Abstract Geometry of Numbers: Linear Forms

TL;DR

This paper extends the geometry of numbers to normed domains of linear type, establishing analogues of Minkowski's theorem and applying the theory to quadratic forms, Nullstellensatz, and small multiple results.

Contribution

It introduces the concept of linear type normed domains and demonstrates that S-integer rings and coordinate rings of algebraic curves are of this type.

Findings

S-integer rings in number fields are of linear type.

Coordinate rings of affine algebraic curves are of linear type.

Applications include a Nullstellensatz and a Small Multiple Theorem for quadratic forms.

Abstract

This paper concerns the \textbf{abstract geometry of numbers}: namely the pursuit of certain aspects of geometry of numbers over a suitable class of normed domains. (The standard geometry of numbers is then viewed as geometry of numbers over Z endowed with its standard absolute value.) In this work we study normed domains of "linear type", in which an analogue of Minkowski's linear forms theorem holds. We show that S-integer rings in number fields and coordinate rings of (nice) affine algebraic curves over an arbitrary ground field are of linear type. The theory is applied to quadratic forms in two ways, yielding a Nullstellensatz and a Small Multiple Theorem.