Polynomial closure in essential domains and pullbacks

Let D be a domain with quotient field [(. Let E k J( be a subset; the ring of D-integer-valued polynomial. over E is Int(E, Dj :; {f E [([X): f(Ej !:; DJ. The polynomial closure in D of a subset E ~ [( is the largest subset F ~ [( containing E such that Int(E, Dj ; Int(F, Dj, and It Is denoted by clD(Ej. We study the polynomial closure of ideals in several classes of domains, including essential domains and domains of strong Krull-type, and we relate it with the t-closure. For domains of Krull-type we also compute the Krull dimension of Int(D).