Why should one study monomial or binomial ideals? Here are two teasers for topics that will be covered in this lecture.

Gröbner bases Everybody knows how to solve linear equations. An insight due to Buchberger and Hironake is much less known: morally the same techniques (triangulating systems, eliminating variables, etc.) also apply to polynomial equations. Gröbner bases (also known as standard bases) are the main technical tool here. A key insight is that polynomial equations can be deformed to monomial equations while preserving enough of their structure to understand certain simplified aspects of the set of solutions to the original equations.

Algebraic combinatorics Monomial and binomial ideals appear naturally in the study of combinatorics. For instance let K be a d-dimensional simplicial complex on n vertices. A natural question is: How many faces can K have in each dimension? The most basic restriction on the face numbers is Euler's formula e-k+f = 2 for three dimensional polyhedra. Today a lot more is known about face numbers of polytopes, spheres, and simplicial complexes. For instance, there is a precise characterization of the set of face-number vectors (so called f-vectors) of all simplicial complexes (the Schützenberger-Kruskal-Katona theorem). There are also precise upper and lower bounds for the individual entries of f-vectors of boundaries of polytopes. What these theorems have in common is that they are intimately connected to algebraic invariants of monomial ideals, and their Hilbert functions. This will be one of the highlights of the first half of the course.

To get inspired, peek into those great books:

• E. Miller and B. Sturmfels "Combinatorial commutative algebra" (Springer, GTM 227)
• Richard P. Stanley "Combinatorics and commutative algebra" (Birkhäuser),
• W. Bruns and J. Herzog "Cohen-Macaulay rings" (Cambridge University Press),
• Herzog and Hibi "Monomial Ideals" (Springer, GTM 260)

Course homepage: http://www.thomas-kahle.de/mabi.html

-- ThomasKahle - 10 Apr 2013

#### Research Unit M9

Department of Mathematics
Boltzmannstraße 3
85748 Garching b. München
Germany
phone:+49 89 289-16858
fax:+49 089 289-16859
sekretariat-m9ma.tum.de

#### Professors

Prof. Dr. Peter Gritzmann
Applied Geometry and Discrete Mathematics

Prof. Dr. Andreas S. Schulz
Mathematics of Operations Research
(affiliated member of M9)

Prof. Dr. Stefan Weltge
Discrete Mathematics

#### News

Jan 25th, 2019
Case Studies 2019: Preliminary Meeting on Wed, Feb 6th, at 16:00 in room MI 03.06.011.