Quartic rings associated to binary quartic forms

TL;DR

This paper establishes a bijection between binary quartic forms and certain quartic rings with monogenic cubic resolvent rings, providing a new parametrization and geometric interpretation of these algebraic structures.

Contribution

It introduces a novel bijection linking binary quartic forms to quartic rings with monogenic cubic resolvents, expanding the understanding of their structure and parametrization.

Findings

Provides a new parametrization of quartic rings with monogenic cubic resolvents.

Relates algebraic structures to geometric interpretations.

Connects binary quartic forms with Bhargava's cubic resolvent rings.

Abstract

We give a bijection between binary quartic forms and quartic rings with a monogenic cubic resolvent ring, relating the rings associated to binary quartic forms with Bhargava's cubic resolvent rings. This gives a parametrization of quartic rings with monogenic cubic resolvents. We also give a geometric interpretation of this parametrization.