> The important thing isn't what algorithm is implemented, but that the
> result is fast(er than Magma).

The important thing is whether users can hope to get an answer at all.
The old Singular factoring code was hopeless and I bet multivariate
gcd
is still hopeless.  And what about correct ?  The new code you linked
to looks like a probabilistic algorithm.  There's no point in being
fast if you're wrong, and developing good routines will take months.
I simply wanted to point out the existence of what appears to be
correct routines, even if they are slow.

Also, to Joel Mohler:
You need "Algorithms for Computer Algebra" by Geddes, Czapor, and
Labahn:
Chapter 5: Chinese Remainder Theorem
Chapter 6: Newton's Iteration and Hensel Lifting
Chapter 7: Polynomial GCD
Chapter 8: Polynomial Factorization

What you are trying to do is not a "weeks long" project, it is one
of the central achievements of the entire field.  It took a
decade
to do the first time, so don't expect to have "industrial strength"
routines soon.  It will realistically take months of full time work.
There's about 100 pages of material in that book, when you take out
the exercises, etc.  You need it all.

The people on this list seem to hilariously underestimate the depth
of this problem, and that concerns me.  I want Sage to succeed, and
you can't with that attitude.  This is a massive undertaking, and
if you treat it like it's not it is ultimately very discouraging.

I'd spell it all out: the things you need to do, the problems and
subproblems, but it's easier and better if you just read the book.
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