Polynomial interpolation over the integers such that all coefficients are from the integers If one is given $t$ different points of a polynomial (all values are from the integers), is it then always possible to construct a polynomial of degree $t$ such that it interpolates all points AND all coefficients are from the integers?
Second: What if some of the points correspond to derivatives? So can the Brikhoff interpolation problem with $t$ points given be used to interpolate a polynomial of degree $t$ such that all coefficients are from the integers?
Note that it is wanted that we are only given $t$ points to interpolate a polynomial of degree $t$. This gives one degree of freedom. Otherwise it is easy to find a counterexample.
First question: Let $x_1,x_2, \ldots, x_t \in \mathbb{N}_0$ such that all $x_i$ are distinct and ordered, i.e., $0\leq x_1 < x_2 < x_3 < \ldots < x_t$. And let let $y_1,y_2, \ldots, y_t \in \mathbb{Z}$. Does there exist a polyonimial $f(x) = a_0 + a_1x + a_2 x^2 + \ldots + a_t x^t$ such that for all $i$ it holds that $f(x_i)=y_i$ and all $a_j \in \mathbb{Z}$.
Second question: Now assume that $c_1^{i_1}, c_2^{i_2}, \ldots, c_t^{i_t} \in \mathbb{Z}$, where $i_j \in \mathbb{N}_0$ is just an indice (not the power). For these indices it holds that $0 \leq i_0 \leq i_1 \leq \ldots \leq i_t < t$ and at least one $i_j > 0$.
Does there exist a polyonimial $f(x) = a_0 + a_1x + a_2 x^2 + \ldots + a_t x^t$ such that for all $j$ it holds that $f^{i_j}(x_j)=c_j^{i_j}$ and all $a_j \in \mathbb{Z}$, where $f^{i_j}(x)$ denotes the $i_j$-th derivative of $f(x)$.

I tried to solve the second question with Birkhoff interpolation. The Birkhoff interpolation can be used to reconstruct the function and also single coefficients: The interpolation of one coefficient is based on a matrix $A$ which is determined by all $x_j$ and $c_j^{i_j}$. Then a coefficient $a_{k-1}$ is computed as $det(A_k)/det(A)$ where $A_k$ is obtained from $A$ by replacing the $k$-th column of $A$ with the $c_j^{i_j}$ in lexicographic order. However, I'm not able to proof that $det(A_k)/det(A) \in \mathbb{Z}$. Note that if we want to interpolate the polynomial of degree $t$ with only $t$ points/derivatives given, then we have to see the birkhoff interpolation problem as a problem where we are given $t+1$ points/derivates but we are allowed to modify one point $(x_z,c_z^{i_z})$ arbitrarily.
The problem is also closely related to determinants, but I have very little knowledge in this area.
Until now, I couldn't construct a counterexample for it or proof it.

A proof, counterexample or any hints where to get additional information would be great! Or maybe someone knows something about the eigenvalues of  the matrix of the Birkhoff interpolation?

In general this is not true.
   Take $t=2, x_1=-1,x_2=1,y_1=0,y_2=1$. Then $a_0-a_1+a_2=0, a_0+a_1+a_2=1$. Adding this equations results in $2(a_0+a_2)=1$ which is impossible for integers $a_0,a_2$.
