Lecture 1: Gröbner Bases

After introducing basic notions such as term orderings and leading terms,  we define Gröbner bases and provide several characterizations of this concept. In particular, Buchberger's criterion allows us to formulate Buchberger's algorithm for computing Gröbner bases. Reduced Gröbner bases and the extended Buchberger algorithm form the underpinnings of many of the later applications.

Lecture 2: Applications of Gröbner Bases

Two types of applications of Gröbner bases are presented. The first one is based on the computation of syzygy modules via lifting of syzygies. It enables linear algebra operations with polynomial ideals and modules such as kernels and images of linear maps, as well as ideal operations such as intersections and colon ideals. The second type of applications is founded on computing elimination ideals. Among others, we discuss the computation of kernels and images of algebra homomorphisms, as well as saturation and homogenization of polynomial ideals.

Lecture 3: Hilbert Functions

After introducing graded rings, homogeneous ideals, and Nakayama's lemma, we present the basics of the theory of Hilbert functions and Hilbert series. Besides algorithms for computing them, we discuss their structure and basic properties.

Lecture 4: Applications of Hilbert Functions

The first applications of Hilbert functions we consider are the computation of important invariants such as dimension, multiplicity, and h-vectors. Then we discuss further topics where Hilbert functions play a prominent role such as Gorenstein rings and the Cayley-Bacharach property. Generalizations to affine algebras and multigraded rings round this lecture off.

Lecture 5: Zero-Dimensional Ideals

Based on the family of multiplication matrices, the structure of zero-dimensional algebras is analyzed in a novel way. They are decomposed into joint generalized eigenspaces and this decomposition is used to solve zero-dimensional polynomial systems.

References

• [1] M. Kreuzer and L. Robbiano, Computational Commutative Algebra 1, Springer-Verlag, Berlin Heidelberg 2000.
• [2] M. Kreuzer and L. Robbiano, Computational Commutative Algebra 2, Springer-Verlag, Berlin Heidelberg 2005.
• [3] M. Kreuzer and L. Robbiano, Computational Linear and Commutative Algebra, Springer International, Cham 2016.

Domains: Computational Commutative Algebra

Sessions: 

• Lorenzo Robbiano: 2 sessions * 120 minutes
• Martin Kreuzer: 3 sessions * 120 minutes

Exercise Sessions for Course 1:

• Exercise Session 1: Selected tutorials from [1], Ch. 1 and 2, e.g., classification of term orderings, implementation and optimization of Buchberger’s Algorithm.
• Exercise Session 2: Selected tutorials from [1], Ch. 3, e.g., computing syzygy modules, implicitization, toric ideals, and graph colouring.
• Exercise Session 3: Selected tutorials from [2], Ch. 5, e.g., mathematical chess puzzles, computing dimension and multiplicities, and affine Hilbert functions.
• Exercise Session 4: Selected tutorials from [2], Ch. 6, e.g., counting magic squares, Ehrhart functions, generic initial ideals, and Hilbert functions of lex-segment ideals
• Exercise Session 5: Selected topics from [3], e.g., commendable families of endomorphisms, kernels and big kernels of ideals, computing joint generalized eigenspaces, and solving polynomial systems using the eigenvector method

Domains: Computational Commutative Algebra

Sessions:

• Nguyen Khanh Linh Tran: 5 sessions * 75 minutes

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