Integer-Valued Polynomials: From Combinatorics to Number Theory, $P$-Adic Analysis, Commutative and Non-Commutative Algebra

This book presents the theory of integer-valued polynomials, as transformed by the work of Manjul Bhargava in the late 1990s. Building from the core ideas in commutative algebra and number theory, the author weaves a panoramic perspective that encompasses results in combinatorics, ultrametric analysis, probability, dynamical systems, and non-commutative algebra. Whether already established in the area or just starting out, readers will find this deep and approachable treatment to be an essential companion to research. Grouped into seven parts, the book begins with the preliminaries of integer-valued polynomials on $\mathbb{Z }$ and subsets of $\mathbb{Z}$. Bhargava's revolutionary orderings and generalized factorials follow, laying the foundation for the modern perspective, before an interlude on algebraic number theory explores the Polya group. Connections between topology and multiplicative ideal theory return the focus to commutative algebra, providing tools for exploring Prufer domains. A part on ultrametric analysis ranges across $p$-adic extensions of the Stone-Weierstrass theorem, new orderings, and dynamics. Chapters on asymptotic densities and polynomials in several variables precede the final part on non-commutative algebra. Exercises and historical remarks engage the reader throughout. A thoroughly modern sequel to the author's 1997 Integer-Valued Polynomials with Paul-Jean Cahen, this book welcomes readers with a grounding in commutative algebra and number theory at the level of Dedekind domains. No specialist knowledge of probability, dynamics, or non-commutative algebra is required.