Thursday, 26 October 2000

Day 14: Math 697R

Applicable Algebraic Geometry
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Make up Day:  Nov 7 (Tu), 8 (Wed)
                 14                16(Th)
       20(Mon)

There will be a makeup day in December

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 - Definition: Standard Monomial
 - Images of Standard Monomials for monomial ideals
 - Macaulay's Theorem for k[T]/I
 - Algorithm for computing in quotient rings

 - Definition: S-polynomials
 
 - Buchberger's Criterion

 - The Buchberger Algorithm

 - Corollary: GB computed without changing the field

 - Example: S_3 invariants; Linear polynomials


SYMBOLIC SOLUTION OF POLYNOMIAL EQUATIONS

 5 questions about a system of equations

 Solutions are V(I).

 Theorem.  V(I) finite iff only finitely many Standard monomials

 Corollary.  # V(I) <= # Standard Monomials

 Definition: degree I

 B\'ezout's Theorem.

 Definition.  Eliminant: monic generator of I\cap K[t_i]

 Theorem. Roots of an eliminants are the set of i-coords of roots of I

 Shape Lemma.