Let $f:\mathbb{R}\to\mathbb{R}$ be a polynomial such that $$|f(x)|<\epsilon\quad\text{for all $x$ with }|x|<1.$$ Can we find an explicit lower bound for the degree of $f$ in terms of $\epsilon$?
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Any polynomial can be scaled to have this property. If you want information about the degree, you have to add a limitation to compensate—for example, we could ask that $f$ be monic, or that $f(2)=1$, or something like this. – Andrew Dudzik Dec 18 '14 at 20:33
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Yes. The lower bound is the degree of the zero polynomial.
copper.hat
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+$1$ for answering the question of whether or not there is a lower bound. – Clayton Dec 18 '14 at 20:34
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I think the best way is just to make $$ f(x) = \epsilon - c$$ $\forall \; 0<c\leq\epsilon \;$ this shows why the lower bound is degree zero.
