Edit detail for Cylindrical Algebraic Decomposition revision 1 of 4

On Wednesday, October 12, 2005 5:29 PM Renaud Rioboo wrote:

Dear Axiom fans,

In the last few years there have been a regain of interest in some Computer Algebra users circles about Cylindrical Algebraic Decomposition (aka CAD). As some of you may know my thesis was about CAD and I am regularly receiving queries about CAD and the Axiom implementation I wrote.

I already privately gave sources or ran particular problems for some people but I have never officially released my sources because I do not think they are at production level and many work has still to be done on them.

Recently Martin Rubey thought it could be a good idea to release them and I have of course no objection on that point nor on sharing these sources with the community. If there is some interest on it you may include it into the Axiom distribution with the same permissions than the rest of the software.

While I wait for my lab to give me the necessary permissions to be able to use Axiom and export this software you may find it's sources at the unlisted url

http://rioboo.free.fr/CadPub/

I would like to add just a few words about that work which is a straightforward implementation of the standard papers by Arnon, Collins and McCallum? which are more than 20 years old now.

My work on Cad is 15 years old and was developped for the defense of my thesis where CAD was an sample application for real algebraic numbers manipulations. By the time I wrote it I also needed some subresultant calculations which were not in Axiom (Scratchpad by that time) and the required machinery to work with real algebraic numbers.

Along the times, real algebraic numbers were included into Axiom and subresultant calculations which are used in several places of the Axiom Library were modified to all use Lionel Ducos's package which enables a performance gain. While making these modifications to Axiom I tried to keep my CAD package up-to-date with the Axiom Library and have used it as a test for other packages. It thus should compile under recent Axiom versions and should compute something that resembles a CAD.

Don't expect this package to be the absolute CAD program, I never found time to further work on it in order to include many enhancements which are present in other CAD packages. While Axiom versions evolved I had to remove what I thought were enhancements and remove some support for generalization. As of today it only includes one projection method which is the McCallum? projection and no reconstruction nor adjacency's are taken into account.

However it did reasonabily compare with more recent techniques such as Rational Univariate Representation (aka RUR) for simple cases though it could not produce results for some more complicated cases. By the time this comparison was made Axiom had some severe memory restrictions which seem to have disappear now. I thus think that many objections to using CAD could now be revised.

For hard problems this package should thus not be worse than any other CAD package except that you cannot fall into an optimzation drawback :-) There is no optimization !

I will of course try to maintain the package to the best of my ability. One thing I should do is include comments describing the different functions, but even that requires going into the code and the algorithms which are quite old for me now.

Let me know if there is some interest on it.

Best regards,

Renaud

)ab pack CADU CylindricalAlgebraicDecompositionUtilities

CylindricalAlgebraicDecompositionUtilities(R,P) : PUB == PRIV where

-- Tese are some standard tools which are needed to compute with univariate
-- polynomials. 
--
-- A gcd basis for a set of polynomials is a set of pairwise relatively prime 
-- polynomials which all divide the original set and whose product is the
-- same than the product of the original set.
--
-- A square free basis for a list of polynomials is just a list
-- of square free polynomials which are pairwise relatively primes and have
-- the same roots than the original polynomials.

  R : GcdDomain
  P : UnivariatePolynomialCategory(R)

  PUB == with
      squareFreeBasis : List(P) -> List(P)
        ++ 
      gcdBasis : List(P) -> List(P)
        ++ decompose a list of polynomials into pairwise relatively 
        ++ prime polynomials
      gcdBasisAdd : (P,List(P)) -> List(P)
        ++ add one polynomial to list of pairwise relatively prime polynomials

  PRIV == add

     squareFreeBasis(lpols) ==
       lpols = [] => []
       pol := first(lpols)
       sqpol := unitCanonical(squareFreePart(pol))
       gcdBasis(cons(sqpol,squareFreeBasis(rest(lpols))))

     gcdBasisAdd(p,lpols) ==
       (degree(p) = 0) => lpols
       null lpols => [unitCanonical p]
       p1 := first(lpols)
       g := gcd(p,p1)
       (degree(g) = 0) => cons(p1,gcdBasisAdd(p,rest lpols))
       p := (p exquo g)::P
       p1 := (p1 exquo g)::P
       basis := gcdBasisAdd(p,rest(lpols))
       if degree(p1) > 0 then basis := cons(p1,basis)
       gcdBasisAdd(g,basis)

     gcdBasis(lpols) ==
       (#lpols <= 1) => lpols
       basis := gcdBasis(rest lpols)
       gcdBasisAdd(first(lpols),basis)

