Products of foldable triangulations

Regular triangulations of products of lattice polytopes are constructed with the additional property that the dual graphs of the triangulations are bipartite. The (weighted) size difference of this bipartition is a lower bound for the number of real roots of certain sparse polynomial systems by recent results of Soprunova and Sottile [E. Soprunova, F. Sottile, Lower bounds for real solutions to sparse polynomial systems, Adv. Math. 204 (1) (2006) 116–151]. Special attention is paid to the cube case.