Nullstellensatz Linear Algebra and Border Spaces Algorithms
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To compile the program, type 'make' in the top level directory. This
builds three executables: 'polysys/border/border', 'polysys/nulla/nulla' and
'polysys/coloring/coloring'.  The program requires that gmp is already
installed with c++ enabled. You may need to modify 'polysys/Makefile.def' to
point to the location where gmp is installed if it is not in the system default
place.

polysys/border/border <filename>:
    This program runs a border space algorithm to compute the vector space
dimension of the quotient ring R/I. Here, R=F[x_1,...,x_n] where F is a field,
and I is an ideal in R. The field 'F', the number of variables 'n' and a set of
generators for the ideal I are given in the input file <filename>. There are
plenty of example input files in the directory 'polysys/test'. It should be
clear (I hope) from the examples what the input format is. Note that the field
is specified as 'FP', where P is the characteristic of the field.  If P is 0,
them the field is the rationals. 'P' must be prime at the moment.

polysys/nulla/nulla -dN [-p] <filename>:
    This program runs the NulLA algorithm for determining infeasibility of the
polynomial system given in <filename>.  The input file format is the same as
for 'border'. Here 'N' is the degree of the Nullstellenstaz certificate that we
wish to find. If the optional argument -p is present, the nulla will print out
the certificate of infeasibility if one exists.

polysys/coloring/coloring -kN -fP <filename>:
    This program prints to stdout a graph coloring ideal given a graph as
input. Here, 'N' is the coloring number to check (default is 3), and 'P' is the
characteristic of the field (default is 0).  The graph given in the input file
<filename> is given in the DIMACS input format. See 'polysys/coloring/test' for
input format examples. After creating the coloring ideal, you can run 'border'
or 'nulla' to determine the feasibility of the system. This program assumes
that the input graph is connected.
