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

Advances in Algebra: Generalised Buchberger and Schreyer Algorithms for Coherent Rings
Prof. Ihsen Yengui (University of Sfax, Tunisia, Visiting Professor INdAM 2025-26)
March 23–April 24, 2026

Scientific sector
MATH-02/A - Algebra

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.

Assessment method
Student project with essay, presentation and discussion

Period
From 23rd March 2026 to 24th April 20256

Number of academic hours
20

Timetable
TBA

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