On centres and direct sum decompositions of higher degree forms

Higher degree forms are homogeneous polynomials of d>2, or equivalently symmetric d-linear spaces. This paper is mainly concerned about the algebraic structure centres higher with applications specifically to direct sum decompositions, namely expressing as sums in disjoint sets variables. We show that centre algebra almost every form ground field, consequently all absolutely indecomposable. If a decomposable, then we provide simple criteria and algorithms for decompositions by its algebra. It shown decomposition problem can be boiled down some standard tasks linear algebra, particular computations eigenvalues eigenvectors. also apply results algebras complete answer classical whether reconstructed from Jacobian ideal.