论文标题
在中心和高度形式的直接总和分解
On Centers and Direct Sum Decompositions of Higher Degree Forms
论文作者
论文摘要
较高度的表格是$ d> 2的同质多项式，或同等对称的$ d $ - 线性空间。本文主要关注高度形式中心的代数结构，其应用专门用于直接总和分解，即在变量差异集中表达较高程度的形式为形式的总和。我们表明，几乎每种形式的中心代数都是地面，因此几乎所有较高程度的形式都是绝对不可分解的。如果更高的形式是可分解的，那么我们提供了简单的标准和算法，以通过其中心代数直接分解。结果表明，直接总和分解问题可以归结为线性代数的某些标准任务，特别是特征值和特征向量的计算。我们还应用了中心代数的结构结果，为是否可以从其雅各布理想中重建更高程度的形式的经典问题提供了完整的答案。
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.
