Let's say I have a polynomial: $$p(x) = \sum_{n=0}^N a_n x^n$$ where $x \in \mathbb C$.
Does there exist theory and/or methods on approximating $p$ as: $$p(x) \approx \hat p(x) = \sum_{m=0}^M b_n x^m$$ where $b_n \in \{0,1\}$ or perhaps $b_n \in \{-1,1\}$? I am prepared to allow $M \gg N$ if that is necessary to get a reasonable approximation.
Bonus generalisation: $b_n$ being discrete, not necessarily with a finite alphabet, would also be interesting.