Explore the fascinating world of real-rooted polynomials in this 33-minute lecture by Jan Vondrák from Stanford University, presented at the Geometry of Polynomials Boot Camp. Delve into key concepts such as the Fundamental Theorem of Algebra, Newton's identities, and the interlacing of roots. Learn techniques for checking real-rootedness and understand the relationship between roots and coefficients. Discover the properties of derivatives of real-rooted polynomials and the reversion of coefficient order. Examine the proof of Newton's inequalities and their connection to independent Bernoulli variables. Investigate combinatorial examples of real-rooted polynomials and uncover a curious fact about random spanning trees in this comprehensive exploration of polynomial geometry.
Overview
Syllabus
Intro
The Fundamental Theorem of Algebra
Roots vs. coefficients
Newton's identities
Checking real-rootedness
Interlacing of roots
Derivative of a real-rooted polynomial
Reversion of the order of coefficients
Proof of Newton's inequalities
Connection with independent Bernoulli variables
Combinatorial examples of real-rooted polynomials
Curious fact about random spanning trees
Taught by
Simons Institute
