Hello Konrad,
I did make a start on setting up a test harness for your algorithm;
all I need to add is the statistics recording; but got sidetracking with
Bill's algorithms. Just building my understanding.
Taking something out of context, you mention that a quiting
criterion for "T" is when it's impossible to factor T=U*V. This is what
the recursion algorithm does. I now understand what Cohn is doing;
although I couldn't pass a test on the real internal material.
I can't afford Mora's volumes; but I have found Cohn's "Free Rings
and Their Relations" and can afford that if you think it would help me.
I do have one (possibly more) of his papers as well. His caveat about
some expressions (in particular rectangular matrices) not being a UFD is
a little disturbing unless one's goal is /a factorization. /I am
presuming that he isn't talkin/g /about things like/
p=x+xyx=x(1+yx)=(1+xy)*x
/Which I got and didn't recognize./
/I am not saying your papers have anything wrong or too hard. I just
like to look into the history of mathematics (even current) to provide
context in my mind. One side of my brain loves the itty-bitty details
and the other wants to know how things came to be./
/ I haven't examined your algorithm in detail but it seems to me that
reducing to ALS, on a term by term bases might be easier that just
extending up matrices with the rules given. Unless the rules of
adding/multiplying to an ALS by a new term automatically produces a new
ALS. Of course, I am also assuming the commutation rules:
ALS(X+Y)=ALS(X+ALS(Y)), etc.. Is this true?
I don't understand your last paragraph. I think I managed to
extend the meaning of GCD to fractions in a manner to keep a large
number of properties (obviously not all :-)) at one time. I don't know
if it's relevant though.
I hope you forgive my ignorance of these matters. It's true, I am
ignorant.:'(
Enjoy
Ray
On 07/20/2018 05:10 AM, Konrad Schrempf wrote:
Dear Bill,
just recently I had a look into the two volumes of
Teo Mora on solving polynomial systems of equations.
There is quite some theory one could consider ;-).
Right now I just would like to add a comment about
more general base rings (rather domains because zero
devisors do not simplify things): The ansatz will
work if we add non-proper left and right subwords.
A simple example is p=6-2x (with integer coefficients),
which is REDUCIBLE because p = 2*(3-x); the only units
in Z are {+1,-1}.
In the case of a (base) field however, ALL elements
of zero degree are units so one can remove extra
dimensions in the solution variety a priori.
(About the latter --in a different context-- Franz
did some nice experiments in FriCAS. Keyword: Primary
Decomposition of ideals.)
Regards,
Konrad
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