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/4986109030324487666-25px001.spad
      using old system compiler.
   CADU abbreviates package CylindricalAlgebraicDecompositionUtilities 
------------------------------------------------------------------------
   initializing NRLIB CADU for CylindricalAlgebraicDecompositionUtilities 
   compiling into NRLIB CADU 
   compiling exported squareFreeBasis : List P -> List P
Time: 0.05 SEC.

   compiling exported gcdBasisAdd : (P,List P) -> List P
Time: 0 SEC.

   compiling exported gcdBasis : List P -> List P
Time: 0.01 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |CylindricalAlgebraicDecompositionUtilities| REDEFINED

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

   Cumulative Statistics for Constructor CylindricalAlgebraicDecompositionUtilities
      Time: 0.06 seconds

   finalizing NRLIB CADU 
   Processing CylindricalAlgebraicDecompositionUtilities for Browser database:
--->-->CylindricalAlgebraicDecompositionUtilities(constructor): Not documented!!!!
--------(squareFreeBasis ((List P) (List P)))---------
--------(gcdBasis ((List P) (List P)))---------
--->-->CylindricalAlgebraicDecompositionUtilities((gcdBasis ((List P) (List P)))): Improper first word in comments: decompose
"decompose a list of polynomials into pairwise relatively prime polynomials"
--------(gcdBasisAdd ((List P) P (List P)))---------
--->-->CylindricalAlgebraicDecompositionUtilities((gcdBasisAdd ((List P) P (List P)))): Improper first word in comments: add
"add one polynomial to list of pairwise relatively prime polynomials"
--->-->CylindricalAlgebraicDecompositionUtilities(): Missing Description
; compiling file "/var/aw/var/LatexWiki/CADU.NRLIB/CADU.lsp" (written 14 JUL 2013 10:28:17 AM):

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

spad

)ab domain SCELL SimpleCell

This domain is made to work with one-dimensional semi-algebraic cells -- ie either an algebraic point, either an interval. The point is specified as -- specification of an algebraic value.
SimpleCell(TheField,ThePols) : PUB == PRIV where TheField : RealClosedField ThePols : UnivariatePolynomialCategory(TheField) O ==> OutputForm B ==> Boolean Z ==> Integer N ==> NonNegativeInteger
-- VARS ==> VariationsPackage(TheField,ThePols) VARS ==> RealPolynomialUtilitiesPackage(TheField,ThePols) LF ==> List(TheField)
PUB == CoercibleTo(O) with
allSimpleCells : (ThePols,Symbol) -> List % allSimpleCells : (List(ThePols),Symbol) -> List % hasDimension? : % -> B samplePoint : % -> TheField stablePol : % -> ThePols variableOf : % -> Symbol separe : (LF,TheField,TheField) -> LF pointToCell : (TheField,B,Symbol) -> %
PRIV == add
Rep := Record(samplePoint:TheField, hasDim:B, varOf:Symbol)
samplePoint(c) == c.samplePoint
stablePol(c) == error "Prout"
hasDimension?(c) == c.hasDim
variableOf(c) == c.varOf
coerce(c:%):O == o : O := ((c.varOf)::O) = ((c.samplePoint)::O) brace [o,(c.hasDim)::O]
separe(liste,gauche,droite) == milieu : TheField := (gauche + droite) / (2::TheField) liste = [] => [milieu] #liste = 1 => [gauche,first(liste),droite] nbe := first(liste) lg :List(TheField) := [] ld :List(TheField) := rest(liste) sg := sign(milieu-nbe) while sg > 0 repeat lg := cons(nbe,lg) ld = [] => return(separe(reverse(lg),gauche,milieu)) nbe := first(ld) sg := sign(milieu-nbe) ld := rest(ld) sg < 0 => append(separe(reverse(lg),gauche,milieu), rest(separe(cons(nbe,ld),milieu,droite))) newDroite := (gauche+milieu)/(2::TheField) null lg => newGauche := (milieu+droite)/(2::TheField) while newGauche >= first(ld) repeat newGauche := (milieu+newGauche)/(2::TheField) append([gauche,milieu],separe(ld,newGauche,droite)) while newDroite <= first(lg) repeat newDroite := (newDroite+milieu)/(2::TheField) newGauche := (milieu+droite)/(2::TheField) null ld => append(separe(reverse(lg),gauche,newDroite),[milieu,droite]) while newGauche >= first(ld) repeat newGauche := (milieu+newGauche)/(2::TheField) append(separe(reverse(lg),gauche,newDroite), cons(milieu,separe(ld,newGauche,droite)))
pointToCell(sp,hasDim?,varName) == [sp,hasDim?,varName]$Rep
allSimpleCells(p:ThePols,var:Symbol) == allSimpleCells([p],var)
PACK ==> CylindricalAlgebraicDecompositionUtilities(TheField,ThePols) allSimpleCells(lp:List(ThePols),var:Symbol) == lp1 := gcdBasis(lp)$PACK null(lp1) => [pointToCell(0,true,var)] b := ("max" / [ boundOfCauchy(p)$VARS for p in lp1 ])::TheField l := "append" / [allRootsOf(makeSUP(unitCanonical(p))) for p in lp1] l := sort(l) l1 := separe(l,-b,b) res : List(%) := [pointToCell(first(l1),true,var)] l1 := rest(l1) while not(null(l1)) repeat res := cons(pointToCell(first(l1),false,var),res) l1 := rest(l1) l1 = [] => return(error "Liste vide") res := cons(pointToCell(first(l1),true,var),res) l1 := rest(l1) reverse! res

