Dear Students,

I am planning to give 6 blocked lectures for the Elimination Theory
course within June 15-30, 1999.

Please let me know as early as possible if you are going to take the
course and what time blocks are not convenient for you. Please also
let me know if you prefer to have the lectures in Hagenberg or on the
campus. I will try to work out an optimized time schedule.

With best regards,
Dongming

## Elimination Theory

#### Dongming Wang

315.452 - Summer 1997
Block course to be held in June 1997
Time and location to be announced

Elimination Theory - the theory of eliminating unknowns from systems of multivariate polynomials - is a classical subject of mathematics that has been studied and extended in the field of symbolic computation and that has numerous applications (e.g., solving algebraic equations, geometric theorem proving, computational algebraic geometry and CAGD). Examples of well-established elimination theory are the theory of resultants, of characteristic sets and of Gröbner bases. This course will provide an introduction to the above-mentioned theory and methods as well as several new developments on the subject with an aim at studying the zero structure of polynomial systems. The contents of the 6 blocked lectures for the course are as follows:
• Polynomial Arithmetic and Zeros
GCD, Pseudo-Division, PRS, Resultants, Subresultants, Field Extension, Factorization, Zeros, Ideals, Nullstellensatz

• Zero Decomposition of Polynomial Systems
Triangular Systems, Characteristic-Set-Based Algorithm, Seidenberg's Algorithm Refined, Subresultant-Based Algorithm

• Projection and Simple Systems
Projection, Zero Decomposition with Projection, Decomposition into Simple Systems, Properties of Simple Systems

• Irreducible Zero Decomposition
Irreducibility of Triangular Sets, Decomposition into Irreducible Triangular Systems, Properties of Irreducible Triangular Systems, Irreducible Simple Systems

• Various Elimination Algorithms
Regular Systems, Canonical Triangular Sets, Gröbner Bases, Resultant Elimination

• Computational Algebraic Geometry and Ideal Theory
Dimension, Decomposition of Algebraic Varieties, Ideal and Radical Ideal Membership, Primary Decomposition of Ideals
Material for the course will be taken from the preliminary version of a book on Elimination Methods by the lecturer. Anyone planning to attend the course is asked to send a note to Dongming.Wang@risc.uni-linz.ac.at as early as possible.

Thu Mar 13 16:57:07 MET 1997