Advances in Computer Algebra

09:00-10:00 Mike Stillman

Kuramoto oscillators: dynamical systems meet algebraic geometry

Abstract. Coupled oscillators appear in a large number of applications: e.g. in biological, chemical sciences, neuro science, power grids, and many more fields. They appear in nature: fireflies flashing in sync with each other is one fun situation.

In 1974, Yoshiki Kuramoto proposed a simple, yet surprisingly effective model for oscillators. We consider homogeneous Kuramoto systems (we will define these notions!). They are determined from a finite graph. In this talk, we describe some of what is known about long term behavior of such systems (do the oscillators self-synchronize? or are there other, "exotic" solutions?), and then relate these systems to systems of polynomial equations. We use algebra, computations in algebraic geometry, and algebraic geometry to study equilibrium solutions to these systems. We will see how computations using algebraic geometry and my computer algebra system Macaulay2 finds all graphs with at most 8 vertices (i.e. 8 oscillators) which have exotic solutions.

Note: we assume essentially NO dynamical systems nor much algebraic geometry in this talk! This is joint work with Heather Harrington and Hal Schenck, and also Steve Strogatz and Alex Townsend.

10:00-10:30 Gert-Martin Greuel

History of Singular and its relation to Zariski’s multiplicity conjecture

10:30-11:00 Gerhard Pfister

About some contributions of Hans to mathematics

Abstract. In the talk three problems will be discussed:

• classification of plane curve singularities with semigroup <np,nq,npq+d>
• complete intersection singularities with exact Poincare complex not being quasihomogeneous
• standard basis computations using the semicontinuity of the highest corner

At the beginning some historical remarks about the development of Singular will be given.

11:30-12:00 Wolfram Decker

Collaborating with Hans — from Singular to OSCAR

Abstract. I had the pleasure of working with Hans for 35 years. I will highlight a few key points from our collaboration, with particular focus on the embedding of the computer algebra system Singular, which was architected by Hans, into the newly evolving system OSCAR which combines specialized tools from different mathematical areas under one umbrella.

12:00-12:30 Magdaleen Marais

The Classification Problem for Corank 2 Singularities with a non-degenerate Newton Boundary

Abstract. In this talk we will give an outline of how we completed the algorithmic classification of corank 2 singularities with a non-degenerate Newton boundary by:

1. proving a normal form theorem: for a mu-constant stratum K which contains a germ with non-degenerate Newton boundary B, a single normal form exists and can be constructed from the vertex points of B and a regular basis of the polynomial formed by these points.
2. developing an algorithmic classifier which computes a normal form for any corank 2 singularity which is equivalent to a germ with a non-degenerate Newton boundary, as well as a specific polynomial in the normal form to which it is equivalent.
3. developing an algorithm to compute lists of normal forms satisfying specified finite modality and/or Milner number bounds.

14:00-15:00 Winfried Bruns

Sagbi bases, defining ideals and algebras of minors

Abstract. This paper extends the article of Bruns and Conca on Sagbi bases and their computation (J. Symb. Comput. 120 (2024)) in two directions. (i) We describe the extension of the Singular library sagbiNormaliz.sing to the computation of defining ideals of subalgebras of polynomial rings. (ii) We give a complete classification of the algebras of minors for which the generating set is a Sagbi basis with respect to a suitable monomial order, and we identify universal Sagbi basis in three cases. The investigation is illustrated by several examples.

This is joint work with Aldo Conca and Francesca Lembo (Bollettino dell’Unione Matematica Italiana 2025, https://doi.org/10.1007/s40574-025-00488-1).

15:00-15:30 Janko Böhm

Continuing Hans’s Legacy: New Developments in Singular

Abstract. In this talk I will discuss recent and ongoing developments in Singular as an effort to continue Hans Schönemann’s legacy. Many of these projects were still initiated while working with Hans and relying on his expertise and judgement.

One focus will be the technical side, with topics ranging from modern software-engineering infrastructure for Singular, through AI-assisted methods for deriving modern documentation from Singular’s code base and documentation, to making Singular available online via WebAssembly. I will also present the AI Computer Algebra Builder, which generates specialized code for Singular and other computer algebra systems. The second focus will be on massively parallel methods combining Singular with GPI-Space, with applications to a massively parallel Buchberger algorithm.

Beyond the technical developments, this talk is also a personal tribute. Hans was a friendly, loyal, and generous colleague whose help was crucial for my algorithmic and theoretical work. He was always eager to help my students and myself, and his influence continues both in computer algebra and among the people he supported.

16:00-16:30 Yang Zhang

My Collaboration with Dr. Hans Schönemann: Syzygies, Feynman Integrals, and NeatIBP — A Memorial from a Physicist

Abstract. As a physicist, I deeply valued my long-standing collaboration with Dr. Hans Schönemann. Our work together greatly advanced the use of syzygy methods in Feynman integral reduction and ultimately contributed to the development of the package NeatIBP. Hans made significant contributions to this direction, bringing his profound expertise in computational algebraic geometry to bottleneck problems in theoretical physics. I will remember his insight, generosity, and dedication with deep respect. May Dr. Hans Schönemann rest in peace.

16:30-17:00 Anna Maria Bigatti

A new iterative algorithm for comprehensive Gröbner systems

Work with Elisa Palezzato and Michele Torielli.

Abstract. A comprehensive Gröbner system for a parametric ideal I in K(A)[X] represents the collection of all Gröbner bases of the ideals I' in K[X] obtained as the values of the parameters A vary in K.

The recent algorithms for computing them consider the corresponding ideal J in K[A,X], and are based on the stability of Gröbner bases of ideals under specializations of the parameters. Starting from a Gröbner basis of J, the computation splits recursively depending on the vanishing of the evaluation of some "coefficients" in K[A].

In this paper, taking inspiration from the algorithm described by Nabeshima, we create a new iterative algorithm to compute comprehensive Gröbner systems. We show how we keep track of the sub-cases to be considered, and how we avoid some redundant computation branches using "comparatively-cheap" ideal-membership tests, instead of radical-membership tests.