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

A polynomial transformation of the real plane ${\mathbb{R}}^2$ 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 ${\mathbb{R}}^2$ is suggested. It is based on quartic forms associated with these transformations.