Exercise 1.

• Try installing SageMath and Macaluay 2 to your computer. Please contact me (instructor) if you need help.

The links: https://www.sagemath.org, http://www2.macaulay2.com/Macaulay2/

• Note that you can use SageMath online via either Cocal or SageMathCell, to which you can acces via the website that is noted above. In case, you face with a problem of installing it, you can use these platforms (temporarily). However, it would be beneficial to carry out the installation as soon as possible.

• Go through the documentation on "ideals in multivariate polynomial rings".

https://doc.sagemath.org/html/en/reference/polynomial_rings/sage/rings/polynomial/multi_polynomial_ideal.html

With its guidance, try out your own examples. Notice that "ideal membership problem" is mentioned very early in the documentation.

• Check the overview on "ideals" for M2:

http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.17/share/doc/Macaulay2/Macaulay2Doc/html/_ideals.html

Starting with "creating an ideal" in the "Menu", check their examples. Make up your examples to explore the class of ideals in M2.

• (From the book "Invitation to Nonlinear Algebra"). #13, #14, #19 from Exercises in Chapter 1; #7, #15 from Exercises in Chapter 2; Example 2.5; investigate all of these exercise with the help of one of the software Sage and M2.

Exercise 2.

• Check through the "Programming" section in the tutoripol of SageMath, and try your own example under its guidance:

https://doc.sagemath.org/html/en/tutorial/programming.html#

• Similarly, find out more about Macaluay 2 language:

http://www2.macaulay2.com/Macaulay2/doc/Macaulay2-1.17/share/doc/Macaulay2/Macaulay2Doc/html/___The_sp__Macaulay2_splanguage.html

• Continue solving the exercises from the textbook (for the moment until Chater 2/Section 5) and write your solutions in our overleaf file.

Exercise 3.

• (From the book "Invitation to Nonlinear Algebra" ~ version of January 10). Excercises #4, #12, #15

• Given positive integers m,n and r<min(m,n). Think of the set of m by n matrices of rank at most r over a field k, say the field of rational numbers. Describe the locus of such matrices. Giving a hint, which polynomial equations would define this set? Once you figured out the polynomials, define the corresponding ideal or the affine variety in one of the software, and study it. For instance, you can try to compute Groebner basis, or find its dimension, or its degree etc. Report your observations.

• Implement your multivariable polynomial division algorithm in one of the software Sage or M2.

Exercise 4.

• Continue solving the exercises from the textbook (IVA) and write your solutions in our overleaf file. Some highlights of the exercises: 2.7.9, 2.7.14, 2.8.5, 2.8.7, 2.8.10, 3.1.2, 3.1.7, 3.1.9, 3.2.4, 3.3.12, 3.4.12, 3.4.12, 3.6.4, 3.6.10, 3.6.11, 3.6.14, 3.6.16, 3.6.19.

• Prove Theorem 2 in 3.2.

• Recall Proposition 5 in 3.6. Write a function in Sage/M2 that outputs A and B of given f and g.

TEXTBOOK:

Ideals, Varieties, and Algorithms (fourth edition) ∼ David A. Cox, John Little, Donal O’Shea

We will collect solutions for the exercises of the textbook, for which you shall use the overleaf file we own.

Some other useful references:

• Invitation to Nonlinear algebra ∼ Mateusz Michalek and Bernd Sturmfels

To download the book: https://www.math.uni-konstanz.de/~michalek/book.html

• Using Algebraic geometry ∼ David A. Cox, John Little, Donal O’Shea

Some definitive references:

• Algebraic Geometry ∼ Robin Hartshorne

• Geometry of Schemes ∼ David Eisenbud, Joe Harris

• 3264 & All That Intersection Theory in Algebraic Geometry ∼ David Eisenbud, Joe Harris - Algebraic Geometry and Arithmetic Curves ∼ Qing Lui

GRADING:

There will be one take home final exam, and an oral exam following the final exam at the end of the semester.

Your conttibution to the course will be considered for the process of fine-tuning.