spad

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/9165941727097331838-25px002.spad
      using old system compiler.
   SCELL abbreviates domain SimpleCell 
------------------------------------------------------------------------
   initializing NRLIB SCELL for SimpleCell 
   compiling into NRLIB SCELL 
   compiling exported samplePoint : $ -> TheField
      SCELL;samplePoint;$TheField;1 is replaced by QVELTc0 
Time: 0.01 SEC.

   compiling exported stablePol : $ -> ThePols
      SCELL;stablePol;$ThePols;2 is replaced by errorProut 
Time: 0 SEC.

   compiling exported hasDimension? : $ -> Boolean
      SCELL;hasDimension?;$B;3 is replaced by QVELTc1 
Time: 0 SEC.

   compiling exported variableOf : $ -> Symbol
      SCELL;variableOf;$S;4 is replaced by QVELTc2 
Time: 0 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0 SEC.

   compiling exported separe : (List TheField,TheField,TheField) -> List TheField
Time: 0.02 SEC.

   compiling exported pointToCell : (TheField,Boolean,Symbol) -> $
      SCELL;pointToCell;TheFieldBS$;7 is replaced by VECTOR 
Time: 0.01 SEC.

   compiling exported allSimpleCells : (ThePols,Symbol) -> List $
Time: 0 SEC.

   processing macro definition PACK ==> CylindricalAlgebraicDecompositionUtilities(TheField,ThePols) 
   compiling exported allSimpleCells : (List ThePols,Symbol) -> List $
Time: 0.02 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |SimpleCell| REDEFINED

;;;     ***       |SimpleCell| REDEFINED
Time: 0.01 SEC.

   Cumulative Statistics for Constructor SimpleCell
      Time: 0.07 seconds

   finalizing NRLIB SCELL 
   Processing SimpleCell for Browser database:
--->-->SimpleCell(constructor): Not documented!!!!
--->-->SimpleCell((allSimpleCells ((List %) ThePols (Symbol)))): Not documented!!!!
--->-->SimpleCell((allSimpleCells ((List %) (List ThePols) (Symbol)))): Not documented!!!!
--->-->SimpleCell((hasDimension? ((Boolean) %))): Not documented!!!!
--->-->SimpleCell((samplePoint (TheField %))): Not documented!!!!
--->-->SimpleCell((stablePol (ThePols %))): Not documented!!!!
--->-->SimpleCell((variableOf ((Symbol) %))): Not documented!!!!
--->-->SimpleCell((separe ((List TheField) (List TheField) TheField TheField))): Not documented!!!!
--->-->SimpleCell((pointToCell (% TheField (Boolean) (Symbol)))): Not documented!!!!
--->-->SimpleCell(): Missing Description
; compiling file "/var/aw/var/LatexWiki/SCELL.NRLIB/SCELL.lsp" (written 14 JUL 2013 10:28:17 AM):

; /var/aw/var/LatexWiki/SCELL.NRLIB/SCELL.fasl written
; compilation finished in 0:00:00.076
------------------------------------------------------------------------
   SimpleCell is now explicitly exposed in frame initial 
   SimpleCell will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/SCELL.NRLIB/SCELL

