Applicable Algebraic Geometry
Math 697R

Day 06, 3 October 2000.

- Invite students to our home on 15 October

- 27 October (Fri) I'll give a CVC at Mt Holyoke on material
     slightly related to the course

-  When is a good time to have an additional meeting to make up
    for the missed classes?


Define: Minimal/ Coprime Factorization

Exercise:   Implement algorithm to find minimal coprime factorization
      Eg.:  C ( sI_n - A )^{-1} B.

Control with linear dynamic/static compensators

    - closed-loop system
    - characteristic equation
    - Use coprime factorization
    - drop to static case, defining linear subspaces L(s) & K
    - Given geometric version of pole placement problem
    - Mention geometric problem studied in 19th C. by Schubert,
      & a goal of this class.
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   Algebraic Geometry

The Algebraic - Geometric Dictionary

  * A note about our fields

  - Today's goal - the precise relation between algebra & geometry
  - Affine n-space, polynomial ring in n variables
  - Define: Affine variety, hypersurface, & give a few examples:
        cubics,
        Mat_nxn, SL_n
        Subvariety
        Product

  - V(.) is inclusion-reversing
*** try to invert the map V
  - Define I(Z), reverses inclusions.   Give the picture.
  - I(Z) is an ideal;   Can V(S) = V( ideal gen. by S ); 
      can restrict to ideals.
  - I V I (Z) = I(Z), can restrict to subvarieties.
  - Observe that, even now, V is not 1-1, and (I is not onto)
     Give examples
  - Clever definition of a radical ideal; define the radical of an 
        ideal.
  - Lemma I(Z) is radical
 
  - Hilbert's Nullstellensatz, Corollary: Precise nature
  - Weak Nullstellensatz & Multivariate Fundamental Theorem of Algebra.