Integer-valued polynomials : from combinatorics to number theory, p-adic analysis, commutative and non-commutative algebra / Jean-Luc Chabert.

Format:

Book

Author/Creator:

Chabert, Jean-Luc, author.

Series:

Colloquium publications (American Mathematical Society) ; v. 69.

American Mathematical Society colloquium publications, 0065-9258 ; volume 69

Language:

English

Subjects (All):

Polynomials.

Ideals (Algebra).

Rings of integers.

Physical Description:

xiv, 611 pages ; 27 cm.

Place of Publication:

[Providence, Rhode Island] : American Mathematical Society, [2025]

Summary:

"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 ℤ and subsets of 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 Pólya group. Connections between topology and multiplicative ideal theory return the focus to commutative algebra, providing tools for exploring Prüfer 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" -- Back cover.

Contents:

The paradigmatic example

Integer-valued polynomials on a subset of ℤ

Bhargava's orderings and generalized factorials

Algebraic number theory : the Pólya group of Galois extensions

Examples of Pólya fields

Class field theory : the Pólya group of non-Galois extensions

Topology : the polynomial closure

Algebra and ultrafilters : the Prüfer property

Commutative ring theory : more algebraic properties

More about orderings in valued fields

Orthonormal bases of spaces of smooth functions

Dynamics : valuative capacity and successor function

Probabilistic number theory

Several indeterminates

I.V.P. on non-commutative algebras, the case of matrices

I.V.P. on division algebras, the case of quaternions.

Notes:

Includes bibliographical references (pages 579-601) and index.

ISBN:

9781470482060

1470482061

OCLC:

1540318991