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In Polynomial, the expression of variables involves only the operations of addition, subtraction, multiplication and non-negative integer exponents of variables. To solve problems in mathematics and science like forming polynomial equations, defining polynomial functions, in calculus and physics to economics and social science, numerical analysis etc. polynomial method is a must. Through this [course_title] you will be able to solve problems in combinatorics.

### Assessment

This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer needs to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.

### Certification

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Course Credit: MIT

### Course Curriculum

 Module: 01 Introduction 00:15:00 The Berlekamp-Welch Algorithm 00:14:00 The Finite Field Nikodym and Kakeya Theorems 00:10:00 The Joints Problem 00:10:00 Why Polynomials? 00:17:00 Introduction to Incidence Geometry 00:08:00 Crossing Numbers and the Szemeredi-Trotter Theorem 00:15:00 Crossing Numbers and Distance Problems 00:16:00 Crossing Numbers and Distinct Distances 00:20:00 Reguli; The Zarankiewicz Problem 00:18:00 The Elekes-Sharir Approach to the Distinct Distance Problem 00:10:00 Module: 02 Degree Reduction 00:23:00 Bezout Theorem 00:22:00 Special Points and Lines of Algebraic Surfaces 00:23:00 An Application to Incidence Geometry 00:25:00 Taking Stock 00:22:00 Introduction to the Cellular Method 00:25:00 Polynomial Cell Decompositions 00:18:00 Using Cell Decompositions 00:15:00 Incidence Bounds in Three Dimensions 00:10:00 What’s Special About Polynomials? (A Geometric Perspective) 00:21:00 Detection Lemmas and Projection Theory 00:26:00 Module: 03 Local to Global Arguments 00:13:00 The Regulus Detection Lemma 00:20:00 Introduction to Thue’s Theorem on Diophantine Approximation 00:24:00 Thue’s Proof (Part I) 00:20:00 Thue’s Proof (Part II) Polynomials of Two Variables 00:21:00 Thue’s Proof (Part III) 00:18:00 Background on Connections Between Analysis and Combinatorics (Loomis-Whitney) 00:17:00 Hardy-Littlewood-Sobolev Inequality 00:13:00 Oscillating Integrals and Besicovitch’s Arrangement of Tubes 00:26:00 Besictovitch’s Construction 00:11:00 The Kakeya Problem 00:12:00 A Version of the Joints Theorem for Long Thin Tubes 00:24:00 Assessment Submit Your Assignment 00:00:00 Certification 00:00:00

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