Some basic properties of Pisot number A Pisot number is  an algebraic integer $>1 $ and all of whose conjugates have modulus $<1$.   First assume that $q$ is  a Pisot number. Denote by $q_1,\ldots, q_d$ the algebraic conjugates of $q$.  Let $P(x)=\sum_{i=0}^n\epsilon_i x^i$ be a polynomial with coefficients in $\{0, \pm 1,\ldots, \pm m\}$. Suppose that $P(q)\neq 0$. Then $P(q_j)\neq 0$ for $1\leq j\leq d$, and $P(q)\prod_{j=1}^dP(q_j)$ is a non-zero integer
How can we prove this result?
 A: The result you ask about has nothing to do with Pisot numbers --- it is true for any algebraic integer. If a polynomial with integer coefficients vanishes at some algebraic number $\alpha$, then the polynomial is a multiple of the minimal polynomial of $\alpha$, so it vanishes at all the conjugates of $\alpha$. And the product of $P(\beta)$ over all the conjugates $\beta$ of $\alpha$ (including $\beta=\alpha$) is an algebraic integer, and is fixed by the Galois group of the rationals adjoin all the conjugates, so it's rational, so it's an integer. 
A: The fundamental theorem of symmetric polynomials says that a symmetric polynomial $Q(x_1,x_2,\ldots,x_n)\in \mathbb Z[x_1,x_2,\ldots,x_n]$ is a polynomial in the elementary symmetric polynomials $\sigma_1,\sigma_2,\ldots,\sigma_n$, i.e. $Q(x_1,x_2,\ldots,x_n)=R(\sigma_1,\sigma_2,\ldots,\sigma_n)$ for a polynomial $R\in\mathbb Z[x_1,x_2,\ldots,x_n]$.
The polynomial $P(q)\prod_{i=1}^dP(q_i)$ is symmetric in $q,q_1,\ldots,q_d$, therefore $P(q)\prod_{i=1}^dP(q_i)=R(\sigma_1,\sigma_2,\ldots,\sigma_n,\sigma_{n+1})$ for a polynomial $R\in\mathbb Z[x_1,x_2,\ldots,x_n,x_{n+1}]$. Since $\sigma_i$, the elementary symmetric polynomials on $q,q_1,\ldots,q_d$, are the coefficients of the minimal polynomial of $q$ (here we use the fact that $q$ is an algebraic integer) it follows that $\sigma_i$ are integers (see this). Therefore $P(q)\prod_{i=1}^dP(q_i)=R(\sigma_1,\sigma_2,\ldots,\sigma_n,\sigma_{n+1})\in\mathbb Z$.
As Gerry Myerson said this has nothing to do with Pisot numbers. It is true for any algebraic integer.
