---
abstract: |
    Exploiting symmetry in Gröbner basis computations is difficult when the symmetry takes the form of a group acting by automorphisms on monomials in finitely many variables. This is largely due to the fact that the group elements, being invertible, cannot preserve a term order. By contrast, inspired by work of Aschenbrenner and Hillar, we introduce the concept of [*equivariant Gröbner basis*]{} in a setting where a [*monoid*]{} acts by [*homomorphisms*]{} on monomials in potentially [*infinitely*]{} many variables. We require that the action be compatible with a term order, and under some further assumptions derive a Buchberger-type algorithm for computing equivariant Gröbner bases.

    Using this algorithm and the monoid of strictly increasing functions ${{\mathbb N}}\to {{\mathbb N}}$ we prove that the kernel of the ring homomorphism $${{\mathbb R}}[y_{ij} \mid i,j \in {{\mathbb N}}, i > j] \to {{\mathbb R}}[s_i,t_i \mid i \in {{\mathbb N}}],\ 
    y_{ij} \mapsto s_is_j + t_it_j$$ is generated by two types of polynomials: [*off-diagonal $3 \times 3$-minors*]{} and [*pentads*]{}. This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian two-factor model from algebraic statistics.
address:
- |
     Department of Mathematics and Computer Science\
    Technische Universiteit Eindhoven\
    P.O. Box 513, 5600 MB Eindhoven, The 