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Projective Geometries Over Finite Fields

Author: James William Peter Hirschfeld

Publisher: Oxford University Press on Demand

Published: 1998

Total Pages: 555

ISBN-13: 9780198502951

I. Introduction 1. Finite fields 2. Projective spaces and algebraic varieties II. Elementary general properties 3. Subspaces 4. Partitions 5. Canonical forms for varieties and polarities III. The line and the plane 6. The line 7. First properties of the plane 8. Ovals 9. Arithmetic of arcs of degree two 10. Arcs in ovals 11. Cubic curves 12. Arcs of higher degree 13. Blocking sets 14. Small planes Appendix Notation References.

Projective Geometries Over Finite Fields

Author: James William Peter Hirschfeld

Publisher: Oxford University Press, USA

Published: 1979

Total Pages: 496

ISBN-13:

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General Galois Geometries

Author: James Hirschfeld

Publisher: Springer

Published: 2016-02-03

Total Pages: 422

ISBN-13: 1447167902

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This book is the second edition of the third and last volume of a treatise on projective spaces over a finite field, also known as Galois geometries. This volume completes the trilogy comprised of plane case (first volume) and three dimensions (second volume). This revised edition includes much updating and new material. It is a mostly self-contained study of classical varieties over a finite field, related incidence structures and particular point sets in finite n-dimensional projective spaces. General Galois Geometries is suitable for PhD students and researchers in combinatorics and geometry. The separate chapters can be used for courses at postgraduate level.

Dynamics, Statistics and Projective Geometry of Galois Fields

Author: V. I. Arnold

Publisher: Cambridge University Press

Published: 2010-12-02

Total Pages: 91

ISBN-13: 1139493442

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V. I. Arnold reveals some unexpected connections between such apparently unrelated theories as Galois fields, dynamical systems, ergodic theory, statistics, chaos and the geometry of projective structures on finite sets. The author blends experimental results with examples and geometrical explorations to make these findings accessible to a broad range of mathematicians, from undergraduate students to experienced researchers.

Introduction to Finite Geometries

Author: F. Kárteszi

Publisher: Elsevier

Published: 2014-05-12

Total Pages: 281

ISBN-13: 148327814X

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North-Holland Texts in Advanced Mathematics: Introduction to Finite Geometries focuses on the advancements in finite geometries, including mapping and combinatorics. The manuscript first offers information on the basic concepts on finite geometries and Galois geometries. Discussions focus on linear mapping of a given quadrangle onto another given quadrangle; point configurations of order 2 on a Galois plane of even order; canonical equation of curves of the second order on the Galois planes of even order; and set of collineations mapping a Galois plane onto itself. The text then ponders on geometrical configurations and nets, as well as pentagon theorem and the Desarguesian configuration, two pentagons inscribed into each other, and the concept of geometrical nets. The publication takes a look at combinatorial applications of finite geometries and combinatorics and finite geometries. Topics include generalizations of the Petersen graph, combinatorial extremal problem, and theorem of closure of the hyperbolic space. The book is a valuable source of data for readers interested in finite geometries.

Finite Projective Spaces of Three Dimensions

Author: James William Peter Hirschfeld

Publisher: Oxford University Press on Demand

Published: 1985

Total Pages: 316

ISBN-13: 9780198535362

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This self-contained and highly detailed study considers projective spaces of three dimensions over a finite field. It is the second and core volume of a three-volume treatise on finite projective spaces, the first volume being Projective Geometrics Over Finite Fields (OUP, 1979). The present work restricts itself to three dimensions, and considers both topics which are analogous of geometry over the complex numbers and topics that arise out of the modern theory of incidence structures. The book also examines properties of four and five dimensions, fundamental applications to translation planes, simple groups, and coding theory.

Algebraic Curves over a Finite Field

Author: J. W. P. Hirschfeld

Publisher: Princeton University Press

Published: 2013-03-25

Total Pages: 717

ISBN-13: 1400847419

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This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.

Projective Geometry

Author: Albrecht Beutelspacher

Publisher: Cambridge University Press

Published: 1998-01-29

Total Pages: 272

ISBN-13: 9780521483643

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Projective geometry is not only a jewel of mathematics, but has also many applications in modern information and communication science. This book presents the foundations of classical projective and affine geometry as well as its important applications in coding theory and cryptography. It also could serve as a first acquaintance with diagram geometry. Written in clear and contemporary language with an entertaining style and around 200 exercises, examples and hints, this book is ideally suited to be used as a textbook for study in the classroom or on its own.

Higher-Dimensional Geometry Over Finite Fields

Author: D. Kaledin

Publisher: IOS Press

Published: 2008-06-05

Total Pages: 356

ISBN-13: 1607503255

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Number systems based on a finite collection of symbols, such as the 0s and 1s of computer circuitry, are ubiquitous in the modern age. Finite fields are the most important such number systems, playing a vital role in military and civilian communications through coding theory and cryptography. These disciplines have evolved over recent decades, and where once the focus was on algebraic curves over finite fields, recent developments have revealed the increasing importance of higher-dimensional algebraic varieties over finite fields. The papers included in this publication introduce the reader to recent developments in algebraic geometry over finite fields with particular attention to applications of geometric techniques to the study of rational points on varieties over finite fields of dimension of at least 2.

An Introduction to Finite Projective Planes

Author: Abraham Adrian Albert

Publisher: Courier Corporation

Published: 2015-02-18

Total Pages: 116

ISBN-13: 0486789942

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Text for both beginning and advanced undergraduate and graduate students covers finite planes, field planes, coordinates in an arbitrary plane, central collineations and the little Desargues' property, the fundamental theorem, and non-Desarguesian planes. 1968 edition.