TITLE: degree of polynomials in nullstellensatz
QUESTION [5 upvotes]: $(A)$ If $K$ is algebraically closed field, then consider $m$ polynomials $f_i$ for $i=\{1,\dots,m\}$ in $K[x_1,\dots,x_n]$ of degree $d_i$ each with no common root. We know there exists $g_i$ for $i=1,\dots,m$ such that $$f_1g_1+\dots+f_mg_m=1$$
What can we say about the minimum degrees of $g_i$? ($(A)$ has been answered below).
$(B)$ If we relax the $1$ requirement or any fixed constant requirement and ask just for polynomials such that  $$f_1g_1+\dots+f_mg_m\neq 0 \mbox{ (but not a constant)}$$ which seems same as $$Z(f_1g_1+\dots+f_mg_m)\bigcap Z((0))=\emptyset$$ can we say anything about the minimum degrees of $g_i$?
Is anything meaningful possible if $K$ is not closed in the relaxed problem $(B)$? Again what are the minimum degrees of $g_i$ here (both lower and upper bounds)?
I am only interested in $K=\Bbb C\mbox{ or }\Bbb R$ here.
related:bound on degree of certain polynomials

REPLY [3 votes]: The bounds in the article by Kollár as refered in the comment of Felipe Voloch are optimal, see Example 2.3 in that article.
To see this concretely, assume that $m \leq n$, and let
$f_1 = x_1^{d_1},f_2 = x_1 x_n^{d_2-1}-x_2^{d_2},...,f_{m-1} = x_{m-2}x_n^{d_{m-1}-1}-x_{m-1}^{d_{m-1}},f_m = 1-x_{m-1} x_m^{d_m-1}$.
By the article of Kollár, one can find $g_1,...,g_m$ such that $f_1 g_1 + ... + f_m g_m = 1$ and $\deg f_i g_i \leq d_1 \cdots d_m$. In the above example, these bounds are optimal, which can be seen as follows:
If we assume that $f_1 g_1 + ... + f_m g_m = 1$, and apply this equality along the curve $x = (t^{d_2... d_m - 1},...,t^{d_m-1},1/t)$, one sees that $\deg_{x_m} g_1 \geq d_1\cdots d_m - d_1$, and thus the bound is optimal. (cf. 
http://www.encyclopediaofmath.org/index.php/Masser-Philippon/Lazard-Mora_example)
(Edit: removed claim with incorrect proof regarding (B))