Minimal value of a polynomial I do not know the following statement is true or not:
Given $1<x_0<2$, there exists $\delta>0$ such that for any n, define  $A=\{ f(x)=\sum\limits_{i=0}^{n}a_ix^i\}$ where $a_i\in\{0\,,1\}$, then for any $f\,,g \in A$ and their degrees are the same, we have $\delta\leq|f(x_0)-g(x_0))|$ or $f(x_0)=g(x_0)$ 

Let $x_0$ be a real root of the polynomial $1-X^2-X^3+X^4$ in the interval $(1,2)$; it exists by the intermediate value theorem because this polynomial has value $0$ and derivative $-1$ at $X=1$, and value $5$ at $X=2$. Then with $f(x)=1+x^4$ and $g(x)=x^2+x^3$ one has $f(x_0)=g(x_0)$, so no positive $\delta\leq|f(x_0)-g(x_0)|=0$ can exist.
I might add that the fact that you get answers that are looking at cases where $f(x_0)-g(x_0)=0$, it is because obviously this is precisely the only thing that can prevent $\delta$ from existing, so the whole formulation with $\delta$ seems a bit pointless.
