On centres and direct sum decompositions of higher degree forms

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 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.

Keywords:

• Higher degree form
• direct sum decomposition
• centre

2010 Mathematics Subject Classifications:

• 15A69
• 11E76
• 14J70

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China [Grant Nos. 11911530172, 11971181 and 11971449].