the ${\mathrm{Inc}({{\mathbb N}})}$-Gröbner basis $B \cap K[y_{ij} \mid i >j]$ is the pentad $P$. The five degree-$3$ polynomials with largest index $6$ in that Gröbner basis form the ${\operatorname{Sym}}({{\mathbb N}})$-orbit of the off-diagonal minor $M$. All $19$ remaining polynomials are already in the ${\mathrm{Inc}({{\mathbb N}})}$-stable ideal generated by these polynomials; this latter statement also follows from the result in [@Drton07] that at least up to $n=9$ the ideal of the two-factor model is generated by pentads and off-diagonal minors. The complete ${\mathrm{Inc}({{\mathbb N}})}$-Gröbner bases of $J_{{\mathbb N}}(K)$ can be downloaded from the second author’s website.

A Gröbner basis of the ideal of the two-factor model $F_{2,n}$ relative to [*circular term orders*]{} was already found in [@Sullivant08]. The proof involves general techniques for determining the ideal of secant varieties, especially of toric varieties; see also [@Sturmfels05]. The Gröbner basis found there, however, does not stabilise as $n$ grows—and indeed, circular term orders are not compatible with the action of ${\mathrm{Inc}({{\mathbb N}})}$. It would be interesting to find a direct translation between Sullivant’s Gröbner basis and ours.

Theorem \[thm:2factorGB\] implies Theorem \[thm:Main\].

It is well known that the $(k+1) \times (k+1)$-minors of the symmetric matrix $(y_{ij})_{i,j=1,\ldots,n}$ generate the ideal of all polynomials 