The Characterization of Finite Elasticities: Factorization Theory in Krull Monoids via Convex Geometry

A new convex geometry theory is developed in this book,generalizing positive bases and related to Carathéodory's Theorem. It combines convex geometry,the combinatorics of infinite subsets of lattice points,and the arithmetic of transfer Krull monoids. This theory is self-contained and has applications in factorization,characterizing when finite elasticity holds for Krull domains with finitely generated class groups.

Format: Paperback / softback
Length: 282 pages
Publication date: 27 October 2022
Publisher: Springer International Publishing AG

This groundbreaking work in convex geometry presents a novel theory that generalizes positive bases and relates to Carathéodory's Theorem. By seamlessly integrating convex geometry, the combinatorics of infinite subsets of lattice points, and the arithmetic of transfer Krull monoids, this theory offers a comprehensive framework for studying convex sets.

The primary motivation behind this innovative theory is its potential applications in factorization, particularly in the context of determining the uniqueness and reliability of factorizations into irreducibles, known as atoms. While factorization into atoms is not always unique, there exist various measures to quantify the extent of this failure. One of the most significant measures is the elasticity, which quantifies the ratio between the maximum and minimum number of atoms in any factorization. Having finite elasticity indicates that factorization, while not strictly unique, is not entirely arbitrary.

Through the development of the material in convex geometry, this book provides a precise characterization of when finite elasticity holds for any Krull domain with finitely generated class group $G$. Furthermore, the results extend to transfer Krull monoids, broadening the scope of its applications.

This book is designed to cater to researchers in the field, while also being accessible to graduate students and general mathematicians with an interest in convex geometry. Its clear and concise writing style makes it an invaluable resource for anyone seeking to delve into this exciting area of mathematics.

Weight: 456g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031148682
Edition number: 1st ed. 2022