Zinovy Reichstein

On the essential dimension of a finite group

Let f(x) = Σaixi be a monic polynomial of degree n whosecoefficients are algebraically independent variables over a base field k of characteristic 0. We say that a polynomial g(x) isgenerating (for the symmetric group) if it can be obtained from f(x) by a nondegenerate Tschirnhaus transformation. We show that the minimal number dk(n) of algebraically independent coefficients of such a polynomial is at least [n/2]. This generalizes a classical theorem of Felix Klein on quintic polynomials and is related to an algebraic form of Hilbert‘s 13th problem.Our approach to this question (and generalizations) is basedon the idea of the ’essential dimension‘ of a finite group G:the smallest possible dimension of an algebraic G-variety over k to which one can ‘compress’ a faithful linear representation of G. We show that dk(n) is just the essential dimension of the symmetricgroup Sn. We give results on the essential dimension ofother groups. In the last section we relate the notion of essential dimension to versal polynomials and discuss their relationship to the generic polynomials of Kuyk, Saltman and DeMeyer.

On the notion of essential dimension for algebraic groups

We introduce and study the notion of essential dimension for linear algebraic groups defined over an algebraically closed fields of characteristic zero. The essential dimension is a numerical invariant of the group; it is often equal to the minimal number of independent parameters required to describe all algebraic objects of a certain type. For example, if our groupG isSn, these objects are field extensions; ifG=On, they are quadratic forms; ifG=PGLn, they are division algebras (all of degreen); ifG=G2, they are octonion algebras; ifG=F4, they are exceptional Jordan algebras. We develop a general theory, then compute or estimate the essential dimension for a number of specific groups, including all of the above-mentioned examples. In the last section we give an exposition of results, communicated to us by J.-P. Serre, relating essential dimension to Galois cohomology.

Essential Dimensions of Algebraic Groups and a Resolution Theorem for G-Varieties

Let G be an algebraic group and let X be a generically free G-variety. We show that X can be trans- formed, by a sequence of blowups with smooth G-equivariant centers, into a G-variety Xwith the following property: the stabilizer of every point of Xis isomorphic to a semidirect product U⋊ A of a unipotent group U and a diagonalizable group A. As an application of this result, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus trans- formation.

Essential dimension of algebraic tori

Abstract The essential dimension is a numerical invariant of an algebraic group G which may be thought of as a measure of complexity of G-torsors over fields. A recent theorem of N. Karpenko and A. Merkurjev gives a simple formula for the essential dimension of a finite p-group. We obtain similar formulas for the essential p-dimension of a broad class of groups, which includes all algebraic tori.

Fields of Definition for Division Algebras

Let

Versality of algebraic group actions and rational points on twisted varieties

A

Stability and equivariant maps

be a finite-dimensional division algebra containing a base field

Equivariant resolution of points of indeterminacy

k

A lower bound on the essential dimension of a connected linear group

in its center

On Tschirnhaus Transformations

F