The Characterization of Finite Elasticities
Factorization Theory in Krull Monoids via Convex Geometry
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Beschreibung
This book develops a new theory in convex geometry, generalizing positive bases and related to Carathéordory’s Theorem by combining convex geometry, the combinatorics of infinite subsets of lattice points, and the arithmetic of transfer Krull monoids (the latter broadly generalizing the ubiquitous class of Krull domains in commutative algebra)
This new theory is developed in a self-contained way with the main motivation of its later applications regarding factorization. While factorization into irreducibles, called atoms, generally fails to be unique, there are various measures of how badly this can fail. Among the most important is the elasticity, which measures the ratio between the maximum and minimum number of atoms in any factorization. Having finite elasticity is a key indicator that factorization, while not unique, is not completely wild. Via the developed material in convex geometry, we characterize when finite elasticity holds for any Krull domain with finitely generated class group $G$, with the results extending more generally to transfer Krull monoids.
This book is aimed at researchers in the field but is written to also be accessible for graduate students and general mathematicians.
This book develops a new theory in convex geometry, generalizing positive bases and related to Carathéordory’s Theorem by combining convex geometry, the combinatorics of infinite subsets of lattice points, and the arithmetic of transfer Krull monoids (the latter broadly generalizing the ubiquitous class of Krull domains in commutative algebra)
This new theory is developed in a self-contained way with the main motivation of its later applications regarding factorization. While factorization into irreducibles, called atoms, generally fails to be unique, there are various measures of how badly this can fail. Among the most important is the elasticity, which measures the ratio between the maximum and minimum number of atoms in any factorization. Having finite elasticity is a key indicator that factorization, while not unique, is not completely wild. Via the developed material in convex geometry, we characterize when finite elasticity holds for any Krull domain with finitely generated class group $G$, with the results extending more generally to transfer Krull monoids.
This book is aimed at researchers in the field but is written to also be accessible for graduate students and general mathematicians.
Develops new convex geometric methods for studying factorization Characterizes finite elasticity for Krull monoids with finitely generated class group Extensively generalizes positive bases in Euclidean space
Autor*in
David J. Grynkiewicz
Themen in »The Characterization of Finite Elasticities«
Carathéordory’s Theorem Krull Domain Zero-sum Lattice Transfer Krull Domain Krull Monoid Catenary Degree Sets of Lengths Convex Cone Simplicial Fan Primitive Partition Identities Well-quasi-ordering Delta Set Minimal Positive Basis Structure Theorem for Unions
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“This work is a well-written and well-organized research monograph which is accessible for the non-specialist. It is a masterpiece of research in factorization theory and convex geometry.” (Alfred Geroldinger, Mathematical Reviews, July, 2024)
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