The following result seems to be fairly well-known.
Suppose $p$ is a polynomial in one variable with positive integer coefficients, and suppose that $p$ is to be determined by giving several inputs in succession, which are allowed to depend on previous outputs. Then only two inputs are needed to determine $p$. [1, 2, 3, 4, 5, 6]
Is it also well-known that it still holds when $p$ is allowed to depend on any finite number of variables?
Suppose $p$ depends on $v$ different variables. Take $B > p(2, \dotsc, 2)$ and $D > \log_2 p (2, \dotsc, 2)$, which are bounds on the coefficients and on the degree, respectively. Now each coefficient of $p$ can be encoded as a digit of the base-$B$ representation of $$ p_2 = p(B^{D^0},B^{D^1}, \dotsc, B^{D^{v-1}}). $$ More specifically, to find the coefficient of $x_0^{d_0} \dotsm x_{v-1}^{d_{v-1}}$, consider its contribution to $p_2$, $$ (B^{D^0})^{d_0} \cdot (B^{D^1})^{d_1} \dotsm (B^{D^{v-1}})^{d_{v-1}} = B^{d_0 D^0 + d_1 D^1 + \dotsb + d_{v-1} D^{v-1}}. $$ This gives a power of $B$ (and therefore a digit in $p_2$) uniquely associated to the desired coefficient precisely because $D$ exceeds the degree of $p$.
