Spaces with Vanishing Characteristic Coefficients

We prove that the maximal dimension of a subspace V of the generic tensor product of m symbol algebras of prime degree p with Tr (v^p-1) =0 for all v V is p^2m-1p-1. The same upper bound is thus obtained for V with Tr (v) =Tr (v²) ==Tr (v^p-1) =0 for all v V. We make use of the fact that for any subset S of Fₚ Fₚ₍ \ ₓ₈₌₄ₒ of |S| > p^n-1p-1, for all u V there exist v, w S and k \![0, p-1\!] such that kv+ (p-1-k) w=u.