Readings

The topics follow the order of the required textbook:

Amazon logo 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.

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