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# Tangency and Discriminants

Published on Jul 19, 2019141 Views

The discriminant of a polynomial with prescribed monomials is an irreducible polynomial in the coefficients vanishing when the corresponding polynomial has multiple roots. It is an important tool in n

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#### Chapter list

Tangency and Discriminants00:00

Goal00:54

Natural concept02:27

The discriminants of univariate polynomials04:13

Algebra vs Geometry06:12

The definition of discriminant - 109:52

The definition of discriminant - 212:03

Example 114:31

Geometry - 115:42

Geometry - 219:47

Projective duality - 120:58

Projective duality - 223:54

Polar geometry:the degree and dimension of the discriminant27:07

Toric projective duality = A-discriminants30:00

Can a discriminant govern multiple roots of systems of polynomials?35:12

Example - 137:11

Tangential intersections37:45

The mixed discriminant39:09

Example - 240:36

One more example: The distance to a variety41:59

Consider now a plane curve43:12

This proves:44:34

Singular intersection of Quadric Surfaces46:53

Two Main Questions47:38

Towards an answer to question 148:24

Answer to question 152:36

Answer to question 254:31

Thank you55:26