Exercise 1.4.15* (Sufficient condition for integer-valued polynomials.)

Proof.

Let $P(x)={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{k=0}^n{c}_k\left(\genfrac{}{}{0.0pt}{}{x}{k}\right)$. Assume that $P(0),P(1),\dots ,P(n)$ are integers.

Then $(c_0,c_1,\dots ,c_n)$ is solution of the linear system

We rewrite this system

The matrix $A$ of this system is $A={(a_{i,j})}_{0\le i,j\le n+1}$, where $a_{i,j}=\left(\genfrac{}{}{0.0pt}{}{i}{j}\right)$. Since $A$ is triangular, with diagonal elements $a_{i,i}=1$, $\det <!--\; FUNCTION\; APPLICATION\; -->(A)=1$, so $A\in {\mathrm{SL}}_{n+1}(\mathbb{Z})$. Therefore $A$ is regular, and $A^{-1}\in {\mathit{SL}}_{n+1}(\mathbb{Z})$ has integer coefficients. So, if $P(0),\dots ,P(n)$ are integers, so are $c_0,\dots ,c_n$. By Exercise 1.4.14, $P$ is integer-valued.

For a more elementary argument, without linear algebra, we note that $c_0=P(0)\in \mathbb{Z}$. Assume for induction that $c_0,\dots c_{i-1}$ are integers for some $i$, where $1\le i\le n$. Then

The induction is done, so $c_0,c_1,\dots ,c_n$ are integers. □