Lower bounds for real solutions to sparse polynomial systems

We show how to construct sparse polynomial systems that have non-trivial lower bounds on their numbers of real solutions. These are unmixed systems associated to certain polytopes. For the order polytope of a poset P this lower bound is the sign-imbalance of P and it holds if all maximal chains of P have length of the same parity. This theory also gives lower bounds in the real Schubert calculus through the sagbi degeneration of the Grassmannian to a toric variety, and thus recovers a result of Eremenko and Gabrielov.