Bill Page wrote:
>
> > On 10/22/18 9:55 AM, Waldek Hebisch wrote:
>
> > > More generally, factorization via equation solving directly
> > > gives absolute factorization, that is factorization over algebraic
> > > closure of base field. To get factorization over base field
> > > one needs to recombine factors.
>
> I am not convinced that this is the case.
>
> > > IIUC 'xdpolyf1.spad' tries
> > > various tricks to avoid algebraic extentions, but this is
> > > very unlikely to work in general.
> > >
>
> The tricks in xdpolyf1 having nothing to do with avoiding algebraic
> extensions. radicalSolve is only called if the base ring has
> RadicalCategory, otherwise SystemSolvePackage only looks for solutions
> in fraction field and xdpolyf1 lifts that solution to the base ring.
>
> Are you claiming that this does not provide the most general
> factorization in the base ring? Are you suggesting that if one looked
> for factorizations over the algebraic closure and then recombined some
> factors it might be possible to general polynomials over the base ring
> that are not considered when directly solving the factorization
> equations over the fraction field?
If you could find solution _in the fraction field_ then the
method would be fine. However, in general finding rational
solutions to polynomial system of equations is uncomputable.
Groebner bases decide if there are solutions in algebraic
closure, but you may have algebraic solutions without
rational solutions. If you say that you can find out
if there is rational solution (= factorization) you should
better justify this and explain what special properties
of system you use.
Classical factorization methods (in commutative case)
avoid uncomputability by recombining algebraic factors.
If you do not want recombination you should say how
you avoid uncomputability.
To be clear: your code apparently makes some assumptions.
Both theory (uncomputablity in general case) an practice
suggest that those assumptions may fail unless we can
prove them.
--
Waldek Hebisch
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