Obtain dimension of multivariate polynomial quotient ring?

Question

Let $\mathbb{C}[z_1,z_2,...,z_n]$ be the ring of multivariate polynomials in the variables $z_1,z_2,...,z_n$ with complex coefficients. This ring is spanned by the countably infinite basis of monomials

$$e_{i_1,i_2,...,i_n}=z_1^{i_1}z_2^{i_2}\cdots z_n^{i_n}$$

for $i_j=0,1,2,...$ where $j\in\{1,2,...,n\}$.

Next, consider taking the quotient ring of $\mathbb{C}[z_1,z_2,...,z_n]$ by an ideal of $n$ known multivariate polynomials $\langle p_1,p_2,...,p_n\rangle$ in variables $z_1,z_2,...,z_n$ with complex coefficients:

$$Q=\frac{\mathbb{C}[z_1,z_2,...,z_n]}{\langle p_1,p_2,...,p_n\rangle}.$$

If $Q$ turns out to have finite dimension, in the sense that it is spanned by a finite subset of monomials $e_{i_1,i_2,...,i_n}$ (not known explicitly), how does one then compute the dimension of $Q$ in general? In other words, how does one compute the overall number of linearly independent $e_{i_1,i_2,...,i_n}$ that are a basis of $Q$?

Answer 1

The ideal $\langle p_1,p_2,...,p_n\rangle$ can be equivalently expressed in terms of the corresponding Groebner basis $G$. In the following we consider the special case where the polynomials $\langle g_1,g_2,...,g_n\rangle$ of the Groebner basis have the following structure:

$$g_1=z_1-P_1(z_n)\\g_2=z_2-P_2(z_n)\\\vdots\\g_{n-1}=z_{n-1}-P_{n-1}(z_n)\\g_n=P_n(z_n)$$

where $P_i(z_n)$ for $i\in\{1,2,...,n\}$ are certain polynomials in the single variable $z_n$, while we assume that $z_n$ is the highest weight variable in the employed monomial ordering.

On the support of the ideal $\langle g_1,g_2,...,g_n\rangle$ we have the simultaneous condition $g_i=0$ for all $i\in\{1,2,...,n\}$. Therefore, $g_1$ through $g_{n-1}$ can be used to eliminate the variables $z_1$ through $z_{n-1}$ in the quotient ring $Q$, producing a new, equivalent, but effectively univariate quotient ring

$$Q\rightarrow Q'=\frac{\mathbb{C}'[z_n]}{\langle g_{n}\rangle}$$

In the univariate case, it is well known that for a quotient ring with an ideal containing a single polynomial of degree $m$, a minimal basis is spanned by the set of univariate monomials $e_i=z_n^{i-1}$ for $i\in\{1,2,...,m\}$ (see i.e. proposition 28 here). Therefore, $Q'$ has dimension $\dim_{\mathbb{C}}(Q')=m$, and by equivalence the same is true for $Q$:

$$\dim_{\mathbb{C}}(Q)=\dim_{\mathbb{C}}(Q')=\deg(P_n(z_n))=m$$

Due to the above, it seems that in order to obtain the dimension of a multivariate quotient polynomial ring over a zero dimensional ideal, one has to determine the degree of the univariate equation in the Groebner basis for the polynomial ideal (if the Groebner basis satisfies the structure above). Which can be rather difficult to compute. But conceptually, this should be a possible way to go about. Any objections anyone?