Re: [Axiom-mail] How to expand a fraction (like Maple does with 'expand')?

[Martin Rubey]

Dear Thomas,

although for the end-user it probably doesn't matter, I'm afraid that the
impression the mails by Bill Page and Francois Maltey give is not quite
"complete".

* in Axiom, it is very important to distinguish between internal representation
  and output.

* "expand" as provided by Maple, Mathematica, etc., is mainly about output, the
  internal representation doesn't matter too much there. It can be interpreted
  as: display the thing with everything multiplied out.

* in Axiom, if you want to have a different output, it is best to write a
  wrapper domain.

* it is very easy to write such a domain in Axiom. In Aldor, it would be even
  easier, but unfortunately, currently the necessary functionality ("extend")
  is not provided by the Axiom-Aldor interface.

Here goes the domain. Put it into a file, for example myexpr.spad, type

)co myexpr.spad

on the axiom command prompt

and 

(((-r1*r2*uoff)+((r2+r1)*r3+r1*r2)*ue)/(r2*r3))::DEXPR INT

should return

        r1 ue   r1 uoff   r1 ue
        ----- - ------- + ----- + ue
          r3       r3       r2


---myexpr.spad ----------------------------------------------------------------
)abb domain DEXPR DistributedExpression
DistributedExpression(R: Join(Ring, OrderedSet)): Exports == Implementation 
where
  AN  ==> AlgebraicNumber
  SUP    ==> SparseUnivariatePolynomial

  EXPRR ==> Expression R

  Exports == FunctionSpace R with
    if R has IntegralDomain then
      AlgebraicallyClosedFunctionSpace R
      TranscendentalFunctionCategory
      CombinatorialOpsCategory
      LiouvillianFunctionCategory
      SpecialFunctionCategory
      reduce: % -> %
        ++ reduce(f) simplifies all the unreduced algebraic quantities
        ++ present in f by applying their defining relations.
      number?: % -> Boolean
        ++ number?(f) tests if f is rational
      simplifyPower: (%,Integer) -> %
        ++ simplifyPower?(f,n) \undocumented{}
      if R has GcdDomain then
        factorPolynomial : SUP  % -> Factored SUP %
           ++ factorPolynomial(p) \undocumented{}
        squareFreePolynomial : SUP % -> Factored SUP %
           ++ squareFreePolynomial(p) \undocumented{}
      if R has RetractableTo Integer then RetractableTo AN

  Implementation == EXPRR add

    Rep := EXPRR

    out: (Polynomial R, %, List %, List %) -> OutputForm 
-- coerces the polynomial to OutputForm completely expanded and replaces the
-- variables in vl with the kernels in kl
    out(p, q, kl, vl) == 
      ex: Fraction Rep := (eval(leadingMonomial(p)::%, vl, kl)::Rep)/(q::Rep)
      if reductum p = 0 
      then coerce(ex)$Fraction(Rep)
      else coerce(ex)$Fraction(Rep) _
           + out(reductum p, q, kl, vl)

    coerce(ex:%):OutputForm == 
      kl := kernels ex
      vl: List % := [subscript('x, [i::OutputForm])::Symbol::% for i in 1..#kl]
      ex1: % := subst(ex, kl, vl)$%
      kl1 := map(coerce(#1)$%, kl)$ListFunctions2(Kernel %, %)
      if R has IntegralDomain then
          out(retract(numerator ex1)@Polynomial(R), 
              denominator eval(ex1, vl, kl1), kl1, vl)
      else
        out(retract(ex1)@Polynomial(R), 1, kl1, vl)
---myexpr.spad ----------------------------------------------------------------

A note for Thomas: the only bit of code that required some work was "out" and
"coerce". Everything else (i.e., everything before and including "Rep:=EXPRR")
is copy and paste. If the Aldor interface would work properly, it wouldn't be
there.

I'm sure that "out" and "coerce" could be simplified and improved, too. I just
didn't bother.

A note for axiom developers:

In my opinion, this is not the right approach. The right approach would be to
generalize DMP and to allow arbitrary variables, just as SMP does.

Martin