I am trying to understand the proof of the following result:
Suppose $F$ is a function field in $k$ variables over an algebraically closed field. Let $f_1,...,f_r \in F[x_1,...,x_n]$ be polynomials without constant term of degrees $d_1,...,d_r$, respectively. Assume that $n>d_1^k+...+d_r^k$. Then $f_1,...,f_r$ have a non-trivial common zero. If $F$ is instead a function field in $k$ variables over a finite field, then these polynomials have a non-trivial common zero provided that $n>d_1^{k+1}+...+d_r^{k+1}$.
This is the Corollary on page 378 from the paper of Lang 1952. The claim of the author is that this is a corollary of the Theorem 6 (that appears on the same page). However, to me, all that Theorem 6 seems to say is that $F$ is a $C_k$ field (for a definition see the same paper page 374) and I do not see this to be enough to obtain our conclusion. Theorem 4 of this paper (page 376) does show this statement but only when $f_i$ are homogeneous of degree $d_i$.
For the second part the author claims that this a corollary of the same theorem and of the following statement by Chevalley:
Let $\mathbb F$ be a finite field. Let $f_1,...,f_r \in \mathbb F[x_1,...,x_n]$ be polynomials without constant term of degrees $d_1,...,d_r$, respectively. Assume that $n>d_1+...+d_r$. Then $f_1,...,f_r$ have a non-trivial common zero.
but again I am struggling to put the ends together. I understand that I am supposed to understand this by myself, but I am new to the subject, and I feel a little bit lost.
I also realize that this seems like a double question, so probably it was inappropriate to post it in this way, but I believe that the reasoning for both parts should be very similar.
