[Math] the ideal of leading terms

Question

Fix a monomial ordering on the polynomial ring $\Bbb{k}[x_1, \dots, x_n] = R$ over a field. What exactly is $LT(I)$ for an ideal $I$ of $R$? How is it defined and does it form an ideal?

Answer 1

If you've fixed a monomial ordering then every polynomial has a leading term, for example in the lexicographic ordering $f = 2x_1^2x_2 + x_3$ has leading term $2x_1^2x_2$ so we write $\mathrm{LT}(f) = 2x_1^2x_2$.

Now $$\mathrm{LT}(I) = \{\mathrm{LT}(f) \ | \ f \in I\}$$ is the set of leading terms of polynomials in $I$. It is not an ideal because every element of $\mathrm{LT}(I)$ is a single term, so it's not closed under addition. But it is closed under multiplication by monomials.

If you want the ideal of leading terms then what you want is the ideal $\langle\mathrm{LT}(I)\rangle$ generated by $\mathrm{LT}(I)$.