Let $(a\_{i})$ be an increasing sequence of positive integers given by a linear recurrence $a\_{i+n}=c\_{n}a\_{i+n-1}+\dots +c\_{1}a\_{i}$ with $c\_{i}\in\{-1,0,1\}$ and $a\_{i}=2^{i}$ for $i=1,\dots n$ such that the characteristic polynomial $p(x)=-x^{n}+c\_{n}x^{n-1}+\dots +c\_{1}x^{0}$ has a unique dominating real root $\alpha> 1$. Let $(s\_{i})\in\{-1,0,1\}^{m}$ with $m\ge n$. 
Is it true that
$\sum\_{i=1}^{m}s\_{i}a\_{i}=0$ 
if and only if $p$ divides the polynomial $\sum\_{i=1}^{m}s\_{i}x^{i}$? Where I can find a proof of this kind of result? 



A question on linear recurrence
I don't know what "dominating" means so perhaps this isn't a counterexample, but how about $-p(x)=x^5 - x^4 - x^3 - x^2 + x + 1=(x^2-x-1)(x^3-1)$ (which has a unique real root greater than 1), with $a\_1,...,a\_5=1,2,3,5,8$ and $s\_1=s\_2=-1$ and $s\_3=1$ giving $\sum s\_ix^i=x^3-x^2-x$?