)ab domain CELL Cell

Cell(TheField) : PUB == PRIV where
  TheField : RealClosedField

  ThePols ==> Polynomial(TheField)

  O           ==> OutputForm
  B           ==> Boolean
  Z           ==> Integer
  N           ==> NonNegativeInteger
  BUP         ==> SparseUnivariatePolynomial(TheField)
  SCELL       ==> SimpleCell(TheField,BUP)

  PUB == CoercibleTo(O) with

     samplePoint : % -> List(TheField)
     dimension : % -> N
     hasDimension? :  (%,Symbol) -> B
     makeCell : List(SCELL) -> %
     makeCell : (SCELL,%) -> %
     mainVariableOf : % -> Symbol
     variablesOf : % -> List Symbol
     projection : % -> Union(%,"failed")

  PRIV == add

    Rep := List(SCELL)

    coerce(c:%):O == 
      paren [sc::O for sc in c]

    projection(cell) ==
      null cell => error "projection: should not appear"
      r := rest(cell)
      null r => "failed"
      r

    makeCell(l:List(SCELL)) == l

    makeCell(scell,toAdd) == cons(scell,toAdd)

    mainVariableOf(cell) == 
      null(cell) => 
        error "Should not appear"
      variableOf(first(cell))

    variablesOf(cell) ==
      null(cell) => []
      cons(mainVariableOf(cell),variablesOf(rest(cell)::%))

    dimension(cell) ==
      null(cell) => 0
      hasDimension?(first(cell)) => 1+dimension(rest(cell))
      dimension(rest(cell))

    hasDimension?(cell,var) ==
      null(cell) => 
        error "Should not appear"
      sc : SCELL := first(cell)
      v := variableOf(sc)
      v = var => hasDimension?(sc)
      v < var => false
      v > var => true
      error "Caca Prout"

    samplePoint(cell) ==
      null(cell) => []
      cons(samplePoint(first(cell)),samplePoint(rest(cell)))

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/6660987790622940247-25px003.spad
      using old system compiler.
   CELL abbreviates domain Cell 
------------------------------------------------------------------------
   initializing NRLIB CELL for Cell 
   compiling into NRLIB CELL 
   compiling exported coerce : $ -> OutputForm
Time: 0 SEC.

   compiling exported projection : $ -> Union($,failed)
Time: 0 SEC.

   compiling exported makeCell : List SimpleCell(TheField,SparseUnivariatePolynomial TheField) -> $
      CELL;makeCell;L$;3 is replaced by l 
Time: 0 SEC.

   compiling exported makeCell : (SimpleCell(TheField,SparseUnivariatePolynomial TheField),$) -> $
      CELL;makeCell;Sc2$;4 is replaced by CONS 
Time: 0 SEC.

   compiling exported mainVariableOf : $ -> Symbol
Time: 0 SEC.

   compiling exported variablesOf : $ -> List Symbol
Time: 0 SEC.

   compiling exported dimension : $ -> NonNegativeInteger
Time: 0 SEC.

   compiling exported hasDimension? : ($,Symbol) -> Boolean
Time: 0 SEC.

   compiling exported samplePoint : $ -> List TheField
Time: 0.01 SEC.

(time taken in buildFunctor:  0)

;;;     ***       |Cell| REDEFINED

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

   Cumulative Statistics for Constructor Cell
      Time: 0.01 seconds

   finalizing NRLIB CELL 
   Processing Cell for Browser database:
--->-->Cell(constructor): Not documented!!!!
--->-->Cell((samplePoint ((List TheField) %))): Not documented!!!!
--->-->Cell((dimension ((NonNegativeInteger) %))): Not documented!!!!
--->-->Cell((hasDimension? ((Boolean) % (Symbol)))): Not documented!!!!
--->-->Cell((makeCell (% (List (SimpleCell TheField (SparseUnivariatePolynomial TheField)))))): Not documented!!!!
--->-->Cell((makeCell (% (SimpleCell TheField (SparseUnivariatePolynomial TheField)) %))): Not documented!!!!
--->-->Cell((mainVariableOf ((Symbol) %))): Not documented!!!!
--->-->Cell((variablesOf ((List (Symbol)) %))): Not documented!!!!
--->-->Cell((projection ((Union % failed) %))): Not documented!!!!
--->-->Cell(): Missing Description
; compiling file "/var/aw/var/LatexWiki/CELL.NRLIB/CELL.lsp" (written 14 JUL 2013 10:28:17 AM):

