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Report on a ``Special Year on Gröbner Bases''

at RISC-Linz, Hagenberg near Linz, Austria, March 1997 -- February 1998
Bruno Buchberger and Franz Winkler
RISC-Linz, Johannes Kepler Universität, A-4040 Linz, Austria
The institute RISC-Linz of the Johannes Kepler Universität in Linz, Austria, organized a Special Year on Gröbner Bases, from March 1997 until February 1998. All events took place in Hagenberg, the home of RISC-Linz. The goals were The culminating events in this special year were an intensive course for graduate students in January and the conference ``33 Years of Gröbner Bases'' in February.

Intensive Course on Gröbner Bases, January 12-30, 1998:

The Intensive Course on Gröbner Bases, organized by Prof. Bruno Buchberger and Prof. Franz Winkler, took place from January 12 through 30. The course was primarily designed as a comprehensive introduction for ph.d. students and young postdocs. The 28 participants came from Austria, Belarus, Czechia, Germany, Hungary, Iran, Italy, Japan, Moldavia, Norway, Romania, Russia, Spain, Sweden, Ukraine, and USA.

List of lecturers and subjects:

B. Buchberger (Linz): Introduction to the theory and construction of Gröbner bases, elementary applications such as ideal membership, computation in residue class rings, solution of algebraic equations, linear syzygies, Hilbert functions, non-linear syzygies, and inverse mappings.

F. Winkler (Linz): Introduction to general rewriting, reductions relations, term rewriting, geometric applications of Gröbner bases, such as implicitization and parametrization of algebraic varieties, computation of dimension of a variety.

F. Schwarz (Bonn): Janet bases as an algebraic tool for investigating partial differential equations, applications of Janet bases for vector fields of Lie groups, for 2nd order ordinary differential equations.

L. Robbiano (Genova): Introduction to the theory of Hilbert functions and the computation of Hilbert functions, how to compute Gröbner bases with the aid of Hilbert functions, using Hilbert functions for computing free resolutions.

W.W. Adams (College Park, MD): Gröbner bases over rings, primality testing using Gröbner bases, primary decomposition of ideals by Gröbner bases.

Conference ``33 Years of Gröbner Bases'', February 2-4, 1998:

The conference ``33 Years of Gröbner Bases'' was held from February 2 through 4. Prof. Bruno Buchberger and Prof. Franz Winkler (both RISC-Linz) were the chairmen of the conference. The 94 participants came from 14 countries.

The conference consisted of one day of tutorial lectures on various applications of the theory of Gröbner bases, and two days of presentations of original research. The proceedings are published as: B.Buchberger and F.Winkler (eds.), Gröbner Bases and Applications, London Math. Soc. LNS 251, Cambridge Univ. Press, 1998.

Titles of the tutorial lectures and research talks in alphabetic order of the authors follow.

Tutorial lectures:

B. Buchberger (Linz): Introduction to Gröbner Bases.

F. Chyzak (Rocquencourt): Gröbner Bases, Symbolic Summation and Symbolic Integration.

W. Decker and T. de Jong (Saarbrücken): Gröbner Bases and Invariant Theory.

G.-M. Greuel and G. Pfister (Kaiserslautern): Gröbner Bases and Algebraic Geometry.

S. Hosten and R. Thomas (Fairfax, VA and College Station, TX): Gröbner Bases and Integer Programming.

H.M. Möller (Dortmund): Gröbner Bases and Numerical Analysis.

L. Robbiano (Genova): Gröbner Bases and Statistics.

D. Wang (Grenoble): Gröbner Bases Applied to Geometric Theorem Proving and Discovering.

Original research talks:

B. Amrhein and O. Gloor (Bern and Tübingen): The Fractal Walk.

M.A. Borges and M. Borges (Santiago de Cuba): Gröbner Bases Property on Elimination Ideal in the Noncommutative Case.

M.-J. Gonzalez-Lopez and L. Gonzalez-Vega (Santander): Newton Identities in the Multivariate Case: Pham Systems.

M. Insa and F. Pauer (Innsbruck): Gröbner Bases in Rings of Differential Operators.

J.B. Little (Worchester, MA): Canonical Curves and the Petri Scheme.

H. Lombardi and H. Perdry (Besançon): The Buchberger Algorithm as a Tool for Ideal Theory of Polynomial Rings in Constructive Mathematics.

K. Madlener and B. Reinert (Kaiserslautern): Gröbner Bases in Non-Commutative Reduction Rings.

J.L. Miller (Bowling Green, KY): Effective Algorithms for Intrinsically Computing SAGBI-Gröbner Bases in a Polynomial Ring over a Field.

Müller-Quade, R. Steinwandt and T. Beth (Karlsruhe): An Application of Gröbner Bases to the Decomposition of Rational Mappings.

P. Nordbeck (Lund): On some Basic Applications of Gröbner Bases in Non-commutative Polynomial Rings.

L. Robbiano and M.P. Rogantin (Genova): Full Factorial Designs and Distracted Fractions.

T. Sauer (Erlangen): Polynomial Interpolation of Minimal Degree and Gröbner Bases.

J. Schicho (Linz): Inversion of Birational Maps with Gröbner Bases.

J. Snellman (Stockholm): Reverse Lexicographic Initial Ideals of Generic Ideals are Finitely Generated.

Q.-N. Tran (Linz): Parallel Computation and Gröbner Bases: An Application of Converting Bases with the Gröbner Walk.

The following invited talks were also given at the conference.

W.W. Adams (College Park, MD): Some Applications of Computational Algebra to Partial Differential Equations.

B. Buchberger (Linz): 33 Years of Gröbner Bases: The Early Years.

F. Schwarz (Bonn): Janet Bases for Symmetry Groups.

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