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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.

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A small observation: the coefficients of $x^n$ and $y^n$ in $M_n$ must be integers; it follows that $M_2=\mathbb{Z}[x^2,xy,y^2]$. – Colin McQuillan Jul 5 '12 at 23:05
I think $M_3=\mathbb Z[x^3,y^3,x^2y,xy(x+y)/2]$ - is that right? – Colin McQuillan Jul 5 '12 at 23:18
up vote 1 down vote accepted

This paper gives a basis for the module of homogeneous, $p$-local integer-valued polynomials in two variables. That "$p$-local" bit might mean this isn't exactly what you want, but you might be able to use the methods in the paper to get what you want; also, a couple of the references at the end of the paper look promising.

As an example, it gives $$\{{y^3,xy^2,x^2(x-y),xy(x-y)/2\}}$$ as a 2-local basis at degree 3.

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