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This package joins with the package NCAlgebra.m
to add powerful automatic methods for handling systems of equations
in noncommuting variables.
All commutative algebra packages contain commands based on
Gröbner Bases and uses of them.
For example in Mathematica the Solve command
put systems of equations in a "cannonical"
form which for simple systems readily yields a solution.
Likewise the Mathematica Eliminate command tries to convert a system of
equations
in unknowns x(1), x(2), ..., x(n)
to a "triangular" form in unknowns,
that is, a new system of equations like
Here the polynomials {q(j) : j=1,..., k}
generate the same ideal that
the polynomials
{p(j) : 1<= j <= k} do. Therefore, the set of solutions to the
system the polynomial equations p(j) = 0 equals the set
of solutions to the system of polynomial equations
q(j) = 0.
This canonical form greatly simplifies
the task of solving systems of polynomial equations
by allowing one to backsolve for x(j) in terms of
{x(1),...,x(j-1)}.A brief mathematical discussion of Gröbner
bases appears in
the appendix in the NCGB document. The user who is not
acquainted at all with Gröbner Basis should still be able to
read and use much of the material which is contained within
this document.
In [FMora], c.f. [TMora], F. Mora described a version
of the Gröbner
basis algorithm which applied to noncommutative
free algebras.
We refer to this algorithm as Mora's algorithm.
It too puts systems of equations into a "cannonical form"
which we believe has considerable possibilities in the
noncommutative
case. This package implements the Mora algorithm and these
applications as well.
We think of this package as being useful for at least 4 things.