Credits: 5

Schedule: 28.10.2019 - 04.12.2019

Contact information for the course (applies in this implementation): 

Office hours by appointment. The instructor can be reached over email at

Details on the course content (applies in this implementation): 

In the first half of the course, we will cover projective algebraic geometry and the dimension of a variety, and time permitting we will also talk about polynomial and rational functions on a variety. The second half of the course is for working on student projects (4-10 pages). The presentations of the project will take place during the last week of the course.

Examples for student projects are:

1) Polynomial systems in economics: Sturmfels "Solving Systems of Polynomial Equations" Chapter 6

2) Low-rank matrix completion: Bernstein, Blekherman, Sinn "Typical and Generic Ranks in Matrix Completion", in particular Section 3

3) Linear partial differential equations with polynomial coefficients: Sattelberger, Sturmfels "D-Modules and Holonomic Functions" (TAKEN)

4) Invariant theory: Cox, Little, O'Shea "Ideals, Varieties and Algorithms" Chapter 7

5) Semidefinite programming and sums of squares: Michalek, Sturmfels "Nonlinear algebra" Chapter 12

6) Polynomial systems in statistics: Sturmfels "Solving Systems of Polynomial Equations" Chapter of 8

7) Numerical algebraic geometry: Bates, Hauenstein, Sommese, Wampler "Numerically Solving Polynomial Systems with Bertini" (TAKEN)

8) Multivariate resultant: Sturmfels "Solving Systems of Polynomial Equations" Chapter of 4

9) Tropical algebra: Michalek, Sturmfels "Nonlinear algebra" Chapter 7

Elaboration of the evaluation criteria and methods, and acquainting students with the evaluation (applies in this implementation): 

20% of the grade is for the homework, 50% of the grade is the written-up student project and 30% of the grade is for the presentation of the student project.

Details on calculating the workload (applies in this implementation): 

Contact hours 16-20h, self-study ca 110h

Details on the course materials (applies in this implementation): 

Cox, Little, O’Shea “Ideals, Varieties and Algorithms”

Michalek, Sturmfels "Nonlinear algebra"

Course Homepage (valid 01.08.2018-31.07.2020):

Grading Scale (valid 01.08.2018-31.07.2020): 


Details on the schedule (applies in this implementation): 

Two homeworks during the first three weeks. The first homework will be posted during the second week and the second homework will be posted during third week. There will be one week to submit homework. Student project topics will be assigned during the first two weeks.


Registration and further information