APPROXIMATE GROBNER BASES A BACKWARDS APPROACH

The main result of this thesis is to give a method for approximating the Grobner basis of an approximate polynomial system. The Grobner basis of a polynomial system is arguably the most fundamental object of exact computation polynomial algebra, as it answers many of the important questions of commutative algebra, such as ideal membership and computation of the Hilbert polynomial. It is traditionally computed using variants of Buchberger’s algorithm. Here, we take a backwards approach, and show that a Grobner basis can be computed using the Hilbert polynomial and another important basis from the jet theory of partial differential equations: an involutive basis. This direction, motivated by approximate systems, will allow us to avoid the strict monomial orderings and ordered elimination (reduction) strategies, at the heart of Buchberger-type methods, which are usually numerically unstable. For the the computation of exact bases for an ideal near to the one from which we began, we make avid use of structured (numerical) linear algebra. Additionally, we introduce approximate leading terms and an approximate reduced row echelon form. Neither of these require Gaussian elimination, unlike the exact case.

Name thesis:degree_name

M Sc

Discipline thesis:degree_discipline

Applied Mathematics

Year dc:date.issued

2006

Author dc:creator

• Scott, Robin J.

Advisor dc:contributor.advisor

• Reid, Greg

• Approximate Coefficients
• Grobner Bases
• Hilbert Polynomial
• Partial Differential Equations
• Involutive Systems
• Nearby Systems
• Numerical Linear Algebra
• Polynomial Algebra
• Reduced Row Echelon Form
• Singular Value Decomposition
• Structured Matrices.

Handle dc:identifier.uri

https://hdl.handle.net/20.500.14721/19954

OAI identifier oai:identifier

oai:uwo.scholaris.ca:20.500.14721/19954