  We investigate a version of Viro's method for constructing polynomial systems
with many positive solutions, based on regular triangulations of the Newton
polytope of the system. The number of positive solutions obtained with our
method is governed by the size of the largest positively decorable subcomplex
of the triangulation. Here, positive decorability is a property that we
introduce and which is dual to being a subcomplex of some regular
triangulation. Using this duality, we produce large positively decorable
subcomplexes of the boundary complexes of cyclic polytopes. As a byproduct we
get new lower bounds, some of them being the best currently known, for the
maximal number of positive solutions of polynomial systems with prescribed
numbers of monomials and variables. We also study the asymptotics of these
numbers and observe a log-concavity property.
