The Product of linear forms over function fields

TL;DR

This paper investigates the maximum minima of products of linear forms over function fields, establishing a connection with algebraic number theory and orbit theory on homogeneous spaces.

Contribution

It provides a new calculation of the maximum minima for small dimensions and links algebraic forms to periodic orbits in a homogeneous space setting.

Findings

Maximum value of minima equals the algebraic number theory bound for small n

Forms correspond to periodic orbits under diagonal group actions

Reduction theory of diagonal group orbits is used in the proof

Abstract

The aim of this paper is to study the product of $n$ linear forms over function fields. We calculate the maximum value of the minima of the forms with determinant one when $n$ is small. The value is equal to the natural bound given by algebraic number theory. Our proof is based on a reduction theory of diagonal group orbits on homogeneous spaces. We also show that the forms defined algebraically correspond to periodic orbits with respect to the diagonal group actions.