Re: [Axiom-developer] Bug in GeneralizedMultivariateFactorize?

[Waldek Hebisch]

Stephen Wilson wrote:
> Is it common enough to encounter integral domains which are not UFD's
> yet admit a computable extension to a field?
>

Well, naive users probably use only things like Integer,
Expression Integer and polynomials over them -- given that they
will probably never hit an integral domain which is not an UFD.

But very simple number theory gives you an integral domain
which is not UFD, namely extend integers adding sqrt(-5).
The resulting ring is not UFD, while the field of fractions
is quite nice.  In general, algebraic extensions of integers
rarely are UFD.

Another example is ring of real trigonometic polynomials, namely
finite sums of the form:

sum a_k*sin(k*x) + sum b_k*cos(k*x)

where a_k and b_k are real (note that if you take _complex_
coefficients you get UFD).

Such ring appear naturally if you try to do computations in
smallest possible ring (and for reasons of efficiency we
want to compute in small rings).


> If so, then can we not use both squarefree factorization and
> Berlekamps algorithm to obtain a complete factorization, apply Hensel
> lifting, and do (in the most general case) an exponential algorithm to
> recombine the factors?
> 

This does not work for rings: modular methods essentially work
on monic polynomials, so you have problems with constant factors
(this problem vanishes if you pass to the field).  Also, at least
theoretically when you try to recombine factors you can get
more factors when you work over field of quotients and there
is no longer clear that factors will combine in the ring in a
unique way.


-- 
                              Waldek Hebisch
address@hidden