I have poked around the forum, and haven't been able to find an answer to the following question:
Given $n$ integral points $(x_i, y_i)$ with distinct $x_i$, is it possible to construct a polynomial $P(x)$ with rational coefficients such that
- all (not necessarily distinct) critical points of $P(x)$ are rational points, and
- all $n$ of the provided integer points are on the polynomial $(P(x_i) = y_i)$ and are not critical points $(P'(x_i) \neq 0)$.
If so, how would one go about constructing $P$?
What if we are additionally given an integer $d > n > 0$, and told $P(x)$ must be degree $d$?