Tuesday, 31 October 2000

Day 15: Math 697R

Applicable Algebraic Geometry
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Wednesday, November 8 2000
    1530 LGRT extra class 5:20 -- 6:30PM



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Theorem  (Refined B\'ezout's Theorem)
         Need K to be algebraically closed

Example (From Macaulay2)

Systems in nature have fewer solutions & challenge is to 
     find better bounds.

Elimination Theory.
  Theorem:  eliminants and roots

  Remark: Eliminants can be computed using a Groebner basis

  Shape Lemma.

  Resulting algorithm for solving & its numerical instability.

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Homotopy methods

 Problem:  Find all isolated solutions to a system

 Step 1 find a homotopy that 
    I) interpolates between given system and 
   II) a trivial system, 
  III) so that all solutions are reached,
   IV) and we avoid singularities

 Step 2  Path-tracing

  Example: Predictor-Corrector

  Definition: Bezout Homotopy
              Optimal Homotopy
  
  Theorem:  Bezout Homotopy is optimal for generic systems.