let $K$ be an algebraically closed field with characteristic $0$ and $E=M_n(K)\times K^n\times K^n$. We consider the algebraic set $$F=\{(B,u,v)\in E|u^TB^kv=tr(B^kvu^T)=0,k=0,\cdots,n-1\}.$$ I think that $F$ has codimension $n$ in $E$, that is true when $n=2,3,4$. Is it true in the general case ? I think that it is linked with the fact (again conjectured) that, for $i\not= j$, the scalars $({B^k}_{i,j}),k=1,\cdots,n-1$ are algebraically independent.
Thanks in advance.
