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This paper provides a unifying approach to recent results linking the fields of integer-valued polynomials and operations in K-theory. Following work of Bhargava, we set up a general framework encompassing several examples of rings of integer-valued polynomials. Our main results give bases for the duals of these rings. The rings and their duals all arise in topology as various kinds of cooperations and operations in complex K-theory. We show how several previously understood examples fit into this framework and we present some new examples.

Integer-valued polynomials and $K$-theory operations