A rough classification of potentially invertible cubic transformations of the real plane

TL;DR

This paper proposes a preliminary classification scheme for cubic polynomial transformations of the real plane, utilizing associated quartic forms to understand their structure and potential invertibility.

Contribution

It introduces a novel classification approach for cubic transformations of the plane based on quartic forms, providing a foundation for further analysis.

Findings

Developed a classification scheme for cubic transformations

Linked transformations to associated quartic forms

Laid groundwork for analyzing invertibility of cubic maps

Abstract

A polynomial transformation of the real plane $\Bbb R^2$ is a mapping $\Bbb R^2\to\Bbb R^2$ given by two polynomials of two variables. Such a transformation is called cubic if the degrees of its polynomials are not greater than three. In the present paper a rough classification scheme for cubic transformations of $\Bbb R^2$ is suggested. It is based on quartic forms associated with these transformations.