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I would like to write down some of my thoughts on "the set of polynomials $p\in\Bbb Q[x]$ which map the integers to the integers" and I would like to know what the proper notation is for discussing this set. My current understanding is that $\Bbb Z[x]$ is "the set of polynomials in $x$ having coefficients in $\Bbb Z$."

After some looking, I think that $\Bbb Z\langle x\rangle$ would be a good way to denote this set, and it could be written as

$$\Bbb Z\langle x\rangle=\left\{\sum_{i=0}^ka_i{x\choose i}\left|\right.\ a_i,k\in\Bbb Z\right\}$$

Note that the set in question is strictly larger than $\Bbb Z[x]$ and strictly smaller than $\Bbb Q[x]$, as $p(x)=\frac {x^2}2-\frac x2={x\choose 2}\in\Bbb Z\langle x\rangle$ and $p(x)\notin\Bbb Z[x]$ while $q(x)=\frac x2\notin\Bbb Z\langle x\rangle$ and $q(x)\in\Bbb Q[x].$

Secondary question: are there other "in-between" polynomial sets like this one, or are the integers unique in this regard?

abiessu
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    The set of polynomials such that $x-y$ is always a divisor of $f(x)-f(y)$ is a subset of your $\mathbb Z\langle x\rangle$, but contains more polynomials that $\mathbb Z[x]$. Essentially, it is the subset of $\mathbb Z\langle x\rangle$ where the $a_i$ is divisible by each of $1,2,3,\dots,i$. This set is also a subring of $\mathbb Q[x]$. – Thomas Andrews Aug 23 '14 at 20:47

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These are known as integer-valued polynomials.. It is a classical result of Polya and Ostrowski (1920) that any integer valued polynomial, i.e. any $\,f(x)\in \mathbb Q[x]\,$ with $\,f(\mathbb Z)\subset \mathbb Z,\,$ is an integral linear combination of binomial coefficients $\,{x \choose k},\,$ see for example Polya And Szego, Problems and theorems in analysis, vol II, Problem 85 p. 129 and its solution on p. 320, or see this answer.

These results have been extended from $\,\Bbb Z\,$ to much more general rings (e.g. Dedekind domains) by Cahen at al. I don't believe that there is any standard notation to denote such rings, though I recall that some ring-theorists use the notation $\,{\rm Int}(D)$ or something similar. A search on "integer-valued polynomials" should locate much interesting literature. This paper is one convenient place to start at: J. L. Chabert, $ $ An overview of some recent developments on integer-valued polynomials. 2010.

Bill Dubuque
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The notation $\mathrm{Int}(D)$ is due to Robert Gilmer in "Sets that determine integer-valued polynomials, J. Number Theory 33 (1989), 95--100" (see Cahen and Chabert, " Old problems and new questions around integer- valued polynomials and factorial sequences, 2006").

T. Ali
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