The root-counting library in PHCv2 is organized as follows :

  Root_Counts          : 4. root counts and homotopy construction
     |-- Product       : 4.1. linear-product start systems
     |-- Implift       : 4.2. implicit lifting
     |-- Stalift       : 4.3. static lifting
     |-- Dynlift       : 4.4. dynamic lifting
     |-- Symmetry      : 4.5. exploitation of symmetry relations

The root counts that are available in the directory Product are
based on Bezout's theorem.  The corresponding start systems are
in general linear-product systems.

Implicit lifting is the name we gave to the algorithm Bernshtein 
used in his proof that the mixed volume of the Newton polytopes
of a polynomial system is a generically exact upper bound on the 
number of its isolated complex solutions with all components different
from zero.

Static lifting is the general procedure to compute mixed volumes
of polytopes.  Subdivisions induced by an integer-valued or
floating-point lifting functions can be computed.
In order to deal with non-fine subdivisions induced by a nongeneric
integer lifting, recursive algorithms have been implemented.

Dynamic lifting allows to have a control of the lifting values to
obtain a numerically stable polyhedral continuation.
When some or all supports are equal, the Cayley trick is
recommended to use.

The Symmetry library provides routines to construct start systems 
that are invariant under a given permutation symmetry.
Hereby symmetric homotopies can be constructed, so that only the
generating solution paths need to be computed.