Is there an explicit nontrivial (= not a constant) polynomial $p \in \mathbb{C}[x_1, \ldots, x_n]$ such that, for any ideal $I \not= \mathbb{C}[x_1, \ldots, x_n]$ generated by $f_1, f_2, \ldots, f_m$ and $\sum_i \deg f_i = n-1$, $p \not\in I$?

If not, is there some $c<n$ such that for all $n$ sufficiently large there is a polynomial $p \in \mathbb{C}[x_1, \ldots, x_n]$ such that , for any ideal $I \not= \mathbb{C}[x_1, \ldots, x_n]$ generated by $f_1, f_2, \ldots, f_m$ and $\sum_i \deg f_i \leq cn$, $p\not \in I$?