Duality for Polygons and the Fundamental Theorem of Algebra - Algebraic Topology Lecture 12

Explore the concepts of polygon duality and the Fundamental Theorem of Algebra in this 46-minute lecture from the Algebraic Topology course. Begin by defining polygon duality in relation to a fixed origin and unit circle, connecting it to cone duality. Delve into a rational formulation of the Fundamental Theorem of Algebra, examining the winding number of concentric circles' images around the origin. Learn about cones, unit circles, dual polygons, and their properties. Investigate applications of winding numbers and the Rotational Polynomial theorem. Conclude with a proof using pictures and winding numbers, gaining essential knowledge for undergraduate mathematics students.

Introduction
Recall T=1 turn angle
Problem 14
Duality of Polygons and cones
Cone
Unit circle and a polygon
Dual polygon
Facts; Exercise on Duality
Applications of Winding numbers
Fundamental theorem of Algebra
Rotational Polynomial theorem
Proof of pictures and Winding numbers