We strengthen a sufficient condition, due to R.S. Vieira, for a (reciprocal) polynomial \(R(x)=(x-\alpha _{1})(x-\alpha _{1}^{-1})\cdots (x-\alpha _{s})(x-\alpha _{s}^{-1})\in \mathbb {Z} [x],\) 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 \((x-(\alpha _{1}^{n}+\alpha _{1}^{-n}))\cdots (x-(\alpha _{s}^{n}+\alpha _{s}^{-n}))\) is a totally real Pisot polynomial not equal to \(x^{2}\pm x-1.\)

由于RS Vieira，我们为（倒数）多项式\（R（x）=（x- \ alpha _ {1}）（x- \ alpha _ {1} ^ {-1}）\\加强了充分条件cdots（X- \阿尔法_ {S}）（X- \阿尔法_ {S} ^ { - 1}）\在\ mathbb {Z} [X]，\）与至少4度，以具有\（（ 2s-2）\）在单位圆上为零。此外，我们表明，- [R是塞伦多项式当且仅当有一个自然数Ñ使得多项式\（（X - （\阿尔法_ {1} ^ {N} + \阿尔法_ {1} ^ { - n}））\ cdots（x-（\ alpha _ {s} ^ {n} + \ alpha _ {s} ^ {-n}））\）是不等于\（x ^ { 2} \ pm x-1。\）