We provide evidence for the proposition that the computational complexity of individual problems, and of whole complexity classes, hinge on the existence of certain solvable polynomial systems that are unlikely to be encountered other than by systematic explorations for them. We consider a minimalist version of Cook?s 3CNF problem, namely that ofmonotone planar 3CNF formulae where each variable occurs twice. We show that counting the number of solutions of these modulo 2 is \oplusP-complete (hence NP-hard) but counting them modulo 7 is polynomial time computable (sic). We also show a similar dichtomy for a vertex cover problem. To derive completeness results we use a new holographic technique for proving completeness results in \oplusP for problems that are in P. For example, we can show in this way that \oplus2CNF, the parity problem for 2CNF, is \oplusP-complete. To derive efficient algorithms we use computer algebra systems to find appropriate holographic gates. In order to explore the limits of holographic techniques we define the notion of an elementary matchgrid algorithm to capture a natural but restricted use of them. We show that for the NP-complete general 3CNF problem no such elementary matchgrid algorithm can exist. We observe, however, that it remains open for many natural #Pcomplete problems whether such elementary matchgrid algorithms exist, and for the general CNF problem whether non-elementary matchgrid algorithms exist.