Computational Algebra

The aim of the course is to introduce Groebner base theory, in order to introduce the student 
to computational algebra and its applications.

In particular, the course aims to provide students with the following skills:

Knowledge and understanding: understanding statements and proofs of fundamental theorems of
 computational algebra; developing mathematical skills in reasoning, manipulation and calculation; 
solving mathematical problems that, although not common, are of a similar nature to others already
 known; having adequate computational skills, including knowledge of specific software.

Applying knowledge and understanding: demonstrating algebraic and geometric results that are 
not identical to those already known, but clearly related to them; constructing rigorous proofs;
 solving algebraic and geometric problems that require original thinking; be able to
 mathematically formalize problems of moderate difficulty, formulated in natural language, 
and to take advantage of this formulation to clarify or solve them;

Making judgements: acquiring an aware autonomy of judgement with reference to the evaluation 
and interpretation of the resolution of an algebraic problem; being able to construct and 
develop logical arguments with a clear identification of assumptions and conclusions; 
being able to recognize correct demonstrations, and to identify fallacious reasoning.

Communication skills: knowing how to communicate information, ideas, problems, solutions 
and their conclusions, as well as the knowledge and rationale underlying them, in a clear 
and unambiguous way; knowing how to present scientific materials and arguments, orally or
 in writing, in a clear and comprehensible way.

Learning skills: having developed a greater degree of autonomy in studying.

Teaching is done on the blackboard in a traditional way. The exercises also include using the computer.

Information for students with disabilities and/or DSA. 
To ensure equal opportunities and in compliance with 
current laws, interested students may request a personal 
interview in order to plan any compensatory and/or dispensatory 
measures, based on their educational objectives and specific needs. 
Students with disabilities and/or DSA must contact the teacher, 
the CInAP contact person of the DMI (Prof. Daniele) and CInAP 
well in advance of the exam date to communicate that they 
intend to take the exam using the appropriate compensatory measures 
(which will be indicated by CInAP).

Vector spaces. Polynomial rings. Quotient rings.

Attendance is strongly recommended (see the CDS regulations).

I. Basic Theory of Groebner Bases. The Linear Case. The One-Variable Case. Monomial Orderings.
 The Division Algorithm. Definition of Groebner Base. S-Polynomials and Buchberger Algorithm. 
Reduced Groebner Bases.

II. Applications of Groebner Bases. Elementary Applications of Groebner Bases. 
Elimination Theory. Polynomial Maps. Some Applications to Algebraic Geometry.

III. Modules. Groebner Bases and Syzygies. Calculation of the modulus of the syzygies of an ideal.

Practical exercises will be assigned to be solved on the computer using a software; 
if solved with at least a sufficient grade, an oral exam will be held. The results of the exercises
carried out during the course will not be taken into account.