TITLE: Quickest way to determine a polynomial with positive integer coefficients
QUESTION [37 upvotes]: Suppose that you are given a polynomial $p(x)$ as a black box (i.e. some oracle, to which you feed $x$ and it returns $p(x)$). It is known that the coefficients of $p(x)$ are all positive integers. How do you determine what $p(x)$ is in the quickest way possible?
(There are 2 metrics for quickness: the number of calls to the oracle and total number of operations. The relationship between the two is not given so we try to minimize both.)

REPLY [17 votes]: Only $1$ input is needed to determine the polynomial: $\pi$. Since $\pi$ is transcendental, in principle we can figure out the polynomial from $f(\pi)$ (in a loose sense, none of the powers of $\pi$ can run into each other).
Note that this also works if we get rid of the positivity constraint.