Context: I am computing the reduced Gröbner bases with respect to degRevLex for the following two ideals:
- $P$ is a homogeneous prime ideal with $36$ generators $\{g_1,\dots,g_{36}\}$ with homogeneous degree at most $5$.
- $Q$ is the homogeneous ideal with generators $\{g_1,\dots,g_5\}$ of least homogeneous degree (equal to $3$) among the generators of $P$.
$P$ and $Q$ share some combinatorial properties, so I am wondering if it is more feasible to work with $Q$ than it is to work with $P$. In principle $Q$ is a "simpler" ideal than $P$, however, computing the reduced Gröbner basis for $Q$ takes much longer than computing the basis for $P$.
My end goal is not a Gröbner basis for $Q$ but a Gröbner basis for at least one minimal prime over $Q$. From what I know, algorithms computing minimal prime ideals are Gröbner basis algorithms (e.g. Gianni-Trager-Zacharias and its refinement by Laplagne), so I am expecting that the computation of a Gröbner basis for a minimal prime over $Q$ would take even longer than the computation for the Gröbner basis for $Q$. And since it seems that the computation is much faster for $P$ than for $Q$, I shouldn't bother working with $Q$ or any of its minimal primes at all.
Let $n\geq 5$ and $R=F[x_1,\dots,x_{\binom n2}]$ a ring of polynomials. Assume that $F\in\mathbb\{\mathbb Q,\overline{\mathbb Q},\mathbb C\}$.
Let $P=\langle g_1,\dots,g_{\binom{n+1}{5}^2}\rangle$ be a prime ideal of $R$. You may in addition assume that each $g_i$ is homogeneous with homogeneous degree $5$. Assume that among the generators of $P$ there are exactly $\binom n4$ generators of least homogeneous degree equal to $3$. Let $Q=\langle g_1,\dots,g_{\binom n4}\rangle$ and let $M$ be a minimal prime over $Q$.
Fix a monomial order on $R$ and let $\mathcal G_P$ and $\mathcal G_M$ denote the reduced Gröbner bases for $P$ and $M$ with respect to the fixed order.
For a large $n$, is there any reason to expect that (in theory) the computation of $\mathcal G_M$ is more feasible than the computation of $\mathcal G_P$?
I am aware that the complexity bounds for a Gröbner basis computation are doubly exponential in the number of indeterminates, so as $n$ goes to infinity, there practically won't be a difference. However, my intuition is that having a much smaller generating set ($\binom n4$ as opposed to $\binom{n+1}{5}^2$, e.g. for $n=14$ that is approx. $10^3$ vs $10^7$ generators) should result in a computation that is easier to handle, but that doesn't appear to be the case for what I am computing.