Polynomial Methods in Combinatorics

Larry Guth, Massachusetts Institute of Technology, Cambridge, MA, USA.

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 Erdos 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"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 "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

• IntroductionFundamental examples of the polynomial methodWhy polynomials?The polynomial method in error-correcting codesOn polynomials and linear algebra in combinatoricsThe Bezout theoremIncidence geometryIncidence geometry in three dimensionsPartial symmetriesPolynomial partitioningCombinatorial structure, algebraic structure, and geometric structureAn incidence bound for lines in three dimensionsRuled surfaces and projection theoryThe polynomial method in differential geometryHarmonic analysis and the Kakeya problemThe polynomial method in number theoryBibliography