Finite polynomial maps and G-variant map germs.
The first half of this thesis is devoted to the study of finite polynomial maps
en --4 en and the use of Grobner bases to determine if a given map is finite. We
begin by examining those maps which have quasihomogeneous components, and
give a simple condition for such maps to be finite. This condition is extended
to those maps which are quasihomogeneous as above, but with extra lower order
terms. Next, we give a general criterion for testing the finiteness of a given
polynomial map and an implementation in the Maple computer algebra system.
Our next step is to generalize our results to regular maps between affine varieties.
Again, a finiteness criterion is given, plus its implementation in Maple. Lastly
in this half, we consider the trace bilinear form associated with a finite map and
show how it may be used to find real roots of a polynomial system.
The second half of the thesis is concerned with the study of G-variant map
germs, which commute with the action of a finite group G on the source and
target spaces. We give a relation between the G-variant degree associated with
a map germ, bilinear forms on the local algebra and preimages of zero under a
perturbation of the original map. We look at both the complex and real affine
space situation. We then give the equivalent results when we do not have a
'good' deformation of the map, when we have two groups acting and when we
use modular representations. Next, we give an invariant of G-variant maps which
is stronger than G-degree, based upon a lattice of vector subspaces. Finally, we
examine the structure of the class of G-variant maps and consider criteria for maps
to have 'good' deformations and to be finite. We then give ways of determining
generators for the class of maps by generalizing theorems of Noether and Molien.