66731 - Algebra and Geometry for applications

Academic Year 2017/2018

  • Docente: Mirella Manaresi
  • Credits: 6
  • SSD: MAT/03
  • Language: Italian
  • Moduli: Mirella Manaresi (Modulo 2) Hans Joachim Rudiger Achilles (Modulo 1)
  • Teaching Mode: Traditional lectures (Modulo 2) Traditional lectures (Modulo 1)
  • Campus: Bologna
  • Corso: Second cycle degree programme (LM) in Mathematics (cod. 8208)

Learning outcomes

The student is supposed to acquire some advanced knowledge in algebra and geometry and to be able to apply them to problems.

Course contents

Polynomials in one variable with coefficients in a field, resultant of two polynomials.
Polynomials in several variables with coefficients in a field and their properties. Monomial orders. Groebner basis of an ideal of the polynomial ring and Buchberger algorithm.
Systems of polinomial equations and elimination theory. Resultants and elimination ideals.
Introduction to the computer algebra systems CoCoA and Singular.
Applications of Groebner basis and elimination theory to the study of singular points of curves and surfaces, envelopes of families of curves, impliticization problems, interpolations problems, cinematic problems of robotics. Applications of Groebner basis to error-correcting linear codes.

Readings/Bibliography

D.Cox - J.Little - D.O'Shea: Ideals, Varieties and Algorithms. 3rd Ed. Undergraduate Texts in Mathematics. Springer Verlag, New York 2007
D.Cox - J.Little - D.O'Shea: Using algebraic geometry. Second edition. Graduate Texts in Mathematics, 185. Springer, New York, 2005.

Teaching methods

Lectures and laboratory exercise sessions. Sheets of exercises will be handed out during the lectures (see http://www.dm.unibo.it/~manaresi/ andhttp://www.dm.unibo.it/~achilles/ ) , in addition to the ones available in the suggested textbooks.

During laboratory sessions some of these exercises will be discussed, some others must be prepared by students before the exam.

In the office hours students will be coached individually.

Assessment methods

Oral exam, starting from the discussion of the exercises that students must solve and give to the teachers at least a week before the exam. For the solution of exercises students need to use COCOA or Singular or some other software for symbolic computation available in the Laboratories.

The date of the exam must be fixed with the teachers of the course.

Teaching tools

Lectures and laboratory exercise sessions. Sheets of exercises will be handed out during the lectures (see http://www.dm.unibo.it/~manaresi/ andhttp://www.dm.unibo.it/~achilles/ ) , in addition to the ones available in the suggested textbooks. In the office hours students will be coached individually.

For Module one will be used the free software CoCoA (see ftp://cocoa.dima.unige.it/cocoa) and Singular (see http://www.singular.uni-kl.de/).

In the students Laboratories of the Department of Mathematics students can be use also Macaulay, Maple, Reduce, Mathematica.

 

In the office hours students will be coached individually.

Links to further information

http://www.dm.unibo.it/~manaresi/

Office hours

See the website of Mirella Manaresi

See the website of Hans Joachim Rudiger Achilles