Homework is assigned from 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.
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 |
