A Valuation Theoretic Approach to Essential Dimension

TL;DR

This paper investigates the essential dimension of algebraic structures using valuation theory, establishing bounds and formulas for specific cases like torus actions, thus advancing understanding in algebraic geometry and field theory.

Contribution

It introduces a valuation-theoretic method to bound and compute the essential dimension, including a new formula for torus orbit functors.

Findings

Valuation rank cannot exceed the transcendence degree.

Established lower bounds for essential dimension in certain cases.

Derived a formula for the essential dimension of torus orbit functors.

Abstract

In this essay we explore the notion of essential dimension using the theory of valuations of fields. Given a field extension K/k and a valuation on K that is trivial on k, we prove that the rank of the valuation cannot exceed the transcendence degree trdeg K. We use this inequality to prove lower bounds on the essential dimension in some interesting situations. We study orbits of a torus action and find a formula for the essential dimension of the functor of these orbits.