; /var/aw/var/LatexWiki/CELL.NRLIB/CELL.fasl written
; compilation finished in 0:00:00.029
------------------------------------------------------------------------
   Cell is now explicitly exposed in frame initial 
   Cell will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/CELL.NRLIB/CELL

)ab pack CAD CylindricalAlgebraicDecompositionPackage

CylindricalAlgebraicDecompositionPackage(TheField) : PUB == PRIV where

  TheField : RealClosedField

  ThePols ==> Polynomial(TheField)
  P ==> ThePols
  BUP ==> SparseUnivariatePolynomial(TheField)
  RUP ==> SparseUnivariatePolynomial(ThePols)

  Z           ==> Integer
  N           ==> NonNegativeInteger

  CELL ==> Cell(TheField)
  SCELL ==> SimpleCell(TheField,BUP)

  PUB == with

      cylindricalDecomposition: List P ->        List CELL
      cylindricalDecomposition: (List(P),List(Symbol)) ->        List CELL
      projectionSet: (List RUP)                ->    List P
      coefficientSet: RUP                ->    List P
      discriminantSet : List RUP -> List(P)
      resultantSet :  List RUP -> List P 
      principalSubResultantSet : (RUP,RUP) -> List P
      specialise : (List(ThePols),CELL) -> List(BUP)

  PRIV == add

     cylindricalDecomposition(lpols) ==
       lv : List(Symbol) := []
       for pol in lpols repeat
         ground?(pol) => "next pol"
         lv := removeDuplicates(append(variables(pol),lv))
       lv := reverse(sort(lv))
       cylindricalDecomposition(lpols,lv)

     cylindricalDecomposition(lpols,lvars) ==
       lvars = [] => error("CAD: cylindricalDecomposition: empty list of vars")
       mv := first(lvars)
       lv := rest(lvars)
       lv = [] =>
         lp1 := [ univariate(pol) for pol in lpols ]
         scells := allSimpleCells(lp1,mv)$SCELL
         [ makeCell([scell]) for scell in scells ]
       lpols1 := projectionSet [univariate(pol,mv) for pol in lpols]
       previousCad := cylindricalDecomposition(lpols1,lv)
       res : List(CELL) := []
       for cell in previousCad repeat
         lspec := specialise(lpols,cell)
         scells := allSimpleCells(lspec,mv)
         res := append(res,[makeCell(scell,cell) for scell in scells])
       res

     PACK1 ==> CylindricalAlgebraicDecompositionUtilities(ThePols,RUP)
     PACK2 ==> CylindricalAlgebraicDecompositionUtilities(TheField,BUP)

     specialise(lpols,cell) ==
       lpols = [] => error("CAD: specialise: empty list of pols")
       sp := samplePoint(cell)
       vl := variablesOf(cell)
       res : List(BUP) := []
       for pol in lpols repeat
         p1 := univariate(eval(pol,vl,sp))
--         zero?(p1) => return(error "Bad sepcialise")
         degree(p1) = 0 => "next pol"
         res := cons(p1,res)
       res

     coefficientSet(pol) ==
       res : List(ThePols) := []
       for c in coefficients(pol) repeat
         ground?(c) => return(res)
         res := cons(c,res)
       res

     SUBRES ==> SubResultantPackage(ThePols,RUP)
     discriminantSet(lpols) ==
       res : List(ThePols) := []
       for p in lpols repeat
         v := subresultantVector(p,differentiate(p))$SUBRES
         not(zero?(degree(v.0))) => return(error "Bad discriminant")
         d : ThePols :=  leadingCoefficient(v.0)
--         d := discriminant p
         zero?(d) => return(error "Non Square Free polynomial")
         if not(ground? d) then res := cons(d,res)
       res

     principalSubResultantSet(p,q) ==
        if degree(p) < degree(q)
        then (p,q) := (q,p)
        if degree(p) = degree(q)
        then (p,q) := (q,pseudoRemainder(p, q))
        v := subresultantVector(p,q)$SUBRES
        [coefficient(v.i,i) for i in 0..(((#v)-2)::N)]

