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 $\mathrm{Tr}(v^{p-1})=0$ for all $v\in V$ is $\frac{p^{2m}-1}{p-1}$. The same upper bound is thus obtained for $V$ with $\mathrm{Tr}(v)=\mathrm{Tr}(v^2)=\cdots =\mathrm{Tr}(v^{p-1})=0$ for all $v\in V$. We make use of the fact that for any subset $S$ of $\underset{n\text{ times}}{\underbrace{{\mathbb{F}}_p\times \cdots \times {\mathbb{F}}_p}}$ of $\mathrm{\mid }S\mathrm{\mid }>\frac{p^n-1}{p-1}$, for all $u\in V$ there exist $v,w\in S$ and $k\in [\text{\hspace{0.17em}⁣}[0,p-1]\text{\hspace{0.17em}⁣}]$ such that $kv+(p-1-k)w=u$.