@alpoge Thank you very much, very helpful remark!

Let us continue this discussion in chat.

All I mean is that degree of $F$ in each of the variables is $1$. Sorry for the confusion -- I made the edit to the question.

No, for example the quadratic form $xy+xz+yz$ is linear in each variable, and still assumes the value $1$ at the point $(1,1,0)$.

@pavl0 can you please clarify? The papers you are referring to all deal with one variable polynomials.

David, great -- thank you very much! In fact, Gang Yu pointed out to me that this result can be obtained by an argument entirely analogous to the proof of the main theorem in: James, R. D.; The Distribution of Integers Represented by Quadratic Forms. Amer. J. Math. 60 (1938), no. 3, 737–744 jstor.org/discover/10.2307/… Here your function $Z_D(s)$ is the first product in the formula for $f(s)$ in Lemma 3 of James' paper.