On Centers and Direct Sum Decompositions of Higher Degree Forms

TL;DR

This paper investigates the algebraic structure of centers of higher degree forms, revealing that most are indecomposable and providing algebraic criteria and algorithms for their decomposition into sums of forms in disjoint variables.

Contribution

It establishes that the center algebra of most higher degree forms is trivial, introduces criteria and algorithms for decomposition, and connects the problem to linear algebra tasks like eigenvalue computation.

Findings

Almost all higher degree forms have trivial center algebra.

Decomposition algorithms reduce to eigenvalue and eigenvector computations.

The structure of center algebras helps determine reconstructibility from Jacobian ideals.

Abstract

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.