Let $k$ be a field and let $f\in k[X_1,\ldots,X_n]$ be a polynomial. Write $$f = \sum_\alpha \underline{X}^{\underline{\alpha}}\qquad \underline{X}^{\underline{\alpha}} \text{ is a monomial in }X_1,\ldots,X_n. $$ For every $\alpha$ appearing above, does there exists a monomial ordering, such that $\mathrm{LM}(f) = \underline{X}^{\underline{\alpha}}$? ($\mathrm{LM}$ stands for leading monomial).
-- Mike
No. Take $f(X)=1+X+X^2$. If you want $X$ to be the single leading monomial, you need
$$X > 1\quad\text{and}\quad X> X^2.$$
But $X> 1\Rightarrow X\cdot X> X\cdot 1\Rightarrow X^2> X$.