Quantum Field Theory over Fq

We consider the number N̄(q) of points in the projective complement of graph hypersurfaces over Fq. We show that the smallest graphs with non-polynomial N̄(q) have 14 edges. We give six examples which fall into two classes. One class has an exceptional prime 2 whereas in the other class N̄(q) depends on the number of cube roots of unity in Fq. At graphs with 16 edges we find examples where N̄(q) can be reduced to the number of points on a (presumably) non-mixed-Tate surface in P. In an outlook we show that applying Feynman-rules in Fq lets the perturbation series terminate for renormalizable and non-renormalizable bosonic quantum field theories.

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