On Invariant Theory

Here we develop a technique of computing the invariants of n−ary forms and systems of forms using the discriminants of corresponding multilinear forms buit of their partial derivatives, which should be cosidered as generalizations of classical discriminants and resultants for binary forms. 0 Introduction Let f(y0, y1, ..., yn) = ∑ α1≤...≤αd cα1...αdyα1 ...yαd be an n + 1−ary form of degree d, which is a polynomial of y of homogeneous degree d. Definition 0.1 The value of coefficients (cα1...αd) is called discriminantal if for this value the system of equations ∂f ∂yα = 0, α = 0, ..., n (0.1) has a solution in Pn, i.e. a nonzero one. If the set of all discriminantal values of cα1...αd is an algebraic manifold of codimension 1 in the space of coefficients, then it is called the discriminant of f , denoted by D(f). In particular case when f is a d−linear form, its coefficients may be viewed as elements of a d−dimensional matrix (ai1...id). In this paper we show how the invariants of n−ary forms can be produced from the discriminants of multilinear forms (determinants of multidimensional matricies), which should be considered as the generalization of the operation of taking classical hessians and resultants. The algorithm of computation of discriminants of multilinear forms is considered in paper [1].