Incompatible double posets and double order polytopes

In 1986 Stanley associated to a poset the order polytope. The close interplay between its combinatorial and geometric properties makes the order polytope an object of tremendous interest. Double posets were introduced in 2011 by Malvenuto and Reutenauer as a generalization of Stanleys labelled posets. A double poset is a finite set equipped with two partial orders. To a double poset Chappell, Friedl and Sanyal (2017) associated the double order polytope. They determined the combinatorial structure for the class of compatible double posets. In this paper we generalize their description to all double posets and we classify the 2-level double order polytopes.Comment: 11 pages, 3 figure

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 sagbi degeneration of the Grassmannian to a toric variety, and thus recovers a result of Eremenko and Gabrielov.Comment: 31 pages. Minor revision