appearing in any of the variables in any of the monomials of $p$.

\[thm:2factorGB\] For any field $K$, let $J_{{\mathbb N}}(K)$ be the ideal in $K[X]$ generated by all $3 \times 3$-minors of the matrix $y$ (recall that we identify $y_{ji}$ for $j<i$ with $y_{ij}$). Relative to the monomial order $\leq$ the ideal $J_{{\mathbb N}}(K)$ has an ${\mathrm{Inc}({{\mathbb N}})}$-Gröbner basis $B$ consisting of $42$ polynomials. The intersection $B \cap K[y_{ij} \mid i >j]$ is an ${\mathrm{Inc}({{\mathbb N}})}$-Gröbner basis of $J_{{\mathbb N}}(K) \cap K[y_{ij} \mid i>j]$ consisting of $20$ polynomials. The largest indices and the degrees of the elements in these bases are summarised in Table \[tab:Bases\].

\[tab:Bases\]

[|l|lllllll|]{} ${l(p)}$ & 3 & 4 & 5 & 6 & 7 & 8 & 9\
$\# p \in B$ & 1 & 6 & 11 & 10 & 8 & 5 & 1\
degrees & $3^1$ & $3^6$ & $3^{10} 5^1$ & $3^5 5^5$ & $5^8$ & $5^5$ & $5^1$\
$\# p \in B \cap K[y_{ij} \mid i>j]$ & & & 1 & 5 & 8 & 5 & 1\
degrees & & & $5^1$ & $3^5 $ & $5^8$ & $5^5$ & $5^1$\

\[re:PentadMinor\] The polynomial with largest index $5$ in 