Tropicalization of witness sets with PHCv2.4.85

This directory is new since release 2.3.48.  The asymptotics of witness sets
have tropical interpretations.  Via the computation of tropisms we develop
polyhedral methods for positive dimensional solution sets.
Added in v2.3.49 are the giftwrapping routines.

A first step is to compute certificates for a regular common factor of
two Laurent polynomials with complex coefficients.

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

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file name                        : short description
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standard_lattice_polygons        : V- and H-representation of Newton polygons
multprec_lattice_polygons        : multiprecision Newton polygons
standard_initial_forms           : initial forms of Laurent polynomials
standard_binomial_factors        : leading term of Puiseux series
standard_binomial_factors_io     : output of leading terms
standard_puiseux_certificates    : second term of Puiseux series
standard_puiseux_certificates_io : output of 1st & 2nd term of Puiseux series
ts_comfac                        : test to find regular common factor 
driver_for_common_factor         : for use in mainfac, for phc -f
standard_lattice_supports        : support functions for lattice polytopes
standard_lattice_3d_facets       : giftwrapping for 3d polytopes
standard_lattice_3d_facets_io    : basic output of facets and lists
standard_lattice_edges           : 3d polytopes as lists of edges
standard_lattice_4d_facets       : giftwrapping for 4d polytopes
standard_integer_orthogonals     : orthogonalization operations
multprec_lattice_supports        : support functions for lattice polytopes
multprec_lattice_3d_facets       : multprec giftwrapping for 3d polytopes
multprec_lattice_3d_facets_io    : basic output of facets and lists
multprec_lattice_edges           : 3d polytopes as lists of edges
multprec_lattice_4d_facets       : multprec giftwrapping for 4d polytopes
multprec_integer_orthogonals     : orthogonalization operations
ts_intorto                       : test on orthogonalization
standard_lattice_polytopes       : convex hull in general dimensions
multprec_lattice_polytopes       : convex hull in general dimensions
convex_hull_methods              : drivers and tests on gift wrapping
ts_convhull                      : testing convex hull operations
standard_pretropical_facets      : giftwrapping for 3d tropisms
ts_pretrop                       : test on standard lattice facets
point_lists_and_strings          : list of tuples formats of point sets
facets_and_strings               : string representations of facets
ts_ptlstr                        : test on conversions
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