  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.
