## COCOA 2007 International School on Computer Algebra RISC Hagenberg-Linz, Austria 18-22 June, 2007

 Preliminary announcement First announcement Second announcement Updates Participants Course materials CoCoA Home Page RISC Summer 2007 Photos!

## COURSE MATERIALS

Classes in the school will begin promptly at 9:00 on Monday morning (18th June 2007).

The School will end with Friday afternoon's tutorials. Following the very successful format we have used for previous schools, there will be two intensive courses (about 6 hours of instruction in each) plus afternoon tutorials Monday-Friday.

 Course 1 Lorenzo Robbiano Approximate Methods in Commutative Algebra tutor: Martin Kreuzer Course 2 Aldo Conca Betti Numbers and Generic Initial Ideals tutor: Anna Bigatti
1819202122
9-10:15lecture 1lecture 1lecture 1lecture 1lecture 1
11-12:15lecture 2lecture 2lecture 2lecture 2lecture 2
14:30-15CoCoACoCoALib 14:45
ApCoCoA
15-16tutorial 1tutorial 2tutorial 1tutorial 2tutorial 1
16-17----16:20
Certificates
group photo
17-18tutorial 2tutorial 1tutorial 2tutorial 1tutorial 2
19dinner

## Approximate Methods in Commutative Algebra

A course (with tutorials) by Lorenzo Robbiano and Martin Kreuzer
1. Polynomial System Solving (Lex-GB method, Eigenvalue method)
2. The Buchberger-Möller Algorithm (vanishing ideals of points, BM algorithm)
3. Border Bases (theory, numerical stability, BB algorithm)
4. Approximate Algorithms (approx. BM algorithm, approx. BB algorithm)

#### References

1. M.Kreuzer and L. Robbiano, Computational Commutative Algebra 1, Springer, Heidelberg 2000 [especially Section 3.7]
2. M.Kreuzer and L. Robbiano, Computational Commutative Algebra 2, Springer, Heidelberg 2000 [especially Sections 6.3 and 6.4]
3. D. Cox, Solving equations via algebras, in: A. Dickenstein and I. Emiris (eds.), Solving Polynomial Equations, Springer, Berlin 2005
4. H. Stetter, Numerical Polynomial Algebra, SIAM, Philadelphia 2004
5. D. Heldt, M. Kreuzer, S. Pokutta and H. Poulisse, Approximate computation of zero-dimensional polynomial ideals, Preprint 2006, available at M. Kreuzer's web pages

## Betti Numbers and Generic Initial Ideals

A course by Aldo Conca, with tutorials by Anna Bigatti
1. Introduction to the basic invariants: Hilbert functions, Betti numbers, regularity. Initial ideals and deformations.
2. Monomial ideals, strongly stable, Borel fixed, lex-segments and their Betti numbers.
3. Generic initial ideals: existence and main properties.
4. Bounds on Betti numbers Macaulay Theorem, Bigatti-Hullett and Pardue Theorem. Rigidity: Herzog-Hibi-Aramova Theorem and extensions.

## Notes

### Course 1

1. Monday slides tutorial
2. Tuesday slides tutorial a tutorial b
3. Wednesday slides cocoa tutorial
4. Thursday slides tutorial
5. Friday slides tutorial

### Course 2

1. Monday slides tutorial
2. Tuesday slides tutorial
3. Wednesday slides tutorial
4. Thursday slides tutorial
5. Friday slides tutorial