Numerical Schubert Calculus using SAGBI homotopies in PHCv2.4.85

This directory was delivered with release 2 of PHCpack, and stowed
away in a separate directory in release 2.2.

SAGBI homotopies are a concatenation of three homotopies:
1) polyhedral homotopies to set up the start system;
2) flat deformation using a Groebner basis for the Grassmannian;
3) Cheater's homotopy towards real target system.

Historically, SAGBI homotopies were the first tools in numerical
Schubert calculus, but were later outperformed (both in mathematical
beauty and efficiency) by the Pieri homotopies.

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

-------------------------------------------------------------------------------
file name                        : short description
-------------------------------------------------------------------------------
brackets                         : representation/manipulation of brackets
brackets_io                      : input/output of brackets
bracket_monomials                : monomials of brackets
bracket_monomials_io             : input/output of bracket monomials
generic_bracket_polynomials      : bracket polynomials over any coeff ring
standard_bracket_polynomials     : polynomials in the brackets, complex coeff
standard_bracket_polynomials_io  : input/output for bracket polynomials
dobldobl_bracket_polynomials     : bracket polynomials with double doubles
dobldobl_bracket_polynomials_io  : i/o for double double bracket polynomials
quaddobl_bracket_polynomials     : bracket polynomials with quad doubles
quaddobl_bracket_polynomials_io  : i/o for quad double bracket polynomials
bracket_polynomial_convertors    : converting bracket polynomials
standard_bracket_systems         : defines systems of bracket polynomials
standard_bracket_systems_io      : output for systems of bracket polynomials
dobldobl_bracket_systems         : systems of double double bracket polynomials
dobldobl_bracket_systems_io      : i/o for double double bracket systems
quaddobl_bracket_systems         : systems of quad double bracket polynomials
quaddobl_bracket_systems_io      : i/o for quad double bracket systems
straightening_syzygies           : implementation of straightening algorithm
bracket_expansions               : expansion of brackets
chebychev_polynomials            : Chebychev polynomials for the conjecture
osculating_planes                : input for the Shapiro^2 conjecture
complex_osculating_planes        : planes for variant of Shapiro^2 conjecture
matrix_homotopies                : management of homotopies between matrices
matrix_homotopies_io             : output for writing matrix homotopies
evaluated_minors                 : determinant computations
minor_computations               : computes all minors to check signs
maximal_minors                   : magnitude computation of maximal minors 
sagbi_homotopies                 : set up of the SAGBI homotopies
main_sagbi_homotopies            : main procedures to run SAGBI homotopies
-------------------------------------------------------------------------------
ts_subsets                       : test generation of subsets
ts_brackets                      : test manipulation of brackets 
ts_brackmons                     : test bracket monomials
ts_brackpols                     : test bracket polynomials
ts_straighten                    : test straightening algorithm
ts_expand                        : test expansion of brackets
ts_local                         : test localization
ts_cheby                         : test working with Chebychev polynomials
ts_topos                         : test on total positivity
ts_shapiro                       : test generation of input planes
ts_eremenko                      : test variant of generation of inputs
ts_detrock                       : test various root counts on (m,p)-system
ts_mathom                        : test on matrix homotopies
ts_sagbi                         : calls the driver to the SAGBI homotopies
-------------------------------------------------------------------------------
The equations in the Groebner homotopies are set up by means of the 
straightening algorithm.  Because the Groebner homotopies are not efficient,
they are only auxiliary to the set up of the SAGBI homotopies, which arise 
from expanding the brackets in the linear equations over the Grasmannian.

These homotopies were used to test some conjectures in real algebraic
geometry, formulated by the Shapiro brothers, with variants by Eremenko
and Gabrielov (who proved a version of the Shapiro conjectures).
