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 $\mathrm{I}\mathrm{n}\mathrm{t}(R)$, 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 $\mathrm{I}\mathrm{n}\mathrm{t}(R)$, 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.

As an operator

We can view the ring of integer-valued polynomials as an operator that takes as input an integral domain and outputs another integral domain (Note: Unlike the polynomial ring, this operator is not functorial). We can then ask what properties of the original integral domain continue to hold in the new ring. It turns out that most good properties, such as Noetherianness and unique factorization, do not hold any more, even when the starting ring is as nice as $\mathbb{Z}$. There are, however, some redeeming features:

• Ring of integer-valued polynomials over normal domain is normal