Math 431/530: Computational Algebraic Geometry // Applied/Computational 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 Campuswire, which you can easily access by logging into the course here, 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 lastminute homework questions; particularly if you have not been participating in the Campuswire 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 21&23  Topic 1: What is applied algebra? Preliminaries: basic introduction to algebraic structures (fields, rings). Polynomials and affine spaces.  CLO: 1.1.  Homework 1, due 8/28. 
August 28&30  Topic 2: Affine varieties and their parametrizations.  CLO: Sections 1.2 (finish) and 1.3.  Homework 2, due 9/4. 
September 4&6  Topic 3: Ideals. 
CLO 1.4.  Homework 3, due 9/11. 
Sep 11&13  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 9/18. 
Sep 18&20  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 9/25. 
Sep 25&27  Topic 6: Chapter 2: Monomial ideals and Dickson's Lemma; The Hilbert basis theorem and consequences (including the ascending chain condition).  CLO 2.4, 2.5  Homework 6, due 10/2. 
October 2&4  Topic 7: Groebner bases and their properties.
Spairs.
10/2: Brief project discussion. 10/5: DEADLINE to form project groups. 
CLO 2.5, 2.6.  Homework 7, due 10/9. 
October 9 & 11  Topic 8: Buchberger's algorithm and first applicatin of
Groebner bases.
QUIZ 2 in class this week *Thursday*, to help you better prepare for the midterm. 10/11: Project Outline due. *EXTENDED TO Friday 10/12, 1pm*! 
CLO 2.7 and 2.8.  Homework 8, due 10/23 (after the midterm!). 
October 16&18  Topic 8: continued from previous week; closing of
chapter 2. Intro to Elimination Theory (start of 3.1).
Planned date for the MIDTERM EXAM: Tuesday October 16th. In class. 
Topics: Chapters 1 and 2, except material covered in class immediately before the exam.  Relevant homework sets: 17. 
October 23&25  Topic 9: Elimination and extension theorems. Geometry of elimination. Time permitting: Implicitization problem.  CLO Sections 3.1, 3.2, 3.3.  Homework 9, due 
Oct 30 & Nov 1  Topic 10: Implicitization problem and algorithms for
polynomial and rational implicitization.
10/30: Project Rough Draft due. No other HW due this week; HW 9 moved by a week. 
CLO section 3.3. Course notes essential; focus on examples!  Homework 10, due 11/13. 
Nov 6&8  Topic 10: Continued: Resultants and proof of Extension Theorem. Topic 11: Hilbert's NullStellenSatz
11/6: Feedback on Project Rough Draft due. 
CLO 3.5, 3.6, and 4.1.  Homework 10, due 11/13. 
Comprehensive Final Exam 
Monday Dec 3, 2018 10:30 a.m. 12:30 p.m. RE 102
