Topic outline

  • General

    Learning outcomes:

    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 from “Ideals, Varieties and Algorithms” by Cox, Little and O’Shea. Two lectures will be spent on applications and 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)

    Applications and numerical algebraic geometry (2 lectures)

    Most results will be presented together with proofs.

    Organization:

    This course is an in-person course.

    • Lectures: Mondays and Wednesdays 14:15-16:00 in M1 (Kaie Kubjas,  kaie.kubjas@aalto.fi)
      • On 13.05 and 27.05 the lecture takes place in U9
      • NB! During the second week, the Wednesday lecture takes place on Thursday 14:15-16:00 in M1
    • Exercise sessions: Fridays 12:15-14:00 in Y342a (Nataliia Kushnerchuk)
    • Exam: Monday, 03.06 at 13:00-16:00 in Y342a
    • Office hours are Wednesdays after the lecture.

    Completing the course:

    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.

    Exam:

    Exam takes place on Monday, 03.06 at 13:00-16:00 in Y342a.

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

    Please see http://math.aalto.fi/exams for individual study arrangements for exams (deadline to submit the form is 1 week before the exam, but the earlier the better).

    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.

    Communication:

    Official announcements will be posted in MyCourses.

    Study material:

    Slides for lectures can be found under Materials. The slides are from 2022 and will be updated as the course progresses. The changes to the slides covering chapters 1-4 in Cox, Little and O’Shea will be minor, major changes can be to the material on applications and numerical algebraic geometry. Video recordings for the 2021 version of the course are available at https://mycourses.aalto.fi/course/view.php?id=29643. Textbooks relevant for the course are:

    • "Ideals, Varieties and Algorithms" by Cox, Little and O’Shea
    • "Numerically solving polynomial systems with Bertini" by Bates, Hauenstein, Sommese and Wampler
    • Further reading: "Invitation to Nonlinear Algebra" by Michalek, Sturmfels