Essential Dimension

Abstract

Essential dimension is a notion that encapsulates the minimal number of parameters necessary to describe an object. For a general polynomial this constitutes the smallest number of algebraically independent coefficients that are needed to define it. The centuries old Tschirnhaus transformations can be used to help determine these values. We moreover employ tools from algebraic geometry. For a finite group G we take a look at the essential dimension of faithful G-varieties, wherefrom we move on to the essential dimension of finite groups. Finally, we make a connection back to general polynomials and prove some results.