Linearizing of n-ic forms and generalized Clifford algebras

Abstract

A homogeneous form fd(x1,…,xn) of degree d with coefficients in a field F has a finite linearization if for some m there are m×m matrices α1,…,αn with entries in F so that is the identity matrix. Finite linearizations of fd correspond to matrix representations of a generalized Clifford algebra C(n,fd) denned by N. Roby; when d = 2 C(n,f2) is the usual Clifford algebra of the quadratic form f2 . Since C(n,f2) has dimension 22 as an F-vector space the left regular representation of C(n,f2) yields a finite linearization of f2 . We prove here however, that for all (n, d), n > 2, d > 3, C(n,f2) is infinite dimensional over F, so the existence of finite linearizations when d > 2 is a nontrivial problem. If f d is the sum of forms each of which has at most two variables or is the product of forms of degree at most two, then f d has a finite linearization. Other cases are open. Finally we relate Roby's C(n,f2) to generalized Clifford algebras studied A. O. Morris, et al., and by Koc, and to prove a conjecture of Koc.