in \mathbb Z[X] : a_k=\pm N,\;0\leq k\leq d-1\}. \] For \(N\geq 2\), J. F. Traub [Iterative methods for the solution of equations. Englewood Cliffs: Prentice-Hall (1964; Zbl 0121.11204)] proved that:
Let \(\gamma_n\) be a Pisot number of degree \(n\), whose minimal polynomial \(P_n(X)\in A_n\). Then \[ P_n(X)=X^n-N X^{n-1} -NX^{n-2}-\dots-NX-N. \] When \(n\to\infty\), the sequence \(\gamma_n\) is strictly increasing and converges to \(N+1\).
Mukunda proved the same for \(N=1\). The author gives an alternative proof of the result of Traub for \(N\geq 2\), with a straight generalization of Mukunda’s proof for \(N=1\).