While attempting to prove some existence theorem for matrices over $\mathbb{F}_{2^k}$ I've come across the following problem concerning fields of definition for closed point of, say affine, varieties.

Let $X/k$ be a variety defined over a field $k$ given by a $k$-algebra $A = k[x_1,\ldots,x_n]/(f_1,..,f_i)$ such that $X = \textrm{Spec}(A)$. Is there any upper bound for the minimal degree $[l:k]$ for a closed point $\textrm{Spec}(l) \to X$, that depends on the field $k$, the degrees of $f_i$'s and the parameter $n$. For example, can we say something for $k$ a $C_r$ field.

In particular, are you familiar with any reference concerning the special case of finite fields or other $C_1$ fields.