MAS 6396 Ideals, Varieties & Algorithms
Course Information
The course provides an introduction into computational techniques of elementary algebraic geometry. The main focus of the course is the close connection between ideals in polynomial rings on the algebraic side and varieties in affine space on the geometric side. The course emphasizes computational aspects and in particular provides an introduction to the theory of Gröbner bases in polynomial rings.
The course assumes familiarity with elementary concepts from algebra, as covered by the qualifying exam in algebra, for instance. Access to a computer algebra system will be provided and homework projects may require the use of a computer. Programming experience is not assumed, however.
The course will be based on the book Ideals, Varieties & Algorithms (David Cox, John Little, Donal O'Shea, Springer), which below is referred to as [CLO97]. I'll use the 2nd edition of the book, but any other edition will be fine, too, and you are not required to buy the book.
More information on the course is available in the syllabus. Your questions and comments are very welcome!
So far the following topics have been addressed in class:
- Jan 8, 2007: basic definitions for ideals and varieties
Literature: [CLS96, Ch. 1.1-1.4]
- Jan 10, 2007: monomial orderings, univariate ideal membership
Literature: [CLS96, Ch. 1.5, Ch. 2.2]
- Jan 12, 2007: an algorithm for multivariate polynomial division
Literature: [CLS96, Ch. 2.3]
- Jan 17, 2007: Dickson's Lemma, Hilbert Basis Theorem
Literature: [CLS96, Ch. 2.4, Ch. 2.5]
- Jan 22, 2007: Hilbert's Nullstellensatz
Literature: [CLS96, Ch. 4.1]
- Jan 24, 2007: Buchberger's S-pair criterion
Literature: [CLS96, Theorem 6 in Ch. 2.6]
- Jan 26, 2007: Buchberger's algorithm
Literature: [CLS96, Ch. 2.7]
- Jan 29, 2007: elimination and extension theorem
Literature: [CLS96, Ch. 3.1]
- Jan 31, 2007: using tag variables
Literature: [CLS96, Ch. 3.3]
- Feb 2, 2007: a short introduction to the computer algebra system Magma
Literature: Magma Computational Algebra System Home Page
- Feb 5, 2007: an ideal-variety correspondence; deciding radical membership; computing the intersection of ideals
Literature: [CLS96, Ch. 4.2, Ch. 4.3]
- Feb 7, 2007: ideal quotients; Homework Project #1 is available
Literature: [CLS96, Ch. 4.4];
- Feb 9, 2007: computing ideal quotients; a correspondence between maximal ideals and points
Literature: [CLS96, Ch. 4.4, Ch. 4.5]
- Feb 12, 2007: decompositions of varieties and ideals
Literature: [CLS96, Ch. 4.6, Ch. 4.7]
- Feb 14, 2007: the coordinate ring of a variety; finding a bound for the size of a finite variety
Literature: [CLS96, Ch. 5.3, Ch. 5.4]
- Feb 16, 2007: some examples
Literature: [CLS96, Ex. 14 in Ch. 4.2, Ch. 5.1]
- Feb 19, 2007: polynomial mappings and isomorphisms between varieties
Literature: [CLS96, Ch. 5.1, Ch. 5.4]
- Feb 21, 2007: computing with field extensions by means of Groebner bases
Literature: Basic algorithms for rational function fields (J. of Symb. Comp. 27(2): 143-179, 1999)
- Feb 23, 2007: computing with field extensions: minimal polynomials and transcendence degree
Literature: Basic algorithms for rational function fields (J. of Symb. Comp. 27(2): 143-179, 1999)
- Feb 26, 2007: finite matrix groups and rings of invariants
Literature: [CLS96, Ch. 7.2]
- Feb 28, 2007: using the Reynolds operator to compute generators for the ring of invariants
Literature: [CLS96, Ch. 7.3]
- Mar 2, 2007: representing an invariant in terms of genrators, examples
Literature: [CLS96, Ch. 7.3]
- Mar 12, 2007: resultant of two univariate polynomials
Literature: [Ch. 3.5]
- Mar 14, 2007: resultants and elimination
Literature: [Ch. 3.5, Ch. 3.6]
- Mar 16, 2007: the forward and inverse kinematic problem for robots
Literature: [CLS96, Ch. 6.1, Ch. 6.2]
- Mar 19, 2007: the inverse kinematic problem; Homework Project #2 is available
Literature: [CLS96, Ch. 6.3]
- Mar 21, 2007: automatic geometric theorem proving
Literature: [CLS96, Ch. 6.4]
- Mar 23, 2007: automatic geometric theorem proving
Literature: [CLS96, Ch. 6.4]
- Mar 26, 2007: Wu's method
Literature: [CLS96, Ch. 6.5]
- Mar 28, 2007: isomorphisms of polynomials problem with one secret
Literature: An attack on the isomorphisms of polynomials problem with one secret
- Mar 30, 2007: dimension of monomial ideals
Literature: [CLS96, Ch. 9.1]
- Apr 2, 2007: coordinate subspaces
Literature: [CLS96, Ch. 9.2]
- Apr 4, 2007: the complement of a monomial ideal
Literature: [CLS96, Ch. 9.2]
- Apr 6, 2007: solution of Homework Project #2, an example for computing monomials in the complement of a monomial ideal
Literature: [CLS96, Ch. 9.2]
- Apr 9, 2007: Hilbert function and dimension of a variety
Literature: [CLS96, Ch. 9.3]
- Apr 11, 2007: dimension and algebraic independence
Literature: [CLS96, Ch. 9.5]
- Apr 13, 2007: Groebner bases and cryptanalysis
- Apr 16, 2007: rational functions on a variety
Literature: [CLS96, Ch. 5.5]
- Apr 18, 2007: birational equivalence and isomorphisms of fields
Literature: [CLS96, Ch. 5.5]
- Apr 20, 2007: exercises in preparation of final exam
- Apr 23, 2007: exercises in preparation of final exam
- Apr 25, 2007: exercises in preparation of final exam
Please do not hesitate to contact me anytime
(see my homepage for email, phone number, etc.).
Apr 26, 2007