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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 $f(x_0)=g(x_0)$

Tao
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1 Answers1

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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.

Marc van Leeuwen
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  • In my research, I need f and g have the same degree. thank you for your comment. – Tao Apr 23 '13 at 11:44
  • Well, nothing prevents you from adding a large degree monomial to both $f$ and $g$. I think you should start thinking: what condition could prevent $f(x_0)=g(x_0)$ for some $x_0\in(1,2)$. The kind of conditions you are giving do not seem to do that. – Marc van Leeuwen Apr 23 '13 at 11:48
  • I have edited my question – Tao Apr 23 '13 at 11:56