Quasi-algebraically closed field

Remarks

Every algebraically closed field is $C_1$. Indeed, let $g(t) = f(t, 1, 0, \ldots, 0)$. If $g$ is not constant, it has a root $t_0$, and $(t_0, 1, 0, \ldots, 0)$ is a nontrivial zero of $f$. If $g$ is constant, the coefficient of $x_1^d$ in $f$ is $0$, so $(1, 0, \ldots, 0)$ is a nontrivial zero of $f$.

Example: a finite field $\F_q$ is $C_1$. Check: by the Chevalley–Warning theorem, the number of zeros in $\F_q^n$ of a polynomial of degree $d < n$ is divisible by the characteristic. The zero vector is one zero, so there is another.

Non-example: $\R$ is not $C_1$. The form $x^2 + y^2 + z^2$ has degree $2$ in $3$ variables, but its only real zero is $(0, 0, 0)$.

More generally, $K$ is a $C_i$ field if every homogeneous polynomial of degree $d \ge 1$ in $n > d^i$ variables has a nontrivial zero; $C_1$ is the case $i = 1$.