Math 430: Applied Algebra
Homework schedule
Homework assignments will be posted at least one week before the due date. It is your responsibility to check the course page on Piazza, which you can easily access by logging into Blackboard, to obtain the assigned problems.You are expected to start working on the homework sets early (not the day they are due or right before). It is extremely difficult to answer last-minute homework questions; particularly if you have not been participating in the Piazza discussion beforehand.
Help with writing up assignments
To improve your mathematical writing quickly, start by writing draft solutions to homework early. A day or two later after you have had time to forget what you wrote, read it. If it doesn’t make sense or convince you, rewrite it. Writing a solution requires saying what you mean and meaning what you say. Be intellectually honest. Intellectual dishonesty includes: 1) stating a “reason” without understanding its relevance. 2) Claiming a conclusion when you know you haven’t proved it. 3) Giving an example and claiming you have proved the statement for all instances.(This text borrowed from Prof. Kaul)
Lecture schedule
You are expected to cover (at least at a high level) the assigned readings before coming to the lecture. This will help you follow the course and organize your notes. In the reading schedule below, "CLO" refers to Cox, Little and O'Shea, Ideals, Varieties and Algorithms.Homework problem sets are naturally related to the material covered in the course; hence, homework numbers are listed next to the corresponding topic.
| Dates | Tentative topics covered | Assigned reading | Related homework |
| August 26&28 | Topic 1: What is applied algebra? Preliminaries: basic introduction to algebraic structures (fields, rings). Polynomials and affine spaces. | CLO: 1.1. | Homework 1, due 9/2. |
| September 2&4 | Topic 2: Affine varieties and their parametrizations. | CLO: Sections 1.2. and 1.3. | Homework 2, due 9/9. |
| September 9&11 | Topic 3: Ideals. |
CLO 1.4. | Homework 3, due 9/16. |
| September 16&18 | Topic 4: Polynomials in one variable. [Or: ways in which
polynomial rings are like integers.] Every ideal in k[x] is principal. Introduction to algorithms/pseudocode. Brief introduction to Chapter 2. QUIZ 1 in class this week. |
CLO 1.5. | Homework 4, due 9/23. |
| September 23&25 | Topic 5: Chapter 2: Background needed for Groebner basics: Monomial orderings and division algorithm in many variables. |
CLO 2.1, 2.2, 2.3. | Homework 5, due 9/30. |
| September 30 & October 2 |
Topic 6: Chapter 2: Monomial ideals and Dicson's Lemma;
the Hilbert basis theorem [intro].
9/30: Brief project discussion. 10/3: DEADLINE to form project groups. |
CLO 2.4, 2.5 | Homework 6, due 10/7. |
| October 7&9 | Topic 7: The Hilbert basis theorem and consequences
(ascending chain condition). Groebner bases and their properties. S-pairs.
10/9: Project Outline due. |
CLO 2.5, 2.6. | Homework 7, due 10/14. |
| October 14&16 | Topic 8: Buchberger's algorithm and first applicatin of Groebner bases. | CLO 2.7 and 2.8. | Homework 8, due 10/28. |
| October 21&23 | Topic 8: continued from previous week; closing of
chapter 2. Intro to Elimination Theory (start of 3.1).
Planned date for the MIDTERM EXAM: October 21. In class, details tbd. |
Topics Chapters 1 and 2, except material covered week immediately before the exam. | Relevant homework sets: 1-7. |
| October 28&30 | Topic 9: Elimination and extension theorems. Geometry of elimination. |
CLO Sections 3.1, 3.2, 3.3. | Homework 9, due 11/4. |
| November 4&6 | Topic 10: Implicitization problem and algorithms for
polynomial and rational implicitization.
11/6: Project Rough Draft due. |
CLO section 3.3. Course notes essential; focus on examples! | Homework 10, due 11/18. |
| November 11&13 | Topic 10: Continued: Resultants and proof of Extension Theorem.
11/13: Feedback on Project Rough Draft due. No other HW due this week! NOTE: Graded project drafts will be available for pickup Friday 11/14 from 10am until 3pm. |
CLO 3.5 and 3.6. | Homework 10, due 11/18. |
| November 18&20 | Topic 11: Hilbert's NullStellenSatz | CLO 4.1 and 4.2. | Homework 11 *extra credit* due 12/2. |
| November 25 & Thanksgiving |
Topic 12: Radical ideals, ideal-variety correspondence, radical membership.
11/25: Final Project Paper due. |
CLO tbd. | Homework 11 *extra credit* due 12/2. |
| December 2&4 | Topic 13: Ideal operations. Irreducible varieties.
12/5: Feedback on Final Project Paper due. |
CLO 4.3 and (part of) 4.5. | Comprehensive Final Exam |
WEDNESDAY, December 10, 8-10 a.m. The take-home part of the final exam will be assigned at least one week before it is due. |