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!