Multiplicative $f$-ic forms on algebraic varieties arising from Thaine's generalized Jacobi sums

We study generalized Jacobi sums, cyclotomic numbers, and $d$-compositions in Thaine’s framework, and prove new multiplicative identities extending Davenport and Hasse’s lifting theorem from the classical prime-power setting to products of prime powers. As applications, we construct multiplicative forms of degree $f\ge 2$, i.e. $f$-ic forms, on complete intersections of $f$-ics. This places Pfister’s theory of multiplicative quadratic forms over fields within the broader setting of multiplicative $f$-ic forms on affine algebraic varieties, where new phenomena arise. Moreover, a dense open subset $W\subset V$ carries the structure of an algebraic torus, and the multiplicative form is compatible with the induced group law on $W$.