On quartic forms associated with cubic transformations of the real plane

TL;DR

This paper explores the connection between cubic polynomial transformations of the real plane and associated quartic forms, defining and analyzing these forms to understand their properties.

Contribution

It introduces and studies the specific quartic forms associated with cubic transformations of the real plane, a novel approach in the field.

Findings

Defined binary and quaternary quartic forms linked to cubic transformations

Analyzed properties of these quartic forms

Established foundational relationships between transformations and forms

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. It turns out that cubic transformations are associated with some binary and quaternary quartic forms. In the present paper these forms are defined and studied.