CMI Silver Jubilee Lecture
Peter Symonds, University of Manchester, UK
Group actions on polynomial rings
Friday, January 16, 2015
We consider a group acting on a polynomial ring over a finite field by linear substitutions and we want to understand the ring as a module for the group.
We consider some examples and some computer calculations that allow us to "spot the pattern" and prove a Structure Theorem that has various interesting consequences. For example only finitely many different isomorphism classes of indecomposable summands can occur.
In the end we also obtain an explicit bound on the degrees of the generators of the generators of the invariants, something that was not previously known over a finite field, even though a version for the real or complex numbers was proved by Noether.