On Tue, 5 Jun 2018, Bill Page wrote:
the domain NCPOLY is only an Algebra, not a field.
NCPOLY(VAR, F) implements the algebra of noncommuting polynomials over a
field F AKA the free associative algebra; mathematically this is exactly
the same as XDPOLY(VARS, F).
However in this approach polynomials are not represented as a sum of
monomials but rather as Schur complements of linear matrix pencils
(a special case of what is called quasideterminants by Gelfand et al).
In the case of a univariate polynomial this boils down to its companion
matrix.
This representation may seem to be an overkill for polynomials, but is
essential when you want to pass to the quotient division algebra, which is
called the free field, and for which this is the only reasonable
representation we have.
But as Konrad has shown, there is a benefit even for polynomials, as it
provides a factorization algorithm over algebraically closed fields, by
reducing it to a system of algebraic equations, which is amenable to the
methods of algebraic geometry.
Even in the univariate case there is a similar benefit:
it is numerically much more efficient to compute the zeros of a univariate
polynomial via the eigenvalues of its companion matrix rather than
directly via bisection or Newton iteration.
Yes, I think that is Konrad's intention. The main issue as I
understand it is minimization,
yes, this is one of the main issues in this field.
Franz
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