Let $K(t_1,...,t_m)[x_1,...,x_n]$ denote a polynomial ring where $K(t_1,...,t_m)$ is a field of rational functions.
Given an equality $f=\sum_{i=1}^sA_jg_j$ in $K(t_1,...,t_m)[x_1,...,x_n]$ and let $a=(a_1,...,a_m)\in K^m$ such that no denominators of the coefficients of $f$, $A_j$, $g_j$ vanish at $a$.
Does this equality still hold in $K[x_1,...,x_n]$ under $(t_1,...,t_m)\to(a_1,...,a_m)\in K^m$? Could anyone provide a proof or a reference for a proof? I would appreciate your help with this situation.
