Polynomial methods in combinatorics

書誌事項

Polynomial methods in combinatorics

Larry Guth

(University lecture series, v. 64)

American Mathematical Society, c2016

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注記

Includes bibliographical references (p. 269-273)

内容説明・目次

内容説明

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 Erdos'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.

目次

Introduction Fundamental examples of the polynomial method Why polynomials? The polynomial method in error-correcting codes On polynomials and linear algebra in combinatorics The Bezout theorem Incidence geometry Incidence geometry in three dimensions Partial symmetries Polynomial partitioning Combinatorial structure, algebraic structure, and geometric structure An incidence bound for lines in three dimensions Ruled surfaces and projection theory The polynomial method in differential geometry Harmonic analysis and the Kakeya problem The polynomial method in number theory Bibliography

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