I have a black-box polynomial function, with $n$ inputs. The only things I know for sure about this polynomial are that (a) it has no constant term, and (b) all coefficients and exponents are integers with magnitude below some $k$. (Exponents are positive, of course.) And I strongly suspect that some of the inputs aren't used at all.
Let's say I get to evaluate this polynomial once, with any set of real number inputs I choose. Is there a clever way of choosing those inputs, such that the answer allows me to unambiguously and straightforwardly recover the polynomial?
One way, of course, would be to make all the inputs unrelated irrational numbers, and then brute-force polynomials until I got a matching answer. If $n$ were $2$, I could feed in (say) $e$ and $\pi$ and see whether the answer matched $5e\pi^4+2e$, $7e^2\pi^2+4\pi$, etc. Obviously that would take some time.