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Your question is related to the study of (generalized) numerical polynomials: If $R$ is an integral domain and $K$ the field of fractions of $R$, then the set ${\rm Int}(R) := \{f \in K[x]: f(R) \subseteq R\}$ is a subdomain of $K[x]$, whose elements are called the numerical polynomials over $R$ (in one variable $x$).

The domain ${\rm Int}(R)$ has been the subject of a great deal of research: Original work in the area was entirely centered on the case where $R$ is the ring of integers, and was motivated by interpolation problems in the early days of calculus. It was only in 1919 that A. Ostrowski and G. Pólya first considered numerical polynomials in their own right, though focused on the case where $R$ is the ring of integers of a number field $K$. In particular, they could show, in this context, that ${\rm Int}(R)$ has a regular basis $(f_k)_{k \ge 0}$ as an $R$-module if and only if the products of prime ideals of $R$ of every given norm are principal, which is certainly true if $R$ is a PID ("regular" means that $\deg f_k = k$ for all $k$): Their proof is constructive, so the answer to your question ("Can we give a precise description of such polynomials etc.?") is yes.

For further details and results, you may want to have a look to P.-J. Cahen and J.-L. Chabert's monograph, Integer-valued polynomials, Math. Surveys Monogr. 48, Amer. Math. Soc., 1997. More specifically, see Remark II.1.5(ii) for an "explicit basis".

Your question is related to the study of (generalized) numerical polynomials: If $R$ is an integral domain and $K$ the field of fractions of $R$, then the set ${\rm Int}(R) := \{f \in K[x]: f(R) \subseteq R\}$ is a subdomain of $K[x]$, whose elements are called the numerical polynomials over $R$ (in one variable $x$).

The domain ${\rm Int}(R)$ has been the subject of a great deal of research: Original work in the area was entirely centered on the case where $R$ is the ring of integers, and was motivated by interpolation problems in the early days of calculus. It was only in 1919 that A. Ostrowski and G. Pólya first considered numerical polynomials in their own right, though focused on the case where $R$ is the ring of integers of a number field. In particular, they could show, in this context, that ${\rm Int}(R)$ has a regular basis $(f_k)_{k \ge 0}$ as an $R$-module if and only if the products of prime ideals of $R$ of every given norm are principal, which is certainly true if $R$ is a PID ("regular" means that $\deg f_k = k$ for all $k$): Their proof is constructive, so the answer to your question ("Can we give a precise description of such polynomials etc.?") is yes.

For further details and results, you may want to have a look to P.-J. Cahen and J.-L. Chabert's monograph, Integer-valued polynomials, Math. Surveys Monogr. 48, Amer. Math. Soc., 1997. More specifically, see Remark II.1.5(ii) for an "explicit basis".

Your question is related to the study of (generalized) numerical polynomials: If $R$ is an integral domain and $K$ the field of fractions of $R$, then the set ${\rm Int}(R) := \{f \in K[x]: f(R) \subseteq R\}$ is a subdomain of $K[x]$, whose elements are called the numerical polynomials over $R$ (in one variable $x$).

The domain ${\rm Int}(R)$ has been the subject of a great deal of research: Original work in the area was entirely centered on the case where $R$ is the ring of integers, and was motivated by interpolation problems in the early days of calculus. It was only in 1919 that A. Ostrowski and G. Pólya first considered numerical polynomials in their own right, though focused on the case where $R$ is the ring of integers of a number field $K$. In particular, they could show, in this context, that ${\rm Int}(R)$ has a regular basis $(f_k)_{k \ge 0}$ as an $R$-module if and only if the products of prime ideals of $R$ of every given norm are principal, which is certainly true if $R$ is factorial ("regular" means that $\deg f_k = k$ for all $k$): Their proof is constructive, so the answer to your question is yes.

For further details and results, you may want to have a look to P.-J. Cahen and J.-L. Chabert's monograph, Integer-valued polynomials, Math. Surveys Monogr. 48, Amer. Math. Soc., 1997. More specifically, see Remark II.1.5(ii) for an "explicit basis".

