that ${\operatorname{im}}\pi_0 \cup {\operatorname{im}}\pi_1=\{1,\ldots,k\}$ for some $k$. For instance, for $b_0=b_1=b$ equal to the polynomial in $B$ with largest index $9$, we having to ${\mathrm{Inc}({{\mathbb N}})}$-reduce $S(\pi_0b,\pi_1b)$ modulo $B$ for all increasing maps $\pi_0,\pi_1:\{1,\ldots,9\} \to \{1,\ldots,18\}$ whose image union is an interval $\{1,\ldots,k\}$. However, if $k=17$ or $k=18$, then $\pi_0b$ and $\pi_1b$ turn out to have leading monomials with gcd $1$, so these cases can be skipped. This reduces the theorem to a finite computation involving polynomials with largest indices up to $16$, which we have implemented directly in [C]{}. Finally, to deduce the result for all base fields—and to speed up the computation—we used the following argument. Since ${\mathrm{Inc}({{\mathbb N}})}B \cap K[y_{ij} \mid 1 \leq j \leq i \leq
n]$ is a subset of the ideal of $3 \times 3$-minors, it is a Gröbner basis if and only if the ideal generated by ${\mathrm{lm}}(B)$ has the same Hilbert series as the ideal generated by $3 \times 3$-minors. Since this Hilbert series is known and does not depend on the field [@Conca94], we may do all our computations over one field and conclude that it holds over all fields. We have verified the equivariant Buchberger criterion over ${{\mathbb F}}_2$, which 