Assignments

Homework is assigned from the required textbook:

Buy at Amazon 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.

Problems with numbers between braces are to be written up formally in TeX and passed in the week after they are assigned.

SES # TOPICS ASSIGNMENTS
1 Polynomials and affine space, affine varieties p. 5: 2, {6b}; p. 12: {6}, 8, 10
2 Parameterizations of affine varieties, ideals p. 22: 1, 3, 4, {11}; p. 34: 3b, {9}, 10
3 Polynomials of one variable, orderings on the monomials in k[x1,...,xn] p. 46: {9}, 10; p. 52: 5; p. 60: {2}, 10
4 A division algorithm in k[x1,...,xn], monomial ideals and Dickson's lemma p. 68: {1}; p. 73: 3, {9}
5 The Hilbert basis theorem and Groebner bases, properties of Groebner bases p. 80: {1}, 10; p. 87: 1, {12}
6 Buchberger's algorithm, first applications of Groebner bases p. 94: 2a, {3a}; p. 100: {1}, 7
7 The elimination and extension theorems, the geometry of elimination p. 121: 1, {6}; p. 127: {3}, 5
8 Implicitization, singular points and envelopes p. 135: 9, {11}; p. 148: 4, {10}
9 Unique factorization and resultants p. 159: 1, {4}; p. 159: 11, {17}
10 Resultants and the extension theorem, the nullstellensatz p. 166: {2}, 3, 8; p. 174: 1, {2}
11 Radical ideals and the ideal-variety correspondence, sums, products, and intersections of ideal p. 182: 2, {7a}; p. 191: 1, {11a-d}
12 Zariski closure and quotients of ideals, irreducible varieties and prime ideals p. 197: {1}, 4; p. 203: 5, {12}
13 Decomposition of a variety into irreducibles, polynomial mappings p. 209: 1, {9}; p. 220: 8, 10
14 Quotients of polynomials R, algorithmic computations in k[x1,...,xn]/I p. 228: 1, 6; p. 237: 6, {8}
15 The coordinate ring of an affine variety, rational functions on a variety p. 246: 8, 9; p. 256: {9}, 10
16 Proof of the Closure theory, geometric description of robots, the forward kinematics problem p. 263: {5}, 8; p. 269: 4; p. 277: {4}
17 The inverse kinematic problem and motion planning, automatic geometric theorem proving p. 287: {3}, 6; p. 303: 9, 13d
18 Wu's method, symmetric polynomials p. 315: 3, 6a; p. 324: {8}, 14
19 Finite matrix groups and rings of invariants, generators for the ring of invariants p. 333: 11, 14; p. 342: {5}, 7
20 Relations among generators and the geometry of orbits, the projective plane, projective space and projective varieties p. 354: 6, 13; p. 375: 4, {7}
21 The projective algebra-geometry dictionary, the projective closure of an affine variety p. 385: 15; p. 385:{10}; p. 391: {2}, 9
22 Projective elimination theory p. 406: 7, {9}; p. 406: 17, 18
23 The geometry of quadric hypersurfaces, the variety of a monomial ideal p. 419: 7, 10, {15}; p. 442: {2}, 5
24 The complement of a monomial ideal, the Hilbert function and the dimension of a variety p. 453: 3, 12; p. 465: 10, {12}
25 Elementary properties of dimension, dimension and algebraic independence p. 474: 6, 10; p. 482: 7, 13
26 Dimension and nonsingularity, the tangent cone p. 493: 13, 17; p. 504: 6, 13