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We give a sheaf theoretic interpretation of Potts models with externalmagnetic field, in terms of constructible sheaves and their Eulercharacteristics. We show that the polynomial countability question for thehypersurfaces defined by the vanishing of the partition function is affected bychanges in the magnetic field: elementary examples suffice to seenon-polynomially countable cases that become polynomially countable after aperturbation of the magnetic field. The same recursive formula for theGrothendieck classes, under edge-doubling operations, holds as in the casewithout magnetic field, but the closed formulae for specific examples likebanana graphs differ in the presence of magnetic field. We give examples ofcomputation of the Euler characteristic with compact support, for the set ofreal zeros, and find a similar exponential growth with the size of the graph.This can be viewed as a measure of topological and algorithmic complexity. Wealso consider the computational complexity question for evaluations of thepolynomial, and show both tractable and NP-hard examples, using dynamicprogramming.

Potts models with magnetic field: arithmetic, geometry, and computation