MATH3349/MATH4349/MATH6209 (Special Topics in Math): Computational Algebraic Geometry

ANU. Semester 1, 2020.

Zoom Lecture: Thursday 3:30 pm -- 5:30 pm.

Zoom Workshop: Wednesday 1:00 pm -- 2:00 pm.


Martin Helmer

Email: martin.helmer at
Office: 4.70, Hanna Neumann Building #145
Zoom Office Hours: Monday 5:00 pm -- 6:00 pm

Markus Hegland

Email: markus.hegland at
Office: 4.78, Hanna Neumann Building #145


The primary course text will be the (draft) textbook Invitation to Nonlinear Algebra below. We will refer to this book as MS, the book by Cox, Little and O'Shea as CLO, and the book by Sommese and Wampler as SW as required in the course outline below.

Invitation to Nonlinear Algebra by Mateusz Michałek and Bernd Sturmfels.

We will also reference the following books for certain portions of the course:

Ideals, varieties, and algorithms : an introduction to computational algebraic geometry and commutative algebra (Fourth Edition) by David Cox, John Little, and Donal O'Shea.

The numerical solution of systems of polynomials arising in engineering and science by Andrew John Sommese and Charles W. Wampler. Note this book can be accessed by ANU students online via the university library.

The references below overlap with our course and interested students may enjoy consulting them on occasion. Note, however, that these references contain some material that we will not cover.

Course Goals:

The goal of this course is to introduce students to algebraic geometry in a hands on manner. We will explore the use of both symbolic and numeric computational techniques in algebraic geometry. Students are encouraged to use computer tools such as Macaulay2 or Sage to explore examples and investigate problems.

The primary object at study will by systems of polynomial equations in n variables. The solutions set of a system of polynomial equations forms a geometric object called a variety; we will see that this corresponds to a (radical) ideal in a polynomial ring. We will explore the geometry of varieties both computationally and abstractly using the algebraic structure of polynomial rings.

A major component of this study will be the theory of Gröbner basis and of homotopy continuation. At the end of the course students will be able to answer such questions as: Does a given system of polynomials have finitely many solutions? If so, what are they? If there are infinitely many solutions, how can can these be described and understood?

Course Schedule and Notes:

The dates of the lectures are approximate and may be adjusted slightly through the course of the term. Hand written notes from lecture will be uploaded here in the corresponding place in the schedule.

Polynomial Rings and Gröbner basis

Varieties Solving and Decomposing Primary Decomposition and Elimination Solving Zero Dimensional Systems and Numerical Algebraic Geometry Floating Point and Dyadic Fractions Nullstellensatze Toric Varieties


Assignments and due dates will be posted here. All assignments are due at the beginning of class on the marked due date. The questions will primarily be taken from exercises in MS and CLO (due dates below are tentative).

Practice Problems

Here are some suggested problems from MS and CLO; you should be prepared to discuss some of these problems in the workshop time.

Workshop Presentation Schedule

March 18: W.C. (Suggested Questions: #6 from Ch. 1 of MS)
April 1: M.T. (Suggested Questions: #15 or both of #16 and #17 from Ch. 1 of MS)
April 22: I.L. (Suggested Questions: #10 of 2.7 in CLO or #12 from Ch. 2 of MS or #10 of Ch. 2 of MS)
April 29: M.S. (Suggested Questions: #11 from Ch. 2 of MS or #12 of Ch. 2 of MS or #8 of 8.2 in CLO or #18 of 8.2 in CLO)
May 6: A.O. (Suggested Questions: #3 from Ch. 3 of MS or #5 of Ch. 3 of MS or #10 of Ch. 3 of MS)
May 13: L.C. (Suggested Questions: #1 and #2 of Ch. 4 of MS or #5 of Ch. 3 of MS or #10 of Ch. 3 of MS)
May 20: A.Y. (Suggested Questions: #2 of Ch. 4 of MS or #10 of Ch. 4 of MS or #5 [Generalized Elimination Theorem] of 3.1 [Chapter 3, Section 1] of CLO)
May 27: A.W. (Suggested Questions: #2 of Ch. 4 of MS or #10 of Ch. 4 of MS or #9 [Application of Extension Theorem] of 3.1 [Chapter 3, Section 1] of CLO [Note the notation used by CLO for elimination ideals is a bit different but a quick glance through the chapter should make it clear])
June 3: P.M. (Suggested Questions: #1 of Ch. 6 of MS or #9 of Ch. 6 of MS or #8 of 4.1 [Chapter 4, Section 1] of CLO [Deals with: Any Variety over a Not Algebraically Closed Field Can be Defined By A Single Eq.])

Term Project/Paper:

This course will involve a term project. The project will require students to independently study a class-related topic. The results of your work and the understanding that you have gained will be summarized in a short paper. Your paper should be self-contained and should be written so that to the other students in our class can understand it. The target length will be approximately 10 pages. If appropriate your project may also have a software component, in such cases the report may be somewhat shorter but should still contain the ideas behind the algorithms present in your software.

Suggested Topics: Time line : Mark break down for project: Group Work: LaTex Example File

Algebra Software:

Macaulay2 (M2 for short) and Sage are both excellent open source computer algebra systems with some very helpful functions for algebra, algebraic geometry and number theory (among other things).

Homework Policy:

There will be four assignments, worth about 11% each. These will consist primarily of problems from the course texts.

Some things to keep in mind when doing your homework:

  1. You are encouraged to discuss problems with your classmates and are free to consult online resources. Working together on math problems can be an excellent way to learn and the internet is a useful resource. However your final written solutions you hand in must be your own work written in your own words, that is your final solutions must be written by yourself without consulting someone else's solution.
  2. All solutions should be written in complete, grammatically correct, English (or at least a very close approximation of this) with mathematical symbols and equations interspersed as appropriate. These solutions should carefully explain the logic of your approach.
  3. All proofs must be complete and detailed for full marks. Avoid the use of phrases such as 'it is easy to see' or 'the rest is straightforward', you will likely be docked marks. Proofs in your homework should be clear and explicit and should be more detailed than textbook proofs.
  4. If the grader is unable to make out your writing then this may hurt your mark.


You will have the option to give a final talk on your term project; this will effect the grade breakdown. Giving a final talk is encouraged, but not required.

If you do give a presentation the grades will be broken down as follows:

If you do not give a presentation the grades will be broken down as follows:

Exam Dates: