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.
Jean-Luc Chabert, University of Picardie, France.
First steps The paradigmatic example: $Int(\mathbb{Z})={f(X)\in\mathbb{Q}[X] f(\mathbb{Z}\subseteq\mathbb{Z}}$ Combinatorics Integer-valued polynomials on a subset of $\mathbb{Z}$ Bhargava's orderings and generalized factorials Number theory Algebraic number theory: The Polya group of Galois extensions Examples of Polya fields (Galois extensions of small degrees) Class field theory: The Polya group of non-Galois extensions Commutative algebra Topology: The polynomial closure Algebra and ultrafilters: The Prufer properties Commutative ring theory: More algebraic properties Ultrametric analysis More about orderings in valued fields Orthonormal bases of spaces of smooth functions Dynamics: Valuative capacity and successor function More about I. V. P.-Asymptotic densities several variables Probabilistic number theory-Using Kempner-Bhargava's formula Several indeterminates Non-commutative algebra I. V. P. on non-commutative algebras-The case of matrices I. V. P. on division algebras-The case of quaternions To go further-Other possible themes around I. V. P. Bibliography Index
