Root Counting and Start Systems Library in PHCpack version 2.4.85

Run "gprbuild root_counts.gpr" to make all test programs.
On windows, type "gprbuild root_counts.gpr -Xos=windows"
at the PowerShell prompt.
The "gprclean root_counts.gpr" removes all files created by gprbuild.

This library formed the main part of the first version of PHCpack.
Modifications in version 2 were mainly due to the rewrite of the
mathematical library using concepts of Ada 95.  Perhaps the most
significant change in release 2.1 was the introduction of ExtSoft,
to connect other software for mixed-volume computation, see MixedVol.
In release 2.2, a black-box solver for fewnomial system was added.
The directory Fewnomials was newly created in version 2.3.06,
and several routines in Product were reorganized for v2.3.07.
The directory MixedVol was released in v2.3.13.  
Better support for stable mixed volumes was added in v2.3.31.
In release 2.3.43, the directory "Fewnomials" was renamed into
"Simpomials" and the directory "Puiseux" was added.
The "Simpomials" directory contains the code for a new general binomial
system solver, released in version 2.3.68, updated for affine solutions
in release 2.3.70.  In release 2.3.71, this directory was renamed into 
"Binomials" because the understanding of binomial systems is fundamental,
whereas a simplicial solver is obtained by just running row reduction on
the coefficient matrix.  The folder "DEMiCs" provides the code for
dynamic enumeration for all mixed cells, added in version 2.4.53.

The root-counting library is organized as follows :

  Root_Counts          : 5. root counts and homotopy construction
     |-- Product       : 5.1. linear-product start systems
     |-- Binomials     : 5.2. solvers for binomial and simplicial systems
     |-- Implift       : 5.3. implicit lifting
     |-- Stalift       : 5.4. static lifting
     |-- Dynlift       : 5.5. dynamic lifting
     |-- Symmetry      : 5.6. exploitation of symmetry relations
     |-- MixedVol      : 5.7. translation of ACM TOMS Algorithm 846
     |-- DEMiCs        : 5.8. interface to the DEMiCs program
     |-- Puiseux       : 5.9. Puiseux series for curves

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.

The MixedVol library contains an Ada translation of ACM TOMS Algorithm 846
developed by Tangan Gao, T.Y. Li, Mengnien Wu, and Li Xing.
The conversion of the source code from C++ down to C was done by Yan Zhuang.
