3 problems

Let $Q_1,\ldots,Q_n$ be a system of integral polynomial equations, and let $\operatorname{rk}_{\mathbb Q}(Q_1,\ldots,Q_n)$ denote its Schmidt rank over $\mathbb Q$. Cook–Magyar con…

Let $d>1$ and let ${\bf k}$ be a field with characteristic greater than $d$ or characteristic zero. For a polynomial $Q\in {\bf k}[x_1,\ldots,x_s]$ of degree $d$, write…

Let ${\mathbf{k}}$ be a non-closed field, let $\overline{{\mathbf{k}}}$ be an algebraic closure of ${\mathbf{k}}$, and let $f\in {\mathbf{k}}[x_1,\ldots,x_n]$ be a homogeneous poly…