Abstract
We strengthen a sufficient condition, due to R.S. Vieira, for a (reciprocal) polynomial R(x) = (x- alpha(1))(x-alpha(-1)(1)) .. (x- alpha(s))(x-alpha(-1)(s)). Z[x], with degree at least 4, to have (2s-2) zeros on the unit circle. Also, we show that R is a Salem polynomial if and only if there is a natural number n such that the polynomial (x-(alpha(n)(1) + alpha(-1)(1))) ... (x-(alpha(n)(s) + alpha(-n)(s))) is a totally real Pisot polynomial not equal to x(2) +/- x - 1.