Abstract of the lecture course

Computing in Multigraded Structures

by Martin Kreuzer (Dortmund)

The purpose of this course is to present the basic theorems and algorithms which allow us to compute in a multigraded environment. Given a field K and a polynomial ring P = K[x_{1}, …, x_{n}], we want to be able to compute in finitely generated modules over P. We try to equip P with a grading which makes the ideals and modules under consideration multihomogeneous. Then we intend to use that additional structure in order to improve the usual algorithms of Computational Commutative Algebra and to compute as many numerical invariants as possible.

The course assumes some knowledge of basic Gröbner basis theory as presented in [1]. With respect to the conventions and notation we follow this book. The material we present is based on parts of the upcoming second volume [2]. The relevant sections will be handed to the students during the course.

Lecture 1: Gradings. If we want to equip the polynomial ring P with gradings which are suitable for actual computer calculations, we need a number of good properties: the graded version of Nakayama’s lemma, finite dimensional homogeneous components, a well-ordering of the set of degrees, and the existence of a degree-compatible term ordering. In this lecture we discuss the relations between those wishes and how we can satisfy them. Source: [2], Section 4.1

Lecture 2: Multihomogenization. In the multigraded world, homogenization is just as easy as in the standard graded case. We present the definition, the basic rules, and the algorithms for multihomogenizing polynomials and ideals. More subtle are the relations between Gröbner bases and homogenization as well as dehomogenization. We shall see that for some statements to hold it is necessary for P to be graded by a single row. Source: [2], Section 4.3

Lecture 3: Multihomogeneous Gröbner Bases. Given a graded submodule of a finitely generated multigraded free P -module, it is advantageous to use the structure given by the grading while computing a Gröbner basis. We present the multihomogeneous Buchberger algorithm and several variants. In particular, we shall se how one can compute a minimal system of generators of a graded module. Source: [2], Sections 4.5 and 4.6

Lecture 4: Hilbert Functions and Hilbert Series. One of the most important applications of the existence of a grading such that the ideals and modules under consideration are homogeneous is the fact that the dimensions of their homogeneous components provide a host of numerical invariants. Using Hilbert functions and Hilbert series, we can get a hold on these invariants. We shall show that the Hilbert series is a rational power series and that we can compute it effectively. Applications include Hilbert polynomials, dimension and multiplicity of affine algebras and finitely generated modules over them. Source: [2], Sections 5.1 and 5.2

Lecture 5: Multigraded Hilbert Functions. When we generalize Hilbert functions and Hilbert series to the multigraded situation, we encounter additional problems. For instance, it is not clear how to find a ring which contains all Hilbert series of finitely generated multigraded modules over P. In this lecture, we study how one can overcome those obstacles and develop the basic properties of multigraded Hilbert functions and Hilbert series. As usual, we insist on being able to compute them, too. Source: [2], Sections 5.8.A and 5.8.B

Lecture 6: Hilbert Bases and Applications. Multigraded Hilbert functions allow us to count intricate sets of objects. For practical applications, it is necessary to be able to control their behaviour under a change of grading, e.g. under refinements. In order to do this, we shall need the concept of Hilbert bases and an algorithm for computing them. Then we can apply to theory to count where no man has counted before... Source: [2], Section 5.8.C and 5.8.D

References

1. M. Kreuzer and L. Robbiano, Computational Commutative Algebra 1, Springer, Heidelberg 2000

2. M. Kreuzer and L. Robbiano, Computational Commutative Algebra 2, in preparation