On norming systems of linear equations

TL;DR

This paper introduces the concept of norming systems of linear equations, establishing conditions for when such systems induce norms on function spaces, with applications to Gowers uniformity norms and classifications of low-rank systems.

Contribution

It systematically studies norming linear systems, providing necessary and sufficient conditions, an isomorphism theorem, and classifying all rank at most two norming systems.

Findings

Necessary and sufficient conditions for a system to be norming.

An isomorphism theorem for the functional $t_L(\

Proof that all norming systems are variable-transitive.

Abstract

A system of linear equations $L$ is said to be norming if a natural functional $t_L(\cdot)$ giving a weighted count for the set of solutions to the system can be used to define a norm on the space of real-valued functions on $\mathbb{F}_q^n$ for every $n>0$. For example, Gowers uniformity norms arise in this way. In this paper, we initiate the systematic study of norming linear systems by proving a range of necessary and sufficient conditions for a system to be norming. Some highlights include an isomorphism theorem for the functional $t_L(\cdot)$, a proof that any norming system must be variable-transitive and the classification of all norming systems of rank at most two.