Journal of the
Korean Mathematical Society JKMS

The main result of this paper shows that every reciprocal Littlewood polynomial, one with $\left\{ -1,\,1\right\}$ coefficients, of odd degree at least $7$ has at least five unimodular roots, and every reciprocal Littlewood polynomial of even degree at least $14$ has at least four unimodular roots, thus improving the result of Mukunda. We also give a sketch of alternative proof of the well-known theorem characterizing Pisot numbers whose minimal polynomials are in $$\mathcal{A}_{N}=\left\{X^{d}+\sum_{k=0}^{d-1}a_{k}X^{k}\in\mathbb{Z} \left[X\right]\, :\, a_{k}=\pm N,\,0\leq k\leq d-1\right\}$$ for positive integer $N\geq 2\,$.

Keywords: Pisot numbers, Littlewood polynomials, unimodular roots, reciprocal polynomials

MSC numbers: 11R06, 11C08, 30C15, 12D10