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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