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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

M. Winter
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Mike V.D.C.
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1 Answers1

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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 $X> 1\Rightarrow X\cdot X> X\cdot 1\Rightarrow X^2> X$.

M. Winter
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  • Thanks @Winter. Now with your example, my own question looks quite stupid... However I am still wondering if it is possible characterize / put some condition on $f$, so that I can have a monomial ordering for every term! – Mike V.D.C. Oct 26 '17 at 10:58
  • @MikeV.D.C. This is a very interesting question. After some initial thought on the problem I concluded that it is non-trivial and I recommend you to post another question to ask this. This question here however is sufficiently answered (assuming you are satisfied with my answer to the original question). When/if you start another post, let me know, I would love to participate. – M. Winter Oct 26 '17 at 12:48