Thursday, 2 November 2000

Day 16: Math 697R

Applicable Algebraic Geometry
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* Wednesday, November 8, Make up lecture at 5:30 PM in 1530 LGRT
* No class on Tuesday, 21 November (Frank is out of town)

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 - Shape Lemma
    * Prove generic form of Lexicographic Groebner Basis
    * Prove fact about reality

    Give an example of the state of the art in 1998
 
 - Symbolic-Numeric algorithm for solving.
   * Numerically unstable.
     One way to improve

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Homotopy methods   
    For computing numerical approximations to complex solutions.

 - Mathematical equivalent of following bad joke

 - Find all isolated solutions to the system F(X)=0.

   Step 1:  Find a homotopy H(X,t) that 
    interpolates between the original system (1) F(X)=H(X,1)
    and a trivial, start system H(X,0) (2) such that all solutions
    are attained (3) [along complex curves connecting them]
    and no singularities occur (4).

   Step 2:  Restrict H(X,t)=0 to a generic smooth path in Complex plane,
     lift to get smooth curves connecting solutions of given system
     to start system, then numerically trace these curves.

  Example:  Predictor-Corrector

  Example: B\'ezout Homotopy

  Define: Optimal Homotopy
H			
  Theorem:  B\'ezout homotopy is optimal, generically.

  State problem solved by Pieri Homotopy by Jan Verschelde in 1998.