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 $\sum_i \deg f_i \leq cn$, $p\not \in I$?

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    $\begingroup$ What if $f_1=1$? $\endgroup$ – Ilya Bogdanov Feb 23 '15 at 23:25
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    $\begingroup$ When $n$ is fixed, "$=o(n)$" is senseless. It would be useful to make the question meaningful by a possible use of quantifiers (which is unclear to me). $\endgroup$ – YCor Feb 24 '15 at 7:46
  • $\begingroup$ Here is an attempt; I don't know if it is what the OP means. Let $c \in (0,1)$. Let $\mathcal{I}_{c,n}$ be the set of all homogenous ideals in $k[x_1, \ldots, x_n]$ whose generators have degrees summing to $<cn$, and where the ideal $(1)$ is forbidden. Is there a homogenous polynomial $p$ of positive degree not in $\bigcup_{I \in \mathcal{I}_{c,n}} I$? $\endgroup$ – David E Speyer Feb 26 '15 at 16:11
  • $\begingroup$ I think $x_1x_2...x_n -1= \Sigma a_i f_i, (f_i)\neq(1) => \Sigma deg f_i \geq n$ but I can prove it only for small n. $\endgroup$ – David Lampert Feb 26 '15 at 20:05
  • $\begingroup$ P.S. I think $x_1^{d_1}+x_2^{d_2}+ ... +x_n^{d_n}$ for suitable large $d_i$ will also require $\Sigma deg f_i \ge n$ (and maybe easiest to prove it with $d_i$ rapidly increasing). $\endgroup$ – David Lampert Feb 26 '15 at 21:33

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