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CommonDenominator for polynomials

Edit detail for CommonDenominator for polynomials revision 1 of 3

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Editor: 127.0.0.1 Time: 2007/11/02 07:50:23 GMT-7
Note: copied for axiom-developer

changed:
-
This package extends UnivariatePolynomialCommonDenominator for arbitrary polynomial categories. In fact, I don't understand why the original package is so restrictive.

\begin{spad}
)abbrev package PCDEN PolynomialCommonDenominator
PolynomialCommonDenominator(R, Q, P, E, VarSet): Exports == Impl where
  R : IntegralDomain
  Q : QuotientFieldCategory R
  E : OrderedAbelianMonoidSup
  VarSet: OrderedSet
  P: PolynomialCategory(Q, E,VarSet)

  Exports ==> with
    commonDenominator: P -> R
      ++ commonDenominator(q) returns a common denominator d for
      ++ the coefficients of q.
    clearDenominator : P -> P
      ++ clearDenominator(q) returns p such that \spad{q = p/d} where d is
      ++ a common denominator for the coefficients of q.
    splitDenominator : P -> Record(num: P, den: R)
      ++ splitDenominator(q) returns \spad{[p, d]} such that \spad{q = p/d} and d
      ++ is a common denominator for the coefficients of q.
 
  Impl ==> add
    import CommonDenominator(R, Q, List Q)
 
    commonDenominator p == commonDenominator coefficients p
 
    clearDenominator p ==
      d := commonDenominator p
      map(numer(d * #1)::Q, p)
 
    splitDenominator p ==
      d := commonDenominator p
      [map(numer(d * #1)::Q, p), d]
\end{spad}

Example use:

\begin{axiom}
)set mess type off
dom:=DMP([x,y], FRAC DMP([z],INT));
p:dom:=x*y^3/(z^2-1) + 3*x*y/(z^3-1)
commonDenominator p
clearDenominator p
splitDenominator p
\end{axiom}

This package extends UnivariatePolynomialCommonDenominator? for arbitrary polynomial categories. In fact, I don't understand why the original package is so restrictive.

spad

)abbrev package PCDEN PolynomialCommonDenominator
PolynomialCommonDenominator(R, Q, P, E, VarSet): Exports == Impl where
  R : IntegralDomain
  Q : QuotientFieldCategory R
  E : OrderedAbelianMonoidSup
  VarSet: OrderedSet
  P: PolynomialCategory(Q, E,VarSet)

  Exports ==> with
    commonDenominator: P -> R
      ++ commonDenominator(q) returns a common denominator d for
      ++ the coefficients of q.
    clearDenominator : P -> P
      ++ clearDenominator(q) returns p such that \spad{q = p/d} where d is
      ++ a common denominator for the coefficients of q.
    splitDenominator : P -> Record(num: P, den: R)
      ++ splitDenominator(q) returns \spad{[p, d]} such that \spad{q = p/d} and d
      ++ is a common denominator for the coefficients of q.

  Impl ==> add
    import CommonDenominator(R, Q, List Q)

    commonDenominator p == commonDenominator coefficients p

    clearDenominator p ==
      d := commonDenominator p
      map(numer(d * #1)::Q, p)

    splitDenominator p ==
      d := commonDenominator p
      [map(numer(d * #1)::Q, p), d]

spad

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/3743324978057018076-25px001.spad
      using old system compiler.
   PCDEN abbreviates package PolynomialCommonDenominator 
------------------------------------------------------------------------
   initializing NRLIB PCDEN for PolynomialCommonDenominator 
   compiling into NRLIB PCDEN 
   importing CommonDenominator(R,Q,List Q)
   compiling exported commonDenominator : P -> R
Time: 0.04 SEC.

   compiling exported clearDenominator : P -> P
Time: 0 SEC.

   compiling exported splitDenominator : P -> Record(num: P,den: R)
Time: 0 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |PolynomialCommonDenominator| REDEFINED

;;;     ***       |PolynomialCommonDenominator| REDEFINED
Time: 0 SEC.

   Cumulative Statistics for Constructor PolynomialCommonDenominator
      Time: 0.04 seconds

   finalizing NRLIB PCDEN 
   Processing PolynomialCommonDenominator for Browser database:
--->-->PolynomialCommonDenominator(constructor): Not documented!!!!
--------(commonDenominator (R P))---------
--------(clearDenominator (P P))---------
--------(splitDenominator ((Record (: num P) (: den R)) P))---------
--->-->PolynomialCommonDenominator(): Missing Description
; compiling file "/var/aw/var/LatexWiki/PCDEN.NRLIB/PCDEN.lsp" (written 22 JUL 2013 03:52:16 PM):

; /var/aw/var/LatexWiki/PCDEN.NRLIB/PCDEN.fasl written
; compilation finished in 0:00:00.027
------------------------------------------------------------------------
   PolynomialCommonDenominator is now explicitly exposed in frame 
      initial 
   PolynomialCommonDenominator will be automatically loaded when needed
      from /var/aw/var/LatexWiki/PCDEN.NRLIB/PCDEN

Example use:

fricas

)set mess type off

dom:=DMP([x,y], FRAC DMP([z],INT));

p:dom:=x*y^3/(z^2-1) + 3*x*y/(z^3-1)

$\label{eq1}{{1 \over{{{z}^{2}}- 1}}\ x \ {{y}^{3}}}+{{3 \over{{{z}^{3}}- 1}}\ x \ y}$

(1)

fricas

commonDenominator p

$\label{eq2}{{z}^{4}}+{{z}^{3}}- z - 1$

(2)

fricas

clearDenominator p

$\label{eq3}{{\left({{z}^{2}}+ z + 1 \right)}\ x \ {{y}^{3}}}+{{\left({3 \ z}+ 3 \right)}\ x \ y}$

(3)

fricas

splitDenominator p

$\label{eq4}\begin{array}{@{}l} \displaystyle \left[{num ={{{\left({{z}^{2}}+ z + 1 \right)}\ x \ {{y}^{3}}}+{{\left({3 \ z}+ 3 \right)}\ x \ y}}}, \: \right. \ \ \displaystyle \left.{den ={{{z}^{4}}+{{z}^{3}}- z - 1}}\right]$

(4)