OVERVIEW

Algebra 3 deals with the basic concepts of commutative algebra and the algorithmic and computational aspects associated with them. In particular, it introduces the notions of Noetherian rings and modules, as well as algebras, followed by an introduction to Groebner bases, which are applied to the computational resolution of problems related to polynomial objects, both theoretically and in the laboratory.

AIMS AND CONTENT

LEARNING OUTCOMES

The aim of the course is to provide students with a solid understanding of the fundamental concepts of commutative algebra and the related computational aspects. In particular, the notions of Noetherian ring, module, Groebner basis are discussed and the some methods to solve of systems of polynomial equations are presented.

AIMS AND LEARNING OUTCOMES

Algebra 3 aims to provide students with fundamental knowledge in the following topics:

1. Noetherian rings and modules.
2. Algebras.
3. Gröbner bases and their use in symbolic computation.
4. Algorithms related to the computation and application of Gröbner bases

Expected learning outcomes are:

1. By the end of Algebra 3, students will be able to recognise a Noetherian ring or module and describe its main properties, including reproducing the principal proofs associated with them.
2. By the end of Algebra 3, students will be able to solve exercises concerning ideals, algebras, and modules with specific structures, such as monomial ideals or modules over PIDs.
3. By the end of Algebra 3, students will be able to explain how Gröbner bases can be used to algorithmically solve problems such as testing whether a polynomial belongs to an ideal or its radical, computing intersections of ideals and their syzygies, eliminating variables, and computing kernels of algebra homomorphisms.
4. By the end of Algebra 3, students will be able to determine algorithmically whether a system of polynomial equations has solutions, and describe these solutions explicitly whenever possible.
5. By the end of Algebra 3, students will be able to use symbolic computation systems such as CoCoA5 to solve concrete computational problems.

PREREQUISITES

The prerequisites include knowledge of basic algebraic structures such as vector spaces, groups, and rings.

TEACHING METHODS

The course includes:

1. Lectures delivered by the instructors, during which theoretical content will be presented, applied to concrete examples, and illustrated through problem-solving exercises.
2. Laboratory sessions in computational algebra, where instructors will demonstrate the practical implementation of the algorithms studied and explained during the theoretical lectures.

Attendance is strongly recommended, as both the lectures and laboratory sessions are essential for understanding the course content. The material covered combines theoretical and practical aspects of algebra and is often guided by heuristic considerations not typically found in textbooks.

SYLLABUS/CONTENT

Rings, ideals, and modules. Noetherian rings and modules. Algebras over a ring. Hilbert’s Basis Theorem. The universal property of the polynomial ring in several variables over a ring. Monomial ideals. The structure theorem for modules over principal ideal domains (PIDs).

Orderings, Gröbner bases, and division with remainder. S-pairs and syzygies. Buchberger’s criterion and algorithm. Compatibility of structural properties with Buchberger’s algorithm. Computational and implementation aspects of Gröbner bases. Algorithmic solutions to the following problems, addressed both theoretically and through computational laboratory work:

1. Testing whether a polynomial belongs to an ideal.
2. Testing whether a polynomial belongs to the radical of an ideal.
3. Computation of syzygies.
4. Intersection of ideals.
5. Polynomial elimination.
6. Computation of kernels of maps between algebras and between modules.
7. Solving systems of polynomial equations

RECOMMENDED READING/BIBLIOGRAPHY

• Computational Commutative Algebra 1, Authors: Martin Kreuzer, Lorenzo Robbiano, Springer, 2000.
• Informal Notes on Computational Algebra, Author: Aldo Conca, 2020 (available on AulaWeb).
• Informal Notes on Commutative Algebra, Author: Matteo Varbaro, 2022 (available on AulaWeb).

TEACHERS AND EXAM BOARD

LESSONS

LESSONS START

The dates of the academic calendar are available on the webpages of the Corso di Studi. Further information regarding the schedule is available here.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The exam consists of a computational laboratory test and an oral examination. The laboratory test must be completed first; passing it gives access the oral exam. The oral exam lasts approximately 60 minutes and is divided into two 30-minute phases, each corresponding to the two main parts of the syllabus.

ASSESSMENT METHODS

The oral examination consists of:

1. A discussion of a fundamental concept from the first part of the course, illustrated with relevant examples and proofs.
2. A discussion and explanation of one of the basic computational problems and its algorithmic solution using Gröbner bases, including the relevant proofs, algorithmic procedures, and significant examples.

The laboratory test consists of writing symbolic computation programs using CoCoA5 to solve two concrete problems.

FURTHER INFORMATION

Compensatory and dispensatory measures Disability/Invalidity/Specific Learning Disorder

Dispensatory measures and compensatory tools are intended to enable students to achieve the same learning objectives as their fellow students, not to facilitate the examination.

The use of compensatory tools and the application of dispensatory measures must be authorised in advance by the teacher in agreement with the Referee.

To take advantage of the adaptations during the examination, fill in the Adaptation request form; the request will be automatically sent by the system to the teacher in charge of the teaching, to the Contact Person of your School/Area/Department and in copy to the Sector; you will also receive a copy of the request sent by e-mail.

The adjustments available to students are as follows:

• Additional time (+30% DSA)
• Additional time (+50% disability/invalidity)
• Additional time during oral exams to organise the answer
• Calculator (programmable and graphing calculators are not allowed)
• Conceptual Maps
• Tables and/or Forms
• Take the exam in written form
• Take the exam in oral form
• Tutor reader (for written tests only)
• Tutor-writer (for written tests only)

Your request for adaptations must be submitted at least 7 working days before the scheduled exam date.

All information for students with disabilities and DSA is available on the webpage: Services for students with disabilities or DSA | UniGe | University of Genoa

Reference for inclusion: Sergio Di Domizio - sergio.didomizio@unige.it