Existence of a polynomial not vanishing on subvarieties

Question

Let $K$ be an algebraically closed field and $X_1,\dots,X_m$ be subvarieties of a variety $X\subset \mathbb A^n$. Let $f_1,\dots,f_k\in K[X]$ such that for every $X_i$ there exists some $f_j$ not vanishing on $X_i$. Prove there exists $g_1,\dots,g_k$ such that $\sum_{a=1}^k f_ag_a$ doesn't vanish on $X_i$ for all $i$.

Answer 1

I suppose your varieties are irreducible. Assume the contrary. Then consider the ideal $I=(f_1,...,f_k)$. Then this implies any element of the ideal vanishes on some $X_i$ and hence is in $\mathscr I (X_i)$ for some $i$. Thus you get $I\subset \bigcup_{i=1} ^n \mathscr I(X_i)$. Prime avoidance tells you $I$ is in some $\mathscr I(X_i)$ which means each $f_j$ vanishes on $X_i$. Contradiction.