Welcome to the Weekly Seminars on Gröbner bases and Applications. The weekly meetings will take place on Wednesdays 10:00-11:00 CE(S)T, at KU Leuven, campus Arenberg 3, in the seminar room B02.18 (full name 200B.02.018; second floor of the Math building 200B).
Organizers: Matthias Orth and Rafael David Mohr
Registration:
Please email Matthias if you are interested in attending the meeting or have any questions.
Overview:
The first goal of the seminar is to give an introduction to Gröbner bases and present some of their applications. Gröbner bases are one of the main tools underlying computer algebra systems such as Macaulay2 and Singular. This illustrates the central role they play for symbolic computation in Algebraic Geometry.
Schedule:
September 30, 2025 - 12:30 CEST - B02.18: Matthias Orth
Introduction to Gröbner bases: In this talk, I will define Gröbner bases and present the basic algorithm for computing them. I will give some context with regard to reduction relations, complexity, and non-commutative rings. We will also have time to discuss which topics to select for the more advanced part of the seminar- you are welcome to contribute ideas!
October 8, 2025 - 10:00 CEST - B02.18: Rafael David Mohr
First Applications of Gröbner bases using OSCAR: Following Matthias' introduction to Gröbner bases, I will present how their basic properties can be used in practice to solve problems coming from different areas of mathematics. For this I will show some concrete computations in the computer algebra system OSCAR which I plan to introduce briefly at the beginning of the session.
If you wish to follow along with the computations on your own computer you can find installation instructions for OSCAR here: https://www.oscar-system.org/install/linux/. At the top of the page you can find the instructions for different operating systems. If you encounter any issues during the installation feel free to contact me.
October 15, 2025 - 10:00 CEST - B02.18: Bence Sógor
Primary Decompositions: Building on the previous talks, which introduced Gröbner bases and their computation, we will look at some applications. By calculating the primary decomposition or the radical of an ideal in a polynomial ring, we can obtain information about the geometry of the variety that the ideal defines. Along the way, we will discuss several elegant algorithms that use Gröbner bases to carry out these computations. Slides for the talk
October 22, 2025: Exceptionally, we won't have a meeting this week. Simon Telen (Max Planck Institute) will give a talk on “Lissajous Varieties” on October 22, 2025, from 10:00 to 11:30 AM. Since the topic is closely related to our seminar series, everyone is encouraged to attend. It should be a very interesting talk from an algebraic perspective. For more information, see set.kuleuven.be/phd/seminars/telen
October 29, 2025 - 10:00 CET - B02.18: Maria Agustina Cagliero
Weight vectors, homogenization, and the Gröbner fan, part 1: Given an ideal, the Gröbner bases of it associated to different monomial orders may have very different properties and can look very distinct from one another. Clearly there are infinitely many monomial orders if we have 2 or more variables. However, if the ideal is fixed, then all monomial orders can be grouped into finitely many equivalence classes, giving us finitely many Gröbner bases. Given an ideal I, one can view the set of finitely many different Gröbner bases by associating a polyhedral fan to the ideal, where each top-dimensional cone corresponds to a different initial ideal of I. This fan is called the Gröbner fan. In this talk, we will prove that the collection of possible Gröbner bases of I is finite, and give examples on how to construct the Gröbner fan for a given ideal. Further information
November 5, 2025 - 10:00 CET - B02.18: Daniel Windisch
Weight vectors, homogenization, and the Gröbner fan, part 2: In this talk, the Gröbner basis conversion algorithm called Gröbner walk was discussed.
November 12 2025 - 10:00 CET - B02.18: Matthias Orth
Module Gröbner bases, syzygies, and free resolutions: After briefly introducing standard graded modules and their maps, I will define module monomial orderings and module Gröbner bases. We will see how these bases can be used to compute syzygy modules of finite sequences of homogeneous polynomials. After that, I will introduce minimal free resolutions and explain how they can be obtained by iteratively computing syzygy modules. Finally, we will discuss important invariants of graded modules, such as the Betti numbers, that can be derived from minimal free resolutions. (Slides, Macaulay2 session)
November 19 2025 - 10:00 CET - B02.18: Sebastian Seemann
Gröbner bases over valued fields: We will discuss how to adapt Gröbner bases to fields with valuations and how this is connected to tropical geometry. We will discuss how to adapt the definition of initials, and in consequence the normal form and Buchberger algorithm. A new problem that appears, is that termination of the adapted normal form algorithm does not immediately follow from termination of the original algorithm. We will end by discussing complexity and implementation issues on concrete examples.
November 26 2025 - 10:00 CET - B02.18: Emiliano Liwski
Homotopy continuation techniques: In this talk, I will present a tropical approach to solving parametrized polynomial systems, following the work of Helminck, Henriksson and Ren. The key idea is to use tropical geometry to construct homotopies that are generically optimal, meaning that the number of solution paths tracked matches the generic number of roots of the system. We will see how this method can be applied to some particular polynomial systems, such as vertically and horizontally parametrized polynomial systems.
December 3 2025 - 10:00 CET - B02.18: Daisie Rock
Questions and conjectures on extremal Hilbert series: We will talk about Hilbert series and some motivations for studying them. Then we will state the long standing conjecture by Fröberg and then examine some (many) cases where the conjecture is true. Next we will move on to look at ideals under a particular monomial ordering and state a conjecture by Moreno-Socías, which turns out to be related to Fröberg's conjecture by a Theorem by Pardue.
December 10 2025 - 10:00 CET - B02.18: Erdenebayar Bayarmagnai
Complexity of Gröbner basis computations: [Abstract TBA]
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The meeting on 10 December will be the last before the Christmas break. We plan to continue the seminar in the new year with more talks on tropical geometry.
January 14 2026 - 10:00 CET - B02.18: Maria Agustina Cagliero
Tropical fan and Gröbner fan: [Abstract TBA]
Books:
Ideals, Varieties and Algorithms by David A. Cox, John Little and Donal O'Shea
Using Algebraic Geometry by David A. Cox, John Little and Donal O'Shea
Introduction to Tropical Geometry by Diane Maclagan and Bernd Sturmfels