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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?

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    $\begingroup$ 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 $\mathbb Q[x]$. $\endgroup$ – 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.

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.

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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").

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