A note on invertible quadratic transformations of the real plane

Abstract

A polynomial transformation of the real plane ${\mathbb{R}}^2$ is a mapping ${\mathbb{R}}^2\to {\mathbb{R}}^2$ 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.

AmSTeX, 21 pages, amsppt style

References in corpus (19)

• Classical Electrodynamics and Theory of Relativity
• Course of linear algebra and multidimensional geometry
• A note on the third cuboid conjecture. Part I
• Perfect cuboids and multisymmetric polynomials
• On an ideal of multisymmetric polynomials associated with perfect cuboids
• On the equivalence of cuboid equations and their factor equations
• A biquadratic Diophantine equation associated with perfect cuboids
• A general rational solution of an equation associated with perfect cuboids
• Inverse problems associated with perfect cuboids
• On singularities of the inverse problems associated with perfect cuboids
• On two algebraic parametrizations for rational solutions of the cuboid equations
• A note on rational and elliptic curves associated with the cuboid factor equations
• On two elliptic curves associated with perfect cuboids
• A note on solutions of the cuboid factor equations
• Two and three descent for elliptic curves associated with perfect cuboids
• Parametric Solutions for a Nearly-Perfect Cuboid
• Asymptotic estimates for roots of the cuboid characteristic equation in the linear region
• Reverse asymptotic estimates for roots of the cuboid characteristic equation in the case of the second cuboid conjecture
• A note on pairs of metrics in a three-dimensional linear vector space