The Stacks project

Definition 59.67.5. A field $K$ is called $C_ r$ if for every $0 < d^ r < n$ and every $f \in K[T_1, \ldots , T_ n]$ homogeneous of degree $d$, there exist $\alpha = (\alpha _1, \ldots , \alpha _ n)$, $\alpha _ i \in K$ not all zero, such that $f(\alpha ) = 0$. Such an $\alpha $ is called a nontrivial solution of $f$.