Suppose $k$ is an algebraically closed field, $I \subset k[x_1, \ldots, x_n]$ is an ideal that contains a polynomial of the form $x_1^{m_1} \cdots x_r^{m_r}+cx_{r+1}^{m_{r+1}}\cdots x_n^{m_n}$, where $c \in k$, and $m_i \in \mathbb{N} \cup\{0\}$ for each $i$ (but at least one of $m_1, \ldots, m_r$ and at least one of $m_{r+1}, \ldots, m_n$ is nonzero). Let $G$ be a Groebner basis for $I$. Does then $G$ necessarily contain a polynomial of the above mentioned form? Can you provide a proof or a counterexample? Where could I read more on this?
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