Sections as a Computational Tool in Invariant Theory

par Evelyne Hubert (Inria)

jeudi 4 juin 2026, 14:00 → 15:00 Europe/Paris

Salle Pellos (1R2 - 207) (IMT)

Invariants are essential for classifying mathematical objects up to a group of transformations. This talk will review several constructions of rational invariants where a section to the orbits is an essential ingredient for tractable computations.

We will start with a general construction to obtain a generating set for the field of invariants. The output of the algorithm further provides rewrite rules so that any other invariants can be written explicitly as a rational function of the generating invariants. 

In the case of linear actions of a torus - a.k.a. scalings - one can determine a section with a single point of intersection with most orbits. Furthermore this rational section, the generating rational invariants and the rewrite rules  can all be computed with  linear algebra over the integers. This has application in parameter reduction in mathematical models.

Another type of sections that proved relevant in a computational context are Seshadri slices. The rational invariants are then characterized uniquely by their restriction to the slice,  and these restrictions are invariants of a subgroup. This allows to address actions of the orthogonal group relevant to applications, exhibiting mininal generating set of invariants, how to evaluate them and examine their separation properties. 

The material presented in this talk is based on the results of collaborations with Irina Kogan (North Carolina State University), George Labahn (University of Waterloo), Paul Görlach  (Otto von Guericke University Magdeburg) and Martin Jalard (Inria Côte d'Azur).