Module page
Istituzioni di Algebra Superiore
| academic year: | 2013/2014 |
| instructor: | Dina Ghinelli |
| degree course: | Mathematics for applications (magistrale) |
| type of training activity: | affine e integrativa |
| credits: | 9 (72 class hours) |
| scientific sector: | MAT/02 Algebra |
| teaching language: | italiano |
| period: | I sem (30/09/2013 - 17/01/2014) |
Lecture meeting time and location
Presence: highly recommended
Module aims:
- To introduce the theory of Groebner basis in order to study some applications in geometry, in modern cryptography and in coding theory.
- To illustrate some of the above applications
- Geometry, Algebra and Algorithms: Polynomials and Affine Space; Affine varieties and parametrizations; Examples; Ideals; Division algorithm and its implications; Euclidean algorithm and Ideal membership problem; Nullstellensatz for polynomials of one variable.
- Groebner Bases: first examples of Groebner bases method; Gauss-Jordan algorithm; parametrization and implicitization of linear varieties; monomials orderings; A division algorithm for polynomials in more than one variable: examples and remarks; Monomial Ideals and Dickson's Lemma; the Hilbert basis Theorem and Groebner bases; S-polynomials; Buchberger's algorithm; First applications of Groebner bases: The Ideal membership problem, the problem of solving polynomial equations, the implicitization problem.
- Elimination theory: the elimination and extension Theorems; the geometry of elimination; examples; the closure Theorem and some consequences; Implicitization algorithm for polynomial and rational parametrizations; the use of Groebner bases for studying singular points and envelopes in the real plane.
- The Algebra-Geometry dictionary: Hilbert's Nullstellensatz; Radical Ideals and the Ideal-Variety correspondence; Sum, Products and Intersections of Ideals; Zariski Closure and Quotients of ideals; Irreducible varieties and Prime Ideals; decomposition of a Variety into irreducibles.
- Introduction to Cryptography and to the theory of error correcting codes.
- Some application in cryptography and coding theory.
- REGISTRAZIONE: Tutti gli studenti sono invitati a registrarsi entro il 21 ottobre 2013 sulla pagina di registrazione, precisando oltre al nome e cognome tutte le altre informazioni (particolarmente importante è inserire anche numero di matricola e Email, anche se nella pagina sono indicati come non obbligatori).
- Pagina del corso
Module subject:
Suggested reading:
- W. Baldoni, C. Ciliberto, G.M. Piacentini Cattaneo, Aritmetica, Crittografia e Codici, Springer-Verlag (2006).
- D. Cox, J. Little, D. O'Shea, Ideals, Varieties, and Algorithms, An introduction to Computational Algebraic Geometry and Commutative Algebra, Springer-Verlag (1992).
- D. A. Gewurz e F. Merola, Concetti essenziali ed esempi d'uso di CoCoA, DMURLS, Roma 2001.
- M. Kreuzer e L. Robbiano, Computational Commutative Algebra I, Springer, 2000.Type of course: standard
Useful links:
Knowledge and understanding: Successful students will be able to study sets of points in n-dimensional affine spaces which are common zeros of a set polynomials in n variables - To use the algorithm of division for polynomials in more than one variable - To know the basic and some problems and classical results in Cryptography and Coding theory.
Skills and attributes: Successful students will be able to establish a sort of dictionary Algebra-Geometry which will allow them to translate algebraic problems into geometric problems and conversely and to use Groebner Basis in the solution of polynomial equations and problems of cryptography and coding theory. They will also be able to express properly concepts and results of the above theories. This may be a useful starting point for further studies of algorithmic nature.
Personal study: the percentage of personal study required by this course is the 65% of the total.