| Abstract: | Lek k be an infinite field and suppose m.i. and n are positive integers such that t m We study the subset of kx 1,x 2, … xm ] which consists of 0 and the homogeneous members t of f of kx 1,x 2, … xm ] of fixed degree n such that there exists homogeneous F 1, F 2, … Ft in kx 1,x 2, … xm ] of degree one and homogenous g 1 g 2, …gt , in kx 1,x 2, … xm ] such that f(x) = F 1(x)g 1(x) + F 2(x)g 2(x) + … + F t (x)g t (x) for each x in k m. In case k is algebrarcally closed we are able to prove that this set is an algebraic variety. Consequently. if k is also of characteristic 0 then we are able to prove that certain collections of symmetric k-valued multilinear functions are algebraic varieties. |
