Elimination Theory
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