Let $F$ be a field and consider the ring $R=F[x_1,...,x_n]$. Let $I$ be a maximal ideal of $R$. Then for all indeterminates $x_j,j=1,...,n$, there exists some $k$ and some $\alpha_{i_1,\cdots,i_n},i_j=1,\cdots,k-1$ such that:
$$x_i^k=\sum_{i_1,\cdots,i_n}\alpha_{i_1,\cdots,i_n}x_1^{i_1}\cdots x_n^{i_n}+q$$ where $q\in I$.
Is this true? If it is not, how can I find a counterexample? If it is, how can I prove it? I was thinking of choosing a Gröbner basis for $I$, but it's a bit messy when I have to choose the order of terms.
Does $I$ need to be maximal? Can we weaken the hypothesis, like "$I$ can be an ideal of the form $\mathcal{I}(\mathcal{Z}(I))$ or a prime ideal"?
Note that for $n=1$, it's the same as saying there exists some $k$ such that
$$x^k=\sum_{i=1}^{k-1}\alpha_i x^i+q$$ where $q\in I$ and this is what I'm trying to generalize.