Ring of integer-valued polynomials

Definition

Let $R$ be an integral domain and let $K$ be its field of fractions. The ring of integer-valued polynomials for $R$, denoted $\operatorname {Int} (R)$, is defined as the subset of the polynomial ring $K[x]$ comprising those polynomials $f$ such that $f(x)\in R$ whenever $x\in R$.

Facts

• In general, $R[x]$ is a subring of $\operatorname {Int} (R)$, which in turn is a subring of $K[x]$.
• When $K$ has characteristic zero, the ring of integer-valued polynomials is contained in the ring generated by binomial polynomials over $R$. For full proof, refer: Ring of integer-valued polynomials is contained in ring generated by binomial polynomials
• For the ring of rational integers $\mathbb {Z}$, the ring of integer-valued polynomials equals the ring generated by binomial polynomials. For full proof, refer: Ring of integer-valued polynomials over rational integers equals ring generated by binomial polynomials
• An interpolation domain is an integral domain for which interpolation using integer-valued polynomials is possible: for any degree $n$, there exist $n+1$ points such that the integer-valued polynomials of degree $n$ can be interpolated from any collection of values at those $n+1$ points.