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The topics follow the order of the required textbook:
Cox, David, John Little, and Donal O'Shea. Ideals, Varieties, and Algorithms. 3rd ed. Undergraduate Texts in Mathematics. New York, NY: Springer, 2007. ISBN: 9780387356518.
Portions of the book are online .
Course readings.
| SES # |
TOPICS |
READINGS |
| 1 |
Polynomials and affine space, affine varieties |
Sections 1-1 and 1-2 |
| 2 |
Parameterizations of affine varieties, ideals |
Sections 1-3 and 1-4 |
| 3 |
Polynomials of one variable, orderings on the monomials in k[x1,...,xn] |
Sections 1-5, 2-1, and 2-2 |
| 4 |
A division algorithm in k[x1,...,xn], monomial ideals and Dickson's lemma |
Sections 2-3 and 2-4 |
| 5 |
The Hilbert basis theorem and Groebner bases, properties of Groebner bases |
Sections 2-5 and 2-6 |
| 6 |
Buchberger's algorithm, first applications of Groebner bases |
Sections 2-7 and 2-8 |
| 7 |
The elimination and extension theorems, the geometry of elimination |
Sections 3-1 and 3-2 |
| 8 |
Implicitization, singular points and envelopes |
Sections 3-3 and 3-4 |
| 9 |
Unique factorization and resultants |
Section 3-5 |
| 10 |
Resultants and the extension theorem, the nullstellensatz |
Sections 3-6 and 4-1 |
| 11 |
Radical ideals and the ideal-variety correspondence, sums, products, and intersections of ideal |
Sections 4-2 and 4-3 |
| 12 |
Zariski closure and quotients of ideals, irreducible varieties and prime ideals |
Sections 4-4 and 4-5 |
| 13 |
Decomposition of a variety into irreducibles, polynomial mappings |
Sections 4-6 and 5-1 |
| 14 |
Quotients of polynomials R, algorithmic computations in k[x1,...,xn]/I |
Sections 5-2 and 5-3 |
| 15 |
The coordinate ring of an affine variety, rational functions on a variety |
Sections 5-4 and 5-5 |
| 16 |
Proof of the Closure theory, geometric description of robots, the forward kinematics problem |
Sections 5-6, 6-1, and 6-2 |
| 17 |
The inverse kinematic problem and motion planning, automatic geometric theorem proving |
Sections 6-3 and 6-4 |
| 18 |
Wu's method, symmetric polynomials |
Sections 6-5 and 7-1 |
| 19 |
Finite matrix groups and rings of invariants, generators for the ring of invariants |
Sections 7-2 and 7-3 |
| 20 |
Relations among generators and the geometry of orbits, the projective plane, projective space and projective varieties |
Sections 7-4, 8-1, and 8-2 |
| 21 |
The projective algebra-geometry dictionary, the projective closure of an affine variety |
Sections 8-3 and 8-4 |
| 22 |
Projective elimination theory |
Section 8-5 |
| 23 |
The geometry of quadric hypersurfaces, the variety of a monomial ideal |
Sections 8-6 and 9-1 |
| 24 |
The complement of a monomial ideal, the Hilbert function and the dimension of a variety |
Sections 9-2 and 9-3 |
| 25 |
Elementary properties of dimension, dimension and algebraic independence |
Sections 9-4 and 9-5 |
| 26 |
Dimension and nonsingularity, the tangent cone |
Sections 9-6 and 9-7 |