Invariants of points system in two-dimensional bilinear-metric spaces over the field of rational numbers

The dissertation is devoted to the study of the equality problems of point systems in two-dimensional bilinear-metric spaces over the field of rational numbers, to the determination of complete invariants of parametric figures in two-dimensional Minkowski space, and to the establishment of complete systems of invariants of figures with respect to the motion groups in two- and three-dimensional Euclidean spaces.
The report will cover the following topics.
A comprehensive analysis of the invariance properties of an m-point system situated in a two-dimensional bilinear-metric space with respect to the groups of rotations and translations. In this section, complete systems of invariants of m points with respect to the fundamental groups SO(2,p,ℚ), O(2,p,ℚ), MSO(2,p,ℚ), and MO(2,p,ℚ) (i.e., with respect to rotations and translations) are presented.
In the second and third sections, the complete systems of invariants of T-objects (figures) in two-dimensional Minkowski space with respect to the groups O⁺(1,1), O(1,1), MO⁺(1,1), and MO(1,1) are obtained — that is, the necessary and sufficient conditions for two figures to be equivalent are derived.
Furthermore, in two- and three-dimensional Euclidean spaces, the complete Euclidean and Galilean invariant systems of parametric figures are determined, providing the necessary and sufficient conditions for the equivalence of two parametric figures with respect to the groups of rotations and translations.