Applicable Algebraic Geometry
Math 697R

Day 07, 5 October 2000.

- Reminder: Dinner at our home, 5:30 PM
   89 N Prospect St, Amh.  256-1223




Recall:  Variety of a subset of polynomials; reverses inclusions

Define: Ideal of a subset of affine space; reverses inclusions

Have two inclusion-reversing maps between subsets of polynomial ring
    and subsets of affine space.    These form the basis of the 
    Algebraic-Geometric Dictionary


- I(Z) is closed under polynomial linear combinations ==> it is an ideal

- V(S)=V(I), where I is the closure of S under polynomial linear combinations,
             the ideal generated by S.

 !  We may restrict the LHS of the dictionary to ideals.

If X = V( I( Z )), then I(X) is equal to I(Z).

 ! We may restrict the RHS of the dictionary to affine varieties.

* The dictionary is still not 1-1
  /R have V(1+x^2)=V(1)=\emptyset
  /C have V(x)=V(x^2)= {0},
     & V( y(y-x^2), y(y+x^2))= V(y)

- Define radical ideal
- Hilbert's Nullstellensatz
- Exact nature of dictionary
- Weak Nullstellensatz
- Multivariate Fundamental Theorem of Algebra

Section 2:  Generic properties of varieties

Thm:  Affine varieties closed under finite unions and arbitrary intersections

  ==> They form a topology, called Zariski topology.


Relate to usual (Euclidean) topology
 * Z-closed/open ==> E-closed/open
 * Non-empty & E-open ==> Z dense
 * Z closed, not all space ==> nowhere E dense
 * R^n is Z_closed in C^n

Ex.  Z-closed subsets of A^1

Definition generic subset/ generic property


Ex.  Invertible matrices

     Polynomials with distinct roots.

     COntrollable & Observable State-Space Realization