We prove that the Hessians of nonzero partial derivatives of the (homogenous) multivariate Tutte polynomial of any matroid have exactly one positive eigenvalue on the positive orthant when $0<q\leq 1$. Consequences are proofs of the strongest conjecture of Mason and negative dependence properties for $q$-state Potts model partition functions.
The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
Galois groups of multivariate Tutte polynomials
Combinatorial applications of the Hodge-Riemann relations
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