## Separating Invariants

##### Abstract

Roughly speaking, a separating algebra is a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this thesis, we introduce the notion of a geometric separating algebra, a more geometric notion of a separating algebra. We find two geometric formulations for the notion of separating algebra which allow us to prove, for geometric separating algebras, the results found in the literature for separating algebras, generally removing the hypothesis that the base field be algebraically closed. Using results from algebraic geometry allows us to prove that, for finite groups, when a polynomial separating algebra exists, the group is generated by reflections, and when a complete intersection separating algebra exists, the group is generated by bireflections. We also consider geometric separating algebras having a small number of generators, giving an upper bound on the number of generators required for a geometric separating algebra. We end with a discussion of two methods for obtaining new separating sets from old. Interesting, and relevant examples are presented throughout the text. Some of these examples provide answers to questions which previously appeared in print.