Interpolation domain

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
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Definition

An integral domain $R$ is termed an interpolation domain if, for any nonnegative integer $n$, there exist elements $(r_{0},r_{1},\dots ,r_{n})\in R^{n+1}$ giving a bijection between the polynomials in $\operatorname {Int} (R)$ (the ring of integer-valued polynomials over $R$) of degree at most $n$ and the elements of $R^{n+1}$ by:

$f\mapsto (f(r_{0}),f(r_{1}),\dots ,f(r_{n}))$.

In other words, for any tuple $(a_{0},a_{1},\dots ,a_{n})\in R^{n+1}$, there exists a unique $f\in \operatorname {Int} (R)$ such that $f(r_{i})=a_{i}$ for every $0\leq i\leq n$.

Examples

• Every field is an interpolation domain. This can be seen in many ways; one of them is the Lagrange interpolation formula.
• The ring of rational integers is an interpolation domain. We can set $r_{i}=i$ for this ring. In other words, given any sequence of $n+1$ integers, we can find an integer-valued polynomial sending $0,1,\dots ,n$ to the elements of that sequence. For full proof, refer: Ring of rational integers is an interpolation domain