  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.
