  Let D be an integral domain with quotient field K. For any set X, the ring
Int(D^X) of integer-valued polynomials on D^X is the set of all polynomials f
in K[X] such that f(D^X) is a subset of D. Using the t-closure operation on
fractional ideals, we find for any set X a D-algebra presentation of Int(D^X)$
by generators and relations for a large class of domains D, including any
unique factorization domain D, and more generally any Krull domain D such that
Int(D) has a regular basis, that is, a D-module basis consisting of exactly one
polynomial of each degree. As a corollary we find for all such domains D an
intrinsic characterization of the D-algebras that are isomorphic to a quotient
of Int(D^X) for some set X. We also generalize the well-known result that a
Krull domain D has a regular basis if and only if the Polya-Ostrowski group of
D (that is, the subgroup of the class group of D generated by the images of the
factorial ideals of D) is trivial, if and only if the product of the height one
prime ideals of finite norm q is principal for every q.
