Topic outline

  • General

    Schedule:

    Lectures: Mondays and Wednesdays 14:15-16:00 in M1 (Kaie Kubjas)

    NB! During the first week, the Monday lecture takes place on Tuesday 14:15-16:00 in M1

    Exercise sessions: Fridays 12:15-14:00 in Y342a (Muhammad Ardiyansyah)

    Exam: Monday, 30.05 at 13:00-16:00 in Y342a

    Organization:

    Almost all lectures and exercise sessions take place in person. There will be a special arrangement for the lectures of the fifth week. Slides for the lectures will be posted on MyCourses. Video recordings for the last year's lectures are available at https://mycourses.aalto.fi/course/view.php?id=29643. 

    Goals:

    At the end of this course, the student can 

    • describe the theory for solving systems of polynomial equations
    • apply the theory to solve systems of polynomial equations symbolically and numerically
    • solve systems of polynomial equations that appear in other fields


    Content:

    We will cover chapters 1-4 and 6 from “Ideals, Varieties and Algorithms” by Cox, Little and O’Shea. One lecture will be spent on numerical algebraic geometry following the book "Numerically solving polynomial systems with Bertini" by Bates, Hauenstein, Sommese and Wampler.

    Chapter 1: Geometry, Algebra and Algorithms (1 week)

    Chapter 2: Groebner Bases (1.5 weeks)

    Chapter 3: Elimination Theory (1 week)

    Chapter 4: The Algebra-Geometry Dictionary (1.5 weeks)

    Chapter 6: Applications (1 lecture)

    Additional topic: Numerical algebraic geometry (1 lecture)

    Most results will be presented together with proofs.

    Homework:

    There will be in total six weekly homework assignments, out of which five best count for the grade (you need to submit solutions only to five homework assignments). Homework assignments contain exercises to be solved by hand or by a computer algebra software. Introduction to a computer algebra software Macaulay2 will be given in the first exercise session.

    All homework is returned through MyCourses as one file in the pdf format. Code can be submitted as a separate file. The homework is posted on Tuesdays and the deadline for homework is one week later on Tuesday.

    Exam:

    Students are allowed to bring one A4 sheet handwritten on one side of the sheet to the exam.

    Grade:

    There will be weekly homework assignments (50% of the grade, 5 best homework grades out of 6 are taken into account) and a final exam at the end of the course (50% of the grade). A necessary (but not sufficient) condition to pass the course is to obtain at least 20% of the points on the exam.

    Communication:

    Official announcements will be posted in MyCourses. The rest of communication for this course takes place in Zulip. Please use the following invitation link to Zulip: https://ms-e1142.zulip.aalto.fi/join/sbe7zv457ururwcgiehovysa/. Please let us know if the invitation link does not work.

    Lecture materials:

    • "Ideals, Varieties and Algorithms" by Cox, Little and O’Shea
    • "Numerically solving polynomial systems with Bertini" by Bates, Hauenstein, Sommese and Wampler
    • Video recordings in the Panopto block of https://mycourses.aalto.fi/course/view.php?id=29643
    • Slides for the lectures will be posted under Materials
    • Further reading: "Invitation to Nonlinear Algebra" by Michalek, Sturmfels