Download or read book Polynomial Methods in Combinatorics written by Larry Guth and published by . This book was released on 2016 with total page 273 pages. Available in PDF, EPUB and Kindle. Book excerpt: 14.3. The Crofton formula in integral geometry -- 14.4. Finding functions with large zero sets -- 14.5. An application of the polynomial method in geometry -- Chapter 15. Harmonic analysis and the Kakeya problem -- 15.1. Geometry of projections and the Sobolev inequality -- 15.2. ^{ } estimates for linear operators -- 15.3. Intersection patterns of balls in Euclidean space -- 15.4. Intersection patterns of tubes in Euclidean space -- 15.5. Oscillatory integrals and the Kakeya problem -- 15.6. Quantitative bounds for the Kakeya problem -- 15.7. The polynomial method and the Kakeya problem -- 15.8. A joints theorem for tubes -- 15.9. Hermitian varieties -- Chapter 16. The polynomial method in number theory -- 16.1. Naive guesses about diophantine equations -- 16.2. Parabolas, hyperbolas, and high degree curves -- 16.3. Diophantine approximation -- 16.4. Outline of Thue's proof -- 16.5. Step 1: Parameter counting -- 16.6. Step 2: Taylor approximation -- 16.7. Step 3: Gauss's lemma -- 16.8. Conclusion -- Bibliography -- Back Cover