Question: Let $k$ be a field, let $f \in k[t_1,\ldots,t_N]$ be a nonzero polynomial. Which monomials $m_a = t_1^{a_1} \cdots t_N^{a_n}$ appearing in $f$ are leadable in the sense that they are the leading monomial with respect to some monomial order $\preceq$?
Guess at the Answer: An obvious necessary condition is that there can't be another monomial $m_b = t_1^{b_1} \cdots t_N^{b_N}$ in $f$ with $a_i \leq b_i$ for all $i$ and $\sum_i a_i < \sum_i b_i$. Indeed, if this holds, then $m_a \prec m_b$ for every monomial ordering $\preceq$. My guess (and hope; this would have some favorable consequences in my current work) is that the converse is also true.
Order-Theoretic Translation: Let $\mathbb{N}$ denote the non-negative integers. On the commutative monoid $\mathbb{N}^N$, let $\leq$ be the natural "additive divisibility" partial ordering: $(x_1,\ldots,x_N) \leq (y_1,\ldots,y_N)$ if and only if $x_i \leq y_i$ for all $i$.
Let $S$ be a finite subset of $\mathbb{N}^N$, and let $a \in S$ be a maximal element. Then $\leq$ extends to an admissible (i.e., compatible with the monoid structure) well-ordering $\preceq$ on $\mathbb{N}^N$ with respect to which $a$ is the top element.
Comments:
My guess is certainly correct when $N = 1$; this is a trivial case.
I finally started learning about Grobner bases this week! Better late than never. So I am very far from being an expert here. I'm aware of a result of Robbiano that parametrizes all monomial orders on $\mathbb{N}^N$ in terms of certain real matrices. So I suppose the idea is to choose the matrix carefully so as to make any $\leq$-maximal monomial the leading monomial. But I haven't even thought carefully about the monomial order associated to a given matrix, so it is not yet clear to me how to proceed. I am however pretty confident that, provided only that my guess is correct, this must be a standard result in the literature, and I would be happy with a reference.
