Polyhedral homotopies by implicit lifting in PHCpack version 2.4.85

This directory contains software for the computation of mixed volumes,
according to the recursive formula Bershtein used in his proof.
Also the homotopy method of Bernshtein has been implemented.

The minor modification in release 2.2 is the addition of a black-box
routine to solve polynomials in one variable, using Durand-Kerner
(a.k.a. the method of Weierstrass), and a black-box solver for systems
with few monomials.  Since 2.3.06 a new fewnomial system solver exists,
and in release 2.3.11, the old fewnomial solvers were removed.

Support for 64-bit integer arithmetic was added in version 2.3.71.
Double double and quad double versions of the method of Weierstrass
(aka Durand-Kerner) were provided in version 2.3.79 and moved to a
new separate directory Curves.

Version 2.3.86 offered a tool to transform a Laurent polynomial system
with support sets of positive corank.

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

-------------------------------------------------------------------------------
file name                           : short description
-------------------------------------------------------------------------------
supports_of_polynomial_systems      : manipulating support sets of polynomials
ts_supports                         : test the power lists
standard_integer32_vertices         : extracting vertices from supports
standard_integer64_vertices         : 64-bit version of vertices computation
main_vertex_points                  : main vertex extraction procedures
ts_mainvpts                         : calls main vertex extraction procedure
-------------------------------------------------------------------------------
standard_integer32_transformations  : unimodular transformations on monomials
standard_integer64_transformations  : 64-bit unimodular transformations
standard_integer_transformations_io : i/o of unimodular transformations
ts_transfo                          : shows transformations from vectors
integer32_vectors_utilities         : utilities to manipulate integer vectors
integer64_vectors_utilities         : utilities for 64-bit integer vectors
transforming_solutions              : transformations on solution lists
transforming_integer32_vector_lists : transforming lists of integer vectors
transforming_integer64_vector_lists : transforming lists of 64-bit vectors
transforming_laurent_systems        : transformations on Laurent systems
ts_tropelim                         : test on transforming Laurent systems
-------------------------------------------------------------------------------
trees_of_vectors                    : data structure to hold outer normals
trees_of_vectors_io                 : input/output of trees of vectors
lists_of_vectors_utilities          : utilities to manipulate lists of vectors
arrays_of_lists_utilities           : utilities to manipulate arrays of lists
volumes                             : compute volumes and mixed volumes
ts_impvol                           : test on mixed-volume computation
-------------------------------------------------------------------------------
mixed_homotopy_continuation         : polyhedral continuation 
set_structures_and_volumes          : combines product and polyhedral methods
generic_position                    : test on genericity of system
drivers_for_coefficient_systems     : set up polyhedral continuation
drivers_for_implicit_lifting        : main driver for implicit lifting
ts_drivimpl                         : calls main driver for implicit lifting
-------------------------------------------------------------------------------
span_of_supports                    : to compute the span of a support set
transformation_of_supports          : to transform positive corank supports
ts_supspan                          : test on span computation
driver_to_rank_supports             : called by maindeco
-------------------------------------------------------------------------------

This is the first implementation of polyhedral homotopy continuation.
Some design mistakes make it inefficient and hard to maintain, although
there is progress: fewnomial system solvers are revised and out.
Also the univariate solvers have been moved out.
