# Monomial and Binomial ideals

# Monomial and Binomial ideals

## Time and location

## Invitation

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.

## Literature

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),
- J. Herzog and T. Hibi "Monomial Ideals" (Springer, GTM 260)