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(vᵖ⁻¹)=0 for all v∈ V is p²ᵐ-1p-1. The same upper bound is thus obtained for V with Tr(v)=Tr(v²)==Tr(vᵖ⁻¹)=0 for all v ∈ V. We make use of the fact that for any subset S of Fₚ × × Fₚn \ times of |S| > pⁿ-1p-1, for all u∈ V there exist v,w∈ S and k∈ [\![0,p-1]\!] such that $kv+(p-1-k)w=u$.