  We introduce the short toric polynomial associated to a graded Eulerian
poset. This polynomial contains the same information as the two toric
polynomials introduced by Stanley, but allows different algebraic
manipulations. The intertwined recurrence defining Stanley's toric polynomials
may be replaced by a single recurrence, in which the degree of the discarded
terms is independent of the rank. A short toric variant of the formula by Bayer
and Ehrenborg, expressing the toric $h$-vector in terms of the $cd$-index, may
be stated in a rank-independent form, and it may be shown using weighted
lattice path enumeration and the reflection principle. We use our techniques to
derive a formula expressing the toric $h$-vector of a dual simplicial Eulerian
poset in terms of its $f$-vector. This formula implies Gessel's formula for the
toric $h$-vector of a cube, and may be used to prove that the nonnegativity of
the toric $h$-vector of a simple polytope is a consequence of the Generalized
Lower Bound Theorem holding for simplicial polytopes.
