Could you elaborate on this? I guess you have better examples, but here is
one for illustration:
(1) -> integrate(1/(1+x+x^3),x)
(1)
+---------------+
| 2 +--+
(\|- 93 %%E0 + 12 - \|31 %%E0)
*
log
+---------------+
+--+ +--+ | 2 2
(2 \|31 %%E0 + \|31 )\|- 93 %%E0 + 12 + 62 %%E0 - 31 %%E0
+
18 x - 4
+
+---------------+
| 2 +--+
(- \|- 93 %%E0 + 12 - \|31 %%E0)
*
log
+---------------+
+--+ +--+ | 2 2
(- 2 \|31 %%E0 - \|31 )\|- 93 %%E0 + 12 + 62 %%E0 - 31 %%E0
+
18 x - 4
+
+--+ 2
2 \|31 %%E0 log(- 62 %%E0 + 31 %%E0 + 9 x + 4)
/
+--+
2 \|31
Type:
Union(Expression(Integer),...)
(2) -> definingPolynomial %%E0
3
31 %%E0 - 3 %%E0 - 1
(2) ---------------------
31
Type:
Expression(Integer)
dim...@gmail.com schrieb am Dienstag, 20. Juli 2021 um 19:26:54 UTC+2:
> On Tue, Jul 20, 2021 at 11:23 AM 'Martin R' via sage-devel
> <sage-...@googlegroups.com> wrote:
> >
> > So, do you have an alternative idea on how to translate these results
> from FriCAS to sage? I guess the most interesting application is in
> symbolic integration, where rootOf objects appear frequently.
>
> As far as integration goes, I only know about cases where one takes
> the sum over the roots.
>
> >
> > Nils Bruin schrieb am Dienstag, 20. Juli 2021 um 02:42:46 UTC+2:
> >>
> >> On Monday, 19 July 2021 at 09:33:39 UTC-7 axio...@yahoo.de wrote:
> >>>
> >>> Dear Nils,
> >>>
> >>> please excuse my late reaction!
> >>>
> >>> I think FriCAS solves this problem as follows: the meaning of its
> rootOf object is "any root of the given (minimal) polynomial". Thus, such
> an object may only appear, when the choice of root does not matter, for
> example, as an integral.
> >>>
> >>> fricas.zerosOf returns a list of zeros of a given polynomial, which
> are given as interdependent rootOf objects. Thus, once you choose any root
> for the first, you get a different root for the second, and so on. Eg
> (sorry, ascii-art follows)
> >>>
> >>> (1) -> z := zerosOf(y^3 + y + a, y)
> >>>
> >>> +------------+ +------------+
> >>> | 2 | 2
> >>> \|- 3 %y0 - 4 - %y0 - \|- 3 %y0 - 4 - %y0
> >>> (1) [%y0, ---------------------, -----------------------]
> >>> 2 2
> >>> Type: List(Expression(Integer))
> >>> (2) -> (z.1)::INFORM
> >>>
> >>> (2) (rootOf (+ a (+ (^ %y0 3) %y0)) %y0)
> >>> Type: InputForm
> >>>
> >>>
> >>> This looks like a pragmatic approach to me.
> >>
> >> It depends a little bit what arithmetic you want how much intermingling
> of results you want to have. I know that maple has an "index" to rootof
> objects that allow you to at least track if two roots of the same
> polynomial are the same root or not necessarily the same one. What this
> basically gives you, if you have roots of polynomials f and g, is
> arithmetic in the ring k[x,y]/(f(x),g(y)). That ring maps onto any field
> extension you may get from adjoining roots of f,g (in an algebraic closure
> of k) to k, but it does not necessarily give you an isomorphism. For
> instance, if f(x)=x^2-2 and g(y)=y^2-8 then the ring Q[x,y]/(f(x),g(y)) has
> zero-divisors, reflecting that g(y) is not irreducible over k[x]/(f(x)).
> This is a fairly fundamental problem: if you really want to compute in a
> field, you'll need additional relations between the RootOf objects. Absent
> that information, indeed computing in this universal ring and accepting the
> zero-divisors is probably the most pragmatic thing to do. You may not even
> run into the zero divisors! However, if you trace back the origin of the
> objects the user creates, there will generally be sufficient information to
> determine the extra relations as well.
> >>
> >> Note that Fricas here makes use of additional information to construct
> the splitting field as a quadratic extension of a cubic one, so these are
> basically nested RootOfs. It does need to know that the (second) rootofs,
> displayed as square-roots here, that occur in the second and third root are
> in fact identical.
> >>
> >> This will be a little more obvious if you ask zerosOf(y^5+y+a,y).
> >>
> > --
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