=ADD=
=reftype=
14
=number=
04-03
=url=
ftp://ftp.risc.uni-linz.ac.at/pub/techreports/2004/04-03.ps.gz
=year=
2004
=month=
03
=author=
Kondratyev; Aleksey
=title=
Numerical computation of Gr{\" o}bner Bases
=abstract=
The method of Gr{\" o}bner Bases is a custom tool in modern computer algebra
to solve systems of polynomial
equations. The topic of this thesis is the theory and computation of Gr{\"
o}bner Bases in a numerical environment.
Particular emphasis is given to an application of Gr{\" o}bner Bases for
solving a zero-dimensional polynomial system by the
eigenvector method. Here, the multiplicative structure of the associated
quotient algebra of the ideal of the system is
exploited to translate the problem of finding the solutions into a matrix
eigenvector problem; such
problems are well-studied in numerical analysis and therefore attractive for
a numerical approach.

We present a completely developed
variation of the theory of Gr{\" o}bner Bases with a modified notion of a
term order and some basic translation laws
which establish a connection with the customary Gr{\" o}bner Bases. An
implementation of the respectively modified
Buchberger algorithm for constructing Gr{\" o}bner Bases adopted for a
numerical environment and
executed in floating-point arithmetic has also been developed.
Our variation of the theory permits the construction of an algorithm
which incorporates an analogue to complete pivoting (as in Gaussian
elimination of a linear system), which
is beyond the frame of classical Gr{\" o}bner Bases where the "direction" of
elimination is rigidly controlled by a chosen
term order. This more flexible pivoting mechanism stabilizes both the course
and the result of the
computation in the sense proposed by Hans Stetter. Our numerical
implementation also employs
a rearrangement of the usual Buchberger algorithm which is attractive from a
numerical point of view;
it has initially been introduced  by J.~C.~Faugere for the customary Gr{\"
o}bner Bases and exact arithmetic. A collection
of numerical examples performed with our implementation is provided.
=note=
PhD Thesis
=sponsor=
Grant P 13266-MAT and the  SFB grant F1304 of the Austrian "Fonds zur Foerderung der wissenschaftlichen Forschung" (FWF)
=keywords=
Groebner bases, polynomial systems, numerical computation