# INSTITUTIONS OF HIGHER ALGEBRAModule MODULO II

**Academic Year 2023/2024**- Teacher:

**VINCENZO MICALE**

## Expected 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.