A note on invertible quadratic 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 quadratic if the degrees of its polynomials are not greater than two. In the present paper an exhaustive description of invertible quadratic transformations of the real plane is given. Their application to the perfect cuboid problem is discussed.

On quartic forms associated with cubic transformations of the real plane

“…The main goal of the present paper is to introduce some tensorial invariants associated with F and study their behavior under non-tensorial transformations given by the formula (1.3). These invariants are analogous to those considered in [14] in the case of quadratic transformations of the real plane R 2 .…”

“…The formulas (4.7), (4.8), (4.9) resemble the formula (3.3) in Definition 3.1. For this reason the analogs of the forms ω [1], ω [2], ω [3], ω [4], ω [5], ω [6] in [14] were called pseudotensors. However, this is not correct with respect to changes of coordinates of the form (2.10) and (2.11).…”

“…Theorem 4.3 is the most important result of the present paper in view of its prospective application to classification of potentially invertible cubic transformations of the real plane R 2 . Such a classification of potentially invertible quadratic transformations can be found in [14].…”

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

“…This classification can further be refined using other invariants and/or methods. The classification of potentially invertible quadratic transformations of R 2 can be found in [13].…”