The space of invariant bilinear forms of the polarization algebra of a polynomial endomorphism: An approach to the problem of Albert

Abstract

In this article we explore an approach to the problem of Albert described by U. Umirbaev. We characterize the space of invariant bilinear forms of the polarization algebra of a polynomial endomorphism F:Kn→Kn, where K is a field with characteristic different from two. We obtain some conjectures expressed in the language of polynomial endomorphisms, which are equivalent to the existence of invariant bilinear forms in a finite-dimensional commutative algebra. We give a characterization of the space of invariant bilinear forms in terms of differential forms in the ring K[x1,…,xn]. We also introduce a new kind of algebra, we call them totally symmetric algebras, and we establish the relationship between these algebras and the existence of invariant bilinear forms in any commutative algebra.