A quadratic inequality among minors is valid for all totally nonnegative matrices iff, for every feasible planar matching, the matching occurs at least as many times on the left side as on the right.
Danilov and A.V Karzanov, On universal quadratic id entities for minors of quantum matrices, J
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On universal quadratic inequalities for minors of TNN matrices
A quadratic inequality among minors is valid for all totally nonnegative matrices iff, for every feasible planar matching, the matching occurs at least as many times on the left side as on the right.