An Upper Bound on the Number of Classes of Perfect Unary Forms in Totally Real Number Fields

TL;DR

This paper establishes an upper bound on the number of classes of perfect unary forms in totally real number fields, relating it to field invariants like discriminant and regulator, and extends results for unit reducible fields.

Contribution

It provides a new upper bound on the classes of perfect unary forms in totally real fields, incorporating field invariants and extending previous methods.

Findings

Bound depends on discriminant and regulator of the field.

For unit reducible fields, the bound simplifies to depend only on discriminant and degree.

The method adapts van Woerden's approach to this specific problem.

Abstract

Let $K$ be a totally real number field of degree $n$ over $\mathbb{Q}$, with discriminant and regulator $\Delta_K, R_K$ respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary forms, up to equivalence and scaling, can be bounded above by $O( \Delta_K \exp(2n \log(n)+f(n,R_K)))$, where $f(n,R_K)$ is a finite value, satisfying $f(n,R_K)=\frac{\sqrt{n-1}}{2}R_K^{\frac{1}{n-1}}+\frac{4}{n-1}\log(\sqrt{|\Delta_K|})^2$ if $n \leq 11$. Moreover, if $K$ is a unit reducible field, the number of classes of perfect unary forms is bound above by $O( \Delta_K \exp(2n \log(n)))$.