A classification of all $D$ such that $Int(D)$ is a Prüfer domain

Let $D$ be an integral domain with quotient field $K$. The ring of integer-valued polynomials $Int(D)$ over $D$ is defined by $Int(D) = \{f(x) \in K[x] \mid f(D) \subseteq D\}$. It is known that if $Int(D)$ is a Prüfer domain, then $D$ is an almost Dedekind domain with all residue fields finite. This condition is necessary and sufficient if $D$ is Noetherian, but has been shown to not be sufficient if $D$ is not Noetherian. Several authors have come close to a complete characterization by imposing bounds on orders of residue fields of $D$ and on normalized values of particular elements of $D$. In this note we give a double-boundedness condition which provides a complete charaterization of all integral domains $D$ such that $Int(D)$ is a Prüfer domain.