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Annotation of OpenXM_contrib/PHC/Ada/Root_Counts/READ_ME, Revision 1.1.1.1

1.1       maekawa     1: The root-counting library in PHCv2 is organized as follows :
                      2:
                      3:   Root_Counts          : 4. root counts and homotopy construction
                      4:      |-- Product       : 4.1. linear-product start systems
                      5:      |-- Implift       : 4.2. implicit lifting
                      6:      |-- Stalift       : 4.3. static lifting
                      7:      |-- Dynlift       : 4.4. dynamic lifting
                      8:      |-- Symmetry      : 4.5. exploitation of symmetry relations
                      9:
                     10: The root counts that are available in the directory Product are
                     11: based on Bezout's theorem.  The corresponding start systems are
                     12: in general linear-product systems.
                     13:
                     14: Implicit lifting is the name we gave to the algorithm Bernshtein
                     15: used in his proof that the mixed volume of the Newton polytopes
                     16: of a polynomial system is a generically exact upper bound on the
                     17: number of its isolated complex solutions with all components different
                     18: from zero.
                     19:
                     20: Static lifting is the general procedure to compute mixed volumes
                     21: of polytopes.  Subdivisions induced by an integer-valued or
                     22: floating-point lifting functions can be computed.
                     23: In order to deal with non-fine subdivisions induced by a nongeneric
                     24: integer lifting, recursive algorithms have been implemented.
                     25:
                     26: Dynamic lifting allows to have a control of the lifting values to
                     27: obtain a numerically stable polyhedral continuation.
                     28: When some or all supports are equal, the Cayley trick is
                     29: recommended to use.
                     30:
                     31: The Symmetry library provides routines to construct start systems
                     32: that are invariant under a given permutation symmetry.
                     33: Hereby symmetric homotopies can be constructed, so that only the
                     34: generating solution paths need to be computed.