formal languages – If \$p(n) := sum_{i=0}^ka_in^i\$ where \$a_iinmathbb{N}, a_k ne 0\$ AND \$k ge 2\$, is \$L = {0^n1^{p(n)} mid ninmathbb{N}}\$ context-free?

I have the really strong feeling it is indeed NOT context-free, since the language $$1^{n^k}$$ for $$kge 2$$ is not context free (proven by the pumping lemma) and, in a sense, “the order of magnitude” of the number of 1s will always supercede the order of the number of 0s, but I really can’t find an exhaustive proof for the global case where $$p(n)$$ is any polynomial of degree $$k$$. Any advice is welcome! Thanks in advance!