Arithmetic matrices for number fields II: Parametrization of rings by binary forms

TL;DR

This paper extends the parametrization of rings by binary forms to degrees less than seven, generalizing Levi's and Bhargava's results, and explores the symbolic matrix calculations involved.

Contribution

It introduces a generalized framework for using binary forms to parametrize rings of degree n less than seven, expanding prior specific cases.

Findings

Binary forms of degree n<7 parameterize rings of rank n

The parametrization involves symbolic matrix calculations

Generalization of Levi's and Bhargava's results

Abstract

We show that binary forms of degree n less than seven parameterize rings, generalizing a result of Levi on binary cubic forms parameterizing cubic rings, which can be related to results of Bhargava. The question of whether or not a binary form of degree n parameterizes a ring of rank $n$ depends on a symbolic calculation with matrices.