Groebner basis in Boolean rings is not polynomial-space

We give an example where the number of elements of a Gröbner basis in a Boolean ring is not polynomially bounded in terms of the bitsize and degrees of the input. 1 Boolean Rings Let Rn = F2[x1, . . . , xn, y1, . . . , yn, z1, . . . , zn] where F2 = Z/(2). If S ⊆ Rn then (S) denotes the ideal in Rn generated by S, and Sol(S) ⊆ K 3n denotes the solution set of S, where K is the algebraic closure of F2. Let Sn := {c 2 − c | c ∈ {x1, . . . , xn, y1, . . . , yn, z1, . . . , zn}}. R n := Rn/(Sn) is a Boolean ring, which means r 2 = r for all r in R n. If I is an ideal in Rn, then Sn ⊆ I if and only if I is radical and Sol(I) ⊆ F 3n 2 . Ideals in Rn that contain Sn are in 1-1 correspondence with ideals in R b n. Thus, Gröbner basis in R n is equivalent to: Gröbner basis in Rn restricted to ideals that contain Sn. We will show with an example that this is not polynomial-space.