Number of monomials in the complement of $\langle LT(I) \rangle$.

Question

I'm trying to find the number of monomials in the complement of $\langle LT(I) \rangle$, with both the lexicographic ($>_{lex}$) and graduated lexicographic ($>_{grlex}$) orders for the ideal $I = \langle x^{4}y - z^{6}, x^{2} - y^{3}z, x^{3}z^{2} - y^{3} \rangle \subset \mathbb{K}[x,y,z]$.

In the first case, the lexicographic ($>_{lex}$), what I get is the following: $$\langle LT(I) \rangle = \langle x^2, xz^6, y^3, yz^6, z^{67} \rangle,$$ whilst in the other, ($>_{grlex}$), what I get is $$\langle LT(I) \rangle = \langle x^9, x^2y^7, y^9, x^7y, x^4y^4, x^5z, z^6, x^3z^2, y^3z \rangle.$$

My question is if there has to be the same number of monomials in the complement of $\langle LT(I) \rangle$ regardless of the order taken to compute the Groebner basis, or it can be a different number. Because when I compute both of them, what I get is 80 monomials and 52 respectively, and I'm not sure if it is well done.

Is there a result which could help me?

Answer 1

The monomials in the complement of $\operatorname{LT}(I)$ form the normal basis $\operatorname{NB}(I)$ of the quotient ring $\mathbb{K}[x,y,z]/I$ with respect to the given monomial ordering, i.e. a $\mathbb{K}$-vector space basis of the quotient ring $\mathbb{K}[x,y,z]/I$. In particular, the cardinality of the complement of $\operatorname{LT}(I)$ is the cardinality of $\operatorname{NB}(I)$, which is the dimension of the $\mathbb{K}$-vector space $\mathbb{K}[x,y,z]/I$. This dimension is independent of any choice of monomial ordering, so the cardinality of the complement of $\operatorname{LT}(I)$ is independent of the choice of monomial ordering, even though the complement of $\operatorname{LT}(I)$ may be different for different monomial orderings.

In the example the cardinality of the complement of $\operatorname{LT}(I)$ is $97$:

sage: R.<x,y,z> = PolynomialRing(QQ, order='lex')
sage: I = R.ideal(x^4*y - z^6, x^2 - y^3*z, x^3*z^2 - y^3)
sage: len(I.normal_basis())
97