Classification of linear actions of algebraic groups on spaces of homogeneous forms

A study was conducted to demonstrate classification of linear actions of algebraic groups on spaces of homogeneous forms. The study investigated the orbits of an algebraic group G acting on the space of rational forms in many variables by linear changes of coordinates. The main result of the paper consisted in finding the field of differential invariants of such an action and obtaining an effective criterion for distinguishing between the orbits of forms with nonzero Hessian. The study presented a method for classifying the G-orbits of regular rational forms of any degree. The obtained classification depended neither on the degree n, on the number p of variables, nor on the group G.