It is known that ${x\choose 0},{x\choose 1},\ldots,{x\choose n}\in\mathbb{Q}[x]$ is a $\mathbb{Z}$-basis for set of polynomials of degree at most $n$ which map $\mathbb{Z}$ into itself.
For fixed positive integer $n$, let $M_n\subset \mathbb{Q}[x,y]$ be the set of homogeneous polynomials of degree $n$, which map $\mathbb{Z}\times\mathbb{Z}$ into $\mathbb{Z}$. My question: is there an explicit description of a $\mathbb{Z}$-basis of $M_n$?
Certainly we have $M_n\subset y^n{x/y\choose 0}\mathbb{Z}+y^n{x/y\choose 1}\mathbb{Z}+\ldots+y^n{x/y\choose n}\mathbb{Z}$, but we don't have equality.