$(A)$ 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.