INSTITUTIONS OF HIGHER ALGEBRAModule MODULO II
Academic Year 2023/2024 - Teacher: VINCENZO MICALEExpected Learning Outcomes
Computational Algebra - 6 CFU - 47 total hours
33 frontal lessons
14 exercises.
The objective of the second module of the course is to introduce the theory of Groebner bases , in order to begin the computational student to algebra and its applications
Course Structure
Teaching is done on the blackboard in a traditional way. The exercises also include using the computer. If the teaching is given in mixed or remote mode, they can be introduced the necessary changes with respect to what was previously stated, in order to comply the planned program and reported in the syllabus.
Information for students with disabilities and / or SLD. To guarantee equal opportunities
and in compliance with the laws in force, interested students can ask for a personal interview
in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives
and specific needs. It is also possible to contact the referent teacher CInAP (Center for Active
and Participated Integration - Services for Disabilities and / or SLD) of our Department,
prof. Filippo Stanco.
Required Prerequisites
Vector spaces. Rings of polynomials. Quotient rings
Attendance of Lessons
Strongly recommended.
Detailed Course Content
I. Basic Theory of Groebner Bases. The linear case. The case of a single variable. Monomial orders. The division algorithm. Definition of Groebner Bases. S - polynomials and Buchberger algorithm. Reduced Groebner bases .
II . Applications of Groebner Bases. Elementary applications of Groebner Bases. Theory of elimination. Polynomial maps. Some applications to Algebraic Geometry .
III . Modules. Groebner bases and Syzygies. Calculation of the module of syzygy of an ideal.
Textbook Information
W.W. Adams, P. Loustaunau, An introduction to Groebner Bases, American Math. Soc, 1994.
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Teoria di base delle Basi di Groebner | 1) |
| 2 | Applicazioni delle Basi di Groebner | 1) |
| 3 | Moduli | 1) |
Learning Assessment
Learning Assessment Procedures
Practical exercises will be assigned to be solved on the computer using software;
if solved with at least a sufficient score, you will move on to an oral exam.
The results of the exercises carried out during the course will not be taken into account.
Verification of learning can also be carried out electronically, should conditions require it.
Examples of frequently asked questions and / or exercises
Characterization theorem of Groebner bases. Applications of Groebner Bases to Algebra
and Algebraic Geometry. Calculation of the modulus of the syzygies of an ideal.