Topic outline

  • General

    Schedule:

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

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

    Exam: February 26 (Friday) 13:00-17:00

    Organization:

    All lectures and exercise sessions of this course take place online via zoom. Video recordings and slides for lectures will be made available after each lecture. 

    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: Robotics (1 lecture)

    Additional topic: Numerical algebraic geometry (1 lecture)

    Most results will be presented together with proofs.

    Homework:

    There will be in total five weekly 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.

    Exam:

    The exam at the end of the course will be an open book exam.

    Grade:

    There will be five weekly homework assignments (50% of the grade) and a final exam at the end of the course (50% of the grade). It is possible to receive extra points for active participation in lectures.

    Optional extra homework:

    You can submit a solution to any exercise from "Invitation to Nonlinear Algebra" by Michalek, Sturmfels as extra homework. Each exercise gives 3 points. Sections 1-4 of the book are most related to this course.

    Communication:

    Official announcements will be posted in MyCourses. The rest of communication for this course takes place in Zulip. Anyone with an Aalto account can join the Zulip channel ms-e1142.zulip.cs.aalto.fi. Please be active asking your questions!

    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 will be posted in the Panopto block on the right just below the completion bar
    • Slides for the lectures will be posted under Materials
    • Further reading: "Invitation to Nonlinear Algebra" by Michalek, Sturmfels