On quartic forms associated with cubic transformations of the real plane

Abstract

A polynomial transformation of the real plane R 2 is a mapping R 2 → 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.2000 Mathematics Subject Classification. 14E05, 15A69, 57S25. 1 We use upper and lower indices according to Einstein's tensorial notation (see [12])

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

“…The property of the associated quartic form ω [1] being indefinite is invariant under passing to an equivalent cubic transformation. This fact follows from Theorems 4.1 and 4.2 in [12].…”

“…The determinants (2.3) are related to six quartic forms ω [1], ω [2], ω [3], ω [4], ω [5], ω [6] introduced in [12]. They are given by the following formulas:…”

“…Theorems 2.1 and 2.2 are proved by means of direct calculations. Theorem 2.1 was formulated in [12]. The formulas (2.11) were also written in [12], though they were not formulated as a theorem.…”

“…Theorem 2.1 was formulated in [12]. The formulas (2.11) were also written in [12], though they were not formulated as a theorem.…”

“…However, in the present paper we study these transformations by themselves. Cubic polynomial transformations of the form (1.1) are associated with some definite quartic forms (see [12]). They are produced using components of the tensor F in (1.1).…”