Interpolation domain

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 defining ingredient::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 $$0 \le i \le n$$.

Examples

 * Every field is an interpolation domain. This can be seen in many ways; one of them is the Lagrange interpolation formula.
 * The particular example::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.