Lecture Notes on Geometry of Numbers
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Beschreibung
This book serves as an illuminating introduction to the intricacies of the geometry of numbers. It commences by exploring basic concepts of convex sets and lattices in Euclidean space and goes on to delve into Minkowski’s fundamental theorem for convex bodies and its applications. It discusses critical determinants and successive minima before explaining the core results of packings and coverings. The text goes on to delve into the significance of renowned conjectures such as Minkowski’s conjecture regarding the product of linear forms, Watson’s conjecture, and the conjecture of Bambah, Dumir, and Hans-Gill concerning non-homogeneous minima of indefinite quadratic forms.
Dedicated to Prof. R.P. Bambah on his 98th birthday, a living legend of number theory in India, this comprehensive book addresses both homogeneous and non-homogeneous problems, while sprinkling in historical insights and highlighting unresolved questions in the field. It is ideally suited for beginners embarking on self-study as well as for use as a text for a one- or two-semester introductory course.
This book serves as an illuminating introduction to the intricacies of the geometry of numbers. It commences by exploring basic concepts of convex sets and lattices in Euclidean space and goes on to delve into Minkowski’s fundamental theorem for convex bodies and its applications. It discusses critical determinants and successive minima before explaining the core results of packings and coverings. The text goes on to delve into the significance of renowned conjectures such as Minkowski’s conjecture regarding the product of linear forms, Watson’s conjecture, and the conjecture of Bambah, Dumir, and Hans-Gill concerning non-homogeneous minima of indefinite quadratic forms.
Dedicated to Prof. R.P. Bambah on his 98th birthday, a living legend of number theory in India, this comprehensive book addresses both homogeneous and non-homogeneous problems, while sprinkling in historical insights and highlighting unresolved questions in the field. It is ideally suited for beginnersembarking on self-study as well as for use as a text for a one- or two-semester introductory course.
Presents as an introductory one- or two-semester course in geometry of numbers Introduces Minkowski's conjecture, Watson's conjecture and the conjecture of Bambah, Dumir, and Hans-Gill Honours Prof. R.P. Bambah on his 98th birthday, a living legend of number theory in India
Autor*in
R. J. Hans-Gill
Themen in »Lecture Notes on Geometry of Numbers«
lattices convex sets Minkowski fundamental theorem critical determinants Minkowski second inequality packing coverings density covering constant quadratic forms homogeneous minima non-homogeneous minima Watson's conjecture Minkowski's conjecture
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“Each individual section in the book comes with a small list of exercises which are designed to be very accessible and aim to integrate the discussed material. ... the book provides a nice and focused introduction to the subject with a special emphasis on the arithmetical aspects and applications of the theory. Except for the last two chapters maybe, it is well suited for an introductory course at both the undergraduate and graduate level.” (Matthias Schymura, Mathematical Reviews, September, 2025)
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