## Prof. Allen Knutson's Math 4370, Spring 2013

Tuesday/Thursday 2:55-4:10 in Malott 224

My office hour: Monday after class, in Malott 515

Anna's office hours: Monday 2-3 (before class), Tuesday TBA

Midterm and final TBA

### class notes, and homework, on the class blog

Catalogue description:
Introduction to Grobner bases theory, which is the foundation of many
algorithms in computational algebra. In this course, students learn
how to compute a Grobner basis for polynomials in many
variables. Covers the following applications: solving systems of
polynomial equations in many variables, solving diophantine equations
in many variables, 3-colorable graphs, and integer programming. Such
applications arise, for example, in computer science, engineering,
economics, and physics.

Please ** print and fill out the class survey, **
and bring to class the first week.

Syllabus:

### Background material (to be covered)

1) Polynomials in one variable over C. The Euclidean Algorithm. Solving a
system of polynomial equations in one variable. Principal ideals.
Background on ideals in commutative rings, e.g. def of prime ideals.
Monomial ideals in polynomial rings.
Solving a system of monomial equations and primary decomposition of
squarefree monomial ideals.
Combinatorial proof that any monomial ideal is finitely generated.
Combinatorial proof of the Ascending Chain Condition for monomial ideals.
Filtrations on vector spaces, and associated gradeds.
Same for rings, and ideals.
### Grobner basics

Definition of Grobner basis of an ideal. Proof of existence.
S-polynomials. Necessity of the Buchberger criterion for Grobner bases.
Big theorem: sufficiency.
The Buchberger algorithm for constructing a Grobner basis.
Big theorem: it terminates.
###
Applications of Grobner bases in algebra

Ideal membership.
Colon ideals.
Hilbert series.
The radical of an ideal.
Decomposing a radical ideal into primes.
###
Applications of Grobner bases outside algebra

3-colorable graphs (geographic maps).
Toric Ideals. Integer programming via Grobner basis.

###
My initial guess at the schedule (by week):

Overview: reduction of studying ideals to studying monomial ideals.
Polynomials in one variable. The Euclidean algorithm. Ideals and
principal ideals. Radical ideals.

Monomial ideals in several variables. Squarefree monomial ideals.
Theorem: monomial ideals are finitely generated.
Ideal membership in monomial ideals.

Theorem: Ascending Chain Condition for monomial ideals.
Hilbert series. Hilbert series of a monomial ideal.

Term orders. Definition of Grobner basis of an ideal.
S-polynomials. Necessity of the Buchberger criterion for Grobner bases.
The Buchberger algorithm for constructing a Grobner basis.

Filtrations on vector spaces, and associated gradeds.
Same for rings, and ideals.
Relation to Grobner bases. Proof of existence of Grobner bases.

Corollary: polynomial ideals are finitely generated and satisfy ACC.
Ideal membership in polynomial ideals.
Theorem: the Buchberger algorithm works.

Applications of Grobner bases:
3-colorable graphs (geographic maps).
Toric Ideals. Integer programming via Grobner basis.

Prime decomposition of squarefree monomial ideals.

** **

Colon ideals.
Introduction to algebraic geometry.

Prime ideals. Prime decomposition of squarefree monomial ideals.
The radical of an ideal.

Decomposing a radical ideal into primes.