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 centres 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 centre 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 centre 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 centre algebras to provide a complete answer to the classical problem of whether a higher degree form can be reconstructed from its Jacobian ideal.

高次形式是次次的齐次多项式$d>\mathrm{2个},$或等效对称d- 线性空间。本文主要关注高次形式中心的代数结构，特别是直接和分解的应用，即将高次形式表示为不相交变量集中的形式之和。我们证明了几乎所有形式的中心代数都是基域，因此几乎所有更高次的形式都是绝对不可分解的。如果更高阶形式是可分解的，那么我们提供简单的标准和算法，用于通过其中心代数进行直和分解。结果表明，直和分解问题可以归结为线性代数的一些标准任务，特别是特征值和特征向量的计算。