Your question is related to the study of (generalized) numerical polynomials: If $R$ is an integral domain and $K$ the field of fractions of $R$, then the set ${\rm Int}(R) := \{f \in K[x]: f(R) \subseteq R\}$ is a subdomain of $K[x]$, whose elements are called the numerical polynomials over $R$ (in one variable $x$).

The domain ${\rm Int}(R)$ has been the subject of a great deal of research: Original work in the area was entirely centered on the case where $R$ is the ring of integers, and was motivated by interpolation problems in the early days of calculus. It was only in 1919 that A. Ostrowski and G. Pólya first considered numerical polynomials in their own right, though focused on the case where $R$ is the ring of integers of a number field $K$. In particular, they could show, in this context, that ${\rm Int}(R)$ has a regular basis $(f_k)_{k \ge 0}$ as an $R$-module if and only if the products of prime ideals of $R$ of every given norm are principal, which is certainly true if $R$ is a PID ("regular" means that $\deg f_k = k$ for all $k$): Their proof is constructive, so the answer to your question ("Can we give a precise description of such polynomials etc.?") is yes.

For further details and results, you may want to have a look to P.-J. Cahen and J.-L. Chabert's monograph, Integer-valued polynomials, Math. Surveys Monogr. 48, Amer. Math. Soc., 1997. More specifically, see Remark II.1.5(ii) for an "explicit basis".

Your question is related to the study of (generalized) numerical polynomials: If $R$ is an integral domain and $K$ the field of fractions of $R$, then the set ${\rm Int}(R) := \{f \in K[x]: f(R) \subseteq R\}$ is a subdomain of $K[x]$, whose elements are called the numerical polynomials over $R$ (in one variable $x$).

The domain ${\rm Int}(R)$ has been the subject of a great deal of research: Original work in the area was entirely centered on the case where $R$ is the ring of integers, and was motivated by interpolation problems in the early days of calculus. It was only in 1919 that A. Ostrowski and G. Pólya first considered numerical polynomials in their own right, though focused on the case where $R$ is the ring of integers of a number field $K$. In particular, they could show, in this context, that ${\rm Int}(R)$ has a regular basis $(f_k)_{k \ge 0}$ as an $R$-module if and only if the products of prime ideals of $R$ of every given norm are principal, which is certainly true if $R$ is factorial ("regular" means that $\deg f_k = k$ for all $k$): Their proof is constructive, so the answer to your question is yes.

For many more details and results, you may want to have a look to P.-J. Cahen and J.-L. Chabert's monograph, Integer-valued polynomials, Math. Surveys Monogr. 48, Amer. Math. Soc., 1997.

Your question is related to the study of (generalized) numerical polynomials: If $R$ is an integral domain and $K$ the field of fractions of $R$, then the set ${\rm Int}(R) := \{f \in K[x]: f(R) \subseteq R\}$ is a subdomain of $K[x]$, whose elements are called the numerical polynomials over $R$ (in one variable $x$).

The domain ${\rm Int}(R)$ has been the subject of a great deal of research: Original work in the area was entirely centered on the case where $R$ is the ring of integers, and was motivated by interpolation problems in the early days of calculus. It was only in 1919 that A. Ostrowski and G. Pólya first considered numerical polynomials in their own right, though focused on the case where $R$ is the ring of integers of a number field $K$. In particular, they could show, in this context, that ${\rm Int}(R)$ has a regular basis $(f_k)_{k \ge 0}$ as an $R$-module if and only if the products of prime ideals of $R$ of every given norm are principal, which is certainly true if $R$ is factorial ("regular" means that $\deg f_k = k$ for all $k$): Their proof is constructive, so the answer to your question is yes.

For further details and results, you may want to have a look to P.-J. Cahen and J.-L. Chabert's monograph, Integer-valued polynomials, Math. Surveys Monogr. 48, Amer. Math. Soc., 1997. More specifically, see Remark II.1.5(ii) for an "explicit basis".