Integer-valued polynomials on the ring of integers have been known for a long time and have been used in calculus. Polya and Ostrowski generalized this notion to rings of integers of number fields. More generally still, one may consider a domain $D$ and the polynomials (with coefficients in its quotient field) mapping $D$ into itself. They form a $D$-algebra - that is, a $D$-module with a ring structure. Appearing in a very natural fashion, this ring possesses quite a rich structure, and the very numerous questions it raises allow a thorough exploration of commutative algebra. Here is the first book devoted entirely to this topic. This book features: thorough reviews of many published works; self-contained text with complete proofs; and numerous exercises.
Coefficients and values Additive structure Stone-Weierstrass Integer-valued polynomials on a subset Prime ideals Multiplicative properties Skolem properties Invertible ideals and the Picard group Integer-valued derivatives and finite differences Integer-valued rational functions Integer-valued polynomials in several indeterminates References List of symbols Index.