The aim of the course is to introduce the students to one of the most
important tools in computational algebra, namely, Groebner bases. This
allows one to concretely study an example of algorithmic approach to
symbolic, and not numerical, manipulation of abstract objects as
polynomial ideals. The goal is firstly to provide the basic concepts on the
topic, but also to show the versatility of Groebner bases in commutative
algebra computations and to ponder on the algorithmic questions related
to them.

Knowledge and understanding: Think about polynomial ideals in a
computational way, in particular concerning the main operations between
them. Know the basics of Groebner basis theory.

Applying knowledge and understanding: Know some basic algorithms for
computational ideal theory (reduction of a polynomial, computation of a
Groebner basis, computation of elimination ideals, intersection of ideals,
radicals).

Making judgements: Be able to assess some classes of problems that
may be attacked via the tools enabled by Groebner bases.

Communication skills: Be able to discuss about the basic topics of
computational ideal theory covered in the course.

Learning skills: Be able to independently study more advanced topics in
Groebner basis theory.

Polynomial rings, ideals, homomorphisms of rings. One may
benefit from prior exposition to basic commutative algebra
and algebraic geometry, but all the related needed concepts
will be defined in the course.

The topic of the course is the study of Groebner bases, a fundamental
tool in computer algebra. The course is structured as follows:
1. Basic concepts: characterizations of Groebner bases, their
existence and Buchberger’s algorithm for their computation.
2. Operations on ideals: how to compute intersections, radicals, primary
decompositions of ideals via Groebner bases.
3. More efficient algorithms for the computation fo Groebner
bases (Groebner walk, FGLM, and F4).
4. Examples via computer algebra software.

1. William W. Adams, Philippe Loustaunau. An introduction to Groebner
bases – Providence : American Mathematical Society.
2. Martin Kreuzer, Lorenzo Robbiano. Computational
Commutative algebra 1 – Springer-Verlag.
3. David Cox, John Little, Donal O’Shea. Ideals, varieties,
and algorithms. An introduction to computational algebraic geometry and
commutative algebra. – Undergraduate Texts in Mathematics. New York:
Springer-Verlag. xi,
513 p. (1992).
4. Bernd Sturmfels. Gröbner Bases and Convex Polytopes
– University Lecture Series. 8. Providece, RI: American
Mathematical Society (AMS). xi, 162 p. (1996).
5. Instructor’s handwritten notes

1. Term orders. Reduction of a polynomial via a family of polynomials.
Definition of Groebner bases. Various characterizations of Groebner
bases (using leading term ideals, using the notion of confluence). Dixon’s
lemma. Derivation of Hilbert’s basis theorem from the existence of
Groebner bases and Dixon’s lemma. S-polynomials. Buchberger’s
theorem and algorithm on the computation of Groebner bases. Definition
and uniqueness theorem for reduced Groebner bases.

2. Resolution of the ideal membership problem and of the problem of
representation of a polynomial with respect to an ideal via Groebner
bases. Determination of a canonical basis of the quotient of the
polynomial ring with respect to an ideal. Computation of elimination
ideals, of the kernel of a homomorphism between polynomial rings, of the
radical of an ideal, of the primary decomposition of an ideal (outline), of
the quotient ideal between two ideals, of the intersection of two ideals.
Brief outline of the geometrical interpretation of these operations.

3. The FGLM algorithm. The Groebner walk algorithm (brief outline). The F4 algorithm
(brief outline).

4. Examples of computations of Groebner bases and operations on ideals
via the software SageMath

The course consists of lectures during which the instructor
discusses the topics covered and answers student’s questions.
The instruction will provide handwritten lecture notes by the end of the
course.

The handwritten lecture notes and other information will be
available through Moodle.

The exam program coincides with the arguments of the lectures. The
exam consists of an oral discussion about the topics covered during the
course, during which the instructor will check
that the student has understood the covered material (definitions,
statements, and proofs) and may ask to solve some exercises.

To obtain a passing grade (18/30), a student should know the basic definitions, results, and proof techniques presented in the course. The grade increases according to the extent and the depth of the knowledge of the student, in terms of technical details and also of quality, clarity and precision of the presentation. To obtain the highest grade, a student should show very good command of the topic, denoted for example by personal reworking of the material or ability to address questions related to the course, but not explicitly taught in the lectures.

To ensure the access to aids at the exam from students with disabilities, specific learning disorder (SLD), or special educational needs (SED), please preemptively contact the University's Disability Service or SLD Service.