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