On the Krull dimension of domains of integer-valued polynomials

Let D be a domain with quotient field K. Let E K be a subset; the ring of D-integer-valued polynomials over E is Int(E;D) := ff 2 K[X]; f(E) Dg. The polynomial closure in D of a subset E K is the largest subset F K containing E such that Int(E;D) = Int(F;D), and it is denoted by clD(E). 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).