Advances in Algebra – Generalised Buchberger and Schreyer Algorithms for Coherent Rings

Abstract:

After a reminder of the key concepts and methods of constructive/computational algebra,
this course will aim at generalised Buchberger and Schreyer algorithms for coherent rings. Chapter 1
focuses on the concept of Gröbner bases over a field as a powerful tool for problem solving in algebraic
geometry. In contrast to typical treatments of Gröbner bases, we give in Chapter 2 a theory of Gröbner
bases in the broader framework of coherent arithmetical rings (possibly with nonzero zero-divisors)
and, more generally in Chapter 3, over coherent strongly discrete rings. Notably, for any finitely
generated submodule M of a free module over a multivariate polynomial ring with coefficients in a
discrete coherent ring, we prove that its module LT(M) of leading terms is countably generated, and
provide an algorithm for computing explicitly a generating set. This result is also useful when LT(M) is
not finitely generated. We further give a general constructive version of Hilbert’s syzygy theorem over
strongly discrete coherent rings, following Schreyer's method. 

Format: The course is offered in a hybrid format, both in person and online. All participants will be provided with the necessary links.

Assessment method: student project with essay, presentation and discussion

Period: from 23rd March 2026 to 24th April 2026

Number of academic hours: 20 

Tentative timetable:

- Moday, March 23, 14:30-17:30, Aula L,
- Wednesday, March 25, 15:30-17:30, Aula M,
- Moday, March 30, 14:30-17:30, Aula L,
- Wednesday, April 1, 15:30-17:30, Aula M,
- Wednesday, April 8, 15:30-17:30, Aula M,
- Moday, April 13, 14:30-17:30, Aula L,
- Wednesday, April 15, 15:30-17:30, Aula M,
- Wednesday, April 22, 15:30-18:30, Aula M.

Venue:

Department of Computer Science
University of Verona
Strada le Grazie 15
37134 Verona

Enroll: LINK