  The BKK theorem states that the mixed volume of the Newton polytopes of a
system of polynomial equations upper bounds the number of isolated torus
solutions of the system. Homotopy continuation solvers make use of this fact to
pick efficient start systems. For systems where the mixed volume bound is not
attained, such methods are still tracking more paths than necessary. We propose
a strategy of improvement by lifting a system to an equivalent system with a
strictly lower mixed volume at the expense of more variables. We illustrate
this idea providing lifting constructions for arbitrary bivariate systems and
certain dense-enough systems.
