Centers of Multilinear Forms and Applications

TL;DR

This paper investigates the algebraic structure of centers of multilinear forms, revealing their finite dimensionality and applications to form decompositions, indecomposability, and polynomial equivalence.

Contribution

It introduces the concept of center algebras for multilinear forms, demonstrating their finite dimensionality and utility in analyzing form decompositions and polynomial equivalences.

Findings

Center of a nondegenerate multilinear form is finite dimensional and commutative.

Almost all multilinear forms are absolutely indecomposable.

The algebraic structure of centers aids in proving polynomial linear equivalence.

Abstract

In this paper we study the center algebras of multilinear forms. It is shown that the center of a nondegenerate multilinear form is a finite dimensional commutative algebra and can be effectively applied to its direct sum decompositions. As an application of the algebraic structure of centers, we also show that almost all multilinear forms are absolutely indecomposable. The theory of centers can be extended to multilinear maps and be applied to their symmetric equivalence. Moreover, with a help of the results of symmetric equivalence, we are able to provide a linear algebraic proof of a well known Torelli type result which says that two complex homogeneous polynomials with the same Jacobian ideal are linearly equivalent.