Forms of higher degree permitting composition

TL;DR

This paper investigates nondegenerate forms of degree d on nonassociative algebras that allow composition, focusing on cubic and quartic cases over various rings and curves, and constructs examples of degenerate forms with this property.

Contribution

It extends the classification of forms permitting composition to cubic and quartic cases over specific rings and curves, including degenerate forms.

Findings

Classified certain cubic and quartic forms over rings and curves.

Constructed examples of highly degenerate cubic forms permitting composition.

Extended Schafer's classification beyond characteristic 0 fields.

Abstract

Nondegenerate forms N of degree d on a unital nonassociative algebra A over a ring R which permit composition, i.e., satisfy N(1)=1 and N(xy)=N(x)N(y) for all x,y in A, are studied. These forms were first classified by Schafer over fields of characteristic 0 or >d. We investigate cubic and quartic nondegenerate forms which permit composition over certain rings and curves. Classes of highly degenerate cubic forms N over fields which permit composition are constructed.