Day 12
Applicable Algebraic Geometry

Wednesday 25 5:00 1530  LGRT

Intrinsic Definition of Tangent Spaces

 (review)  d_x : K[X] --> dual of tangent space

 - Theorem:   This is an isomorphism on m_x/(m_x)^2.

 - New Definition of tangent space.
 - Functoriality
 - Exercise:  Composition
              Behavior under isomorphism
 - Example:  V(y^2-x^3) \not \sim A^1
 - Theorem:   Algebraic groups are smooth

 Definition of Dimension
  Ex:  dim A^n = n
       dim cubic curves = 1
     Def:  Curve, Surface
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Chapter 2: Effective and computational geometry

 Two algorithms for simplifying polynomials:
 - Euclidean
 - Gaussian Elimination

 What to do for arbitrary polynomials?

 Definition:  Monomial; Total order
    A *monomial order* is a total order on monomials with
    - 1 minimal,  and - respects multiplication by monomials.