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Polynomial Methods in Combinatorics

Larry Guth Massachusetts Institute of Technology, Cambridge, MA
Available Formats:
Softcover ISBN: 978-1-4704-2890-7
Product Code: ULECT/64
List Price: $48.00 MAA Member Price:$43.20
AMS Member Price: $38.40 Electronic ISBN: 978-1-4704-3214-0 Product Code: ULECT/64.E List Price:$48.00
MAA Member Price: $43.20 AMS Member Price:$38.40
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List Price: $72.00 MAA Member Price:$64.80
AMS Member Price: $57.60 Click above image for expanded view Polynomial Methods in Combinatorics Larry Guth Massachusetts Institute of Technology, Cambridge, MA Available Formats:  Softcover ISBN: 978-1-4704-2890-7 Product Code: ULECT/64  List Price:$48.00 MAA Member Price: $43.20 AMS Member Price:$38.40
 Electronic ISBN: 978-1-4704-3214-0 Product Code: ULECT/64.E
 List Price: $48.00 MAA Member Price:$43.20 AMS Member Price: $38.40 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$72.00 MAA Member Price: $64.80 AMS Member Price:$57.60
• Book Details

University Lecture Series
Volume: 642016; 273 pp
MSC: Primary 05;

This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book.

The subject is a beautiful one that has seen contributions by many leading mathematicians, including the author. The applications of the polynomial method covered in the book are quite wide-ranging, and run from combinatorial geometry to diophantine equations.

One particular feature of the book—and of the author's writing more generally—is the very inviting and discursive style, emphasising ideas where possible. This is *not* a dry monograph.

Ben Joseph Green, Oxford University

Some of the greatest advances in geometric combinatorics and harmonic analysis in recent years have been accomplished using the polynomial method. Larry Guth gives a readable and timely exposition of this important topic, which is destined to influence a variety of critical developments in combinatorics, harmonic analysis and other areas for many years to come.

Alex Iosevich, University of Rochester, author of “The Erdős Distance Problem” and “A View from the Top”

It is extremely challenging to present a current (and still very active) research area in a manner that a good mathematics undergraduate would be able to grasp after a reasonable effort, but the author is quite successful in this task, and this would be a book of value to both undergraduates and graduates.

Terence Tao, University of California, Los Angeles, author of “An Epsilon of Room I, II” and “Hilbert's Fifth Problem and Related Topics”

Graduate students and research mathematicians interested in combinatorial incidence geometry, algebraic geometry, and harmonic analysis.

• Chapters
• Chapter 1. Introduction
• Chapter 2. Fundamental examples of the polynomial method
• Chapter 3. Why polynomials?
• Chapter 4. The polynomial method in error-correcting codes
• Chapter 5. On polynomials and linear algebra in combinatorics
• Chapter 6. The Bezout theorem
• Chapter 7. Incidence geometry
• Chapter 8. Incidence geometry in three dimensions
• Chapter 9. Partial symmetries
• Chapter 10. Polynomial partitioning
• Chapter 11. Combinatorial structure, algebraic structure, and geometric structure
• Chapter 12. An incidence bound for lines in three dimensions
• Chapter 13. Ruled surfaces and projection theory
• Chapter 14. The polynomial method in differential geometry
• Chapter 15. Harmonic analysis and the Kakeya problem
• Chapter 16. The polynomial method in number theory

• Reviews

• One of the strengths that combinatorial problems have is that they are understandable to non-experts in the field...One of the strengths that polynomials have is that they are well understood by mathematicians in general. Larry Guth manages to exploit both of those strengths in this book and provide an accessible and enlightening drive through a selection of combinatorial problems for which polynomials have been used to great effect.

Simeon Ball, Jahresbericht der Deutschen Mathematiker-Vereinigung
• In the 273 page long book, a huge number of concepts are presented, and many results concerning them are formulated and proved. The book is a perfect presentation of the theme.

Béla Uhrin, Mathematical Reviews
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 642016; 273 pp
MSC: Primary 05;

This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. One of the first results in this story is a short elegant solution of the Kakeya problem for finite fields, which was considered a deep and difficult problem in combinatorial geometry. The author also discusses in detail various problems in incidence geometry associated to Paul Erdős's famous distinct distances problem in the plane from the 1940s. The proof techniques are also connected to error-correcting codes, Fourier analysis, number theory, and differential geometry. Although the mathematics discussed in the book is deep and far-reaching, it should be accessible to first- and second-year graduate students and advanced undergraduates. The book contains approximately 100 exercises that further the reader's understanding of the main themes of the book.

The subject is a beautiful one that has seen contributions by many leading mathematicians, including the author. The applications of the polynomial method covered in the book are quite wide-ranging, and run from combinatorial geometry to diophantine equations.

One particular feature of the book—and of the author's writing more generally—is the very inviting and discursive style, emphasising ideas where possible. This is *not* a dry monograph.

Ben Joseph Green, Oxford University

Some of the greatest advances in geometric combinatorics and harmonic analysis in recent years have been accomplished using the polynomial method. Larry Guth gives a readable and timely exposition of this important topic, which is destined to influence a variety of critical developments in combinatorics, harmonic analysis and other areas for many years to come.

Alex Iosevich, University of Rochester, author of “The Erdős Distance Problem” and “A View from the Top”

It is extremely challenging to present a current (and still very active) research area in a manner that a good mathematics undergraduate would be able to grasp after a reasonable effort, but the author is quite successful in this task, and this would be a book of value to both undergraduates and graduates.

Terence Tao, University of California, Los Angeles, author of “An Epsilon of Room I, II” and “Hilbert's Fifth Problem and Related Topics”

Graduate students and research mathematicians interested in combinatorial incidence geometry, algebraic geometry, and harmonic analysis.

• Chapters
• Chapter 1. Introduction
• Chapter 2. Fundamental examples of the polynomial method
• Chapter 3. Why polynomials?
• Chapter 4. The polynomial method in error-correcting codes
• Chapter 5. On polynomials and linear algebra in combinatorics
• Chapter 6. The Bezout theorem
• Chapter 7. Incidence geometry
• Chapter 8. Incidence geometry in three dimensions
• Chapter 9. Partial symmetries
• Chapter 10. Polynomial partitioning
• Chapter 11. Combinatorial structure, algebraic structure, and geometric structure
• Chapter 12. An incidence bound for lines in three dimensions
• Chapter 13. Ruled surfaces and projection theory
• Chapter 14. The polynomial method in differential geometry
• Chapter 15. Harmonic analysis and the Kakeya problem
• Chapter 16. The polynomial method in number theory
• One of the strengths that combinatorial problems have is that they are understandable to non-experts in the field...One of the strengths that polynomials have is that they are well understood by mathematicians in general. Larry Guth manages to exploit both of those strengths in this book and provide an accessible and enlightening drive through a selection of combinatorial problems for which polynomials have been used to great effect.

Simeon Ball, Jahresbericht der Deutschen Mathematiker-Vereinigung
• In the 273 page long book, a huge number of concepts are presented, and many results concerning them are formulated and proved. The book is a perfect presentation of the theme.

Béla Uhrin, Mathematical Reviews
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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