     resultantSet(lpols) ==
       res : List(ThePols) := []
       laux := lpols
       for p in lpols repeat
         laux := rest(laux)
         for q in laux repeat
           r : ThePols :=  first(principalSubResultantSet(p,q))
--           r := resultant(p,q)
           zero?(r) => return(error "Non relatively prime polynomials")
           if not(ground? r) then res := cons(r,res)
       res

     projectionSet(lpols) ==
       res : List(ThePols) := []
       for p in lpols repeat
         c := content(p)
         ground?(c) => "next p"
         res := cons(c,res)
       lp1 := [primitivePart p for p in lpols]
       f : ((RUP,RUP) -> Boolean) := (degree(#1) <= degree(#2))
       lp1 := sort(f,lp1)
       lsqfrb := squareFreeBasis(lp1)$PACK1
       lsqfrb := sort(f,lsqfrb)
       for p in lp1 repeat
         res := append(res,coefficientSet(p))
       res := append(res,discriminantSet(lsqfrb))
       append(res,resultantSet(lsqfrb))

   Compiling FriCAS source code from file 
      /var/lib/zope2.10/instance/axiom-wiki/var/LatexWiki/1235315989953270759-25px004.spad
      using old system compiler.
   CAD abbreviates package CylindricalAlgebraicDecompositionPackage 
------------------------------------------------------------------------
   initializing NRLIB CAD for CylindricalAlgebraicDecompositionPackage 
   compiling into NRLIB CAD 
   compiling exported cylindricalDecomposition : List Polynomial TheField -> List Cell TheField
Time: 0 SEC.

   compiling exported cylindricalDecomposition : (List Polynomial TheField,List Symbol) -> List Cell TheField
Time: 0.02 SEC.

   processing macro definition PACK1 ==> CylindricalAlgebraicDecompositionUtilities(Polynomial TheField,SparseUnivariatePolynomial Polynomial TheField) 
   processing macro definition PACK2 ==> CylindricalAlgebraicDecompositionUtilities(TheField,SparseUnivariatePolynomial TheField) 
   compiling exported specialise : (List Polynomial TheField,Cell TheField) -> List SparseUnivariatePolynomial TheField
Time: 0.02 SEC.

   compiling exported coefficientSet : SparseUnivariatePolynomial Polynomial TheField -> List Polynomial TheField
Time: 0 SEC.

   processing macro definition SUBRES ==> SubResultantPackage(Polynomial TheField,SparseUnivariatePolynomial Polynomial TheField) 
   compiling exported discriminantSet : List SparseUnivariatePolynomial Polynomial TheField -> List Polynomial TheField
Time: 0.02 SEC.

   compiling exported principalSubResultantSet : (SparseUnivariatePolynomial Polynomial TheField,SparseUnivariatePolynomial Polynomial TheField) -> List Polynomial TheField
Time: 0.01 SEC.

   compiling exported resultantSet : List SparseUnivariatePolynomial Polynomial TheField -> List Polynomial TheField
Time: 0 SEC.

   compiling exported projectionSet : List SparseUnivariatePolynomial Polynomial TheField -> List Polynomial TheField
****** comp fails at level 8 with expression: ******
error in function projectionSet 

(SEQ
 (LET (|:| |res| (|List| (|Polynomial| |TheField|)))
   (|construct|))
 (REPEAT (IN |p| |lpols|)
         (SEQ
          (LET |c|
            (|content| |p|))
          (LET #1=#:G798
            (|ground?| |c|))
          (|exit| 1
           (IF #1#
               "next p"
               (LET |res|
                 (|cons| |c| |res|))))))
 (LET |lp1|
   (COLLECT (IN |p| |lpols|) (|primitivePart| |p|)))
 (LET (|:|
       |f|
       (|Mapping|
        (|Boolean|)
        (|SparseUnivariatePolynomial| (|Polynomial| |TheField|))
        (|SparseUnivariatePolynomial| (|Polynomial| |TheField|))))
   (<= (|degree| |#1|) (|degree| (|#| 2))))
 (LET |lp1|
   (|sort| |f| |lp1|))
 (LET |lsqfrb|
   ((|elt|
     (|CylindricalAlgebraicDecompositionUtilities| (|Polynomial| |TheField|)
                                                   (|SparseUnivariatePolynomial|
                                                    (|Polynomial| |TheField|)))
     |squareFreeBasis|)
    |lp1|))
 (LET |lsqfrb|
   (|sort| |f| |lsqfrb|))
 (REPEAT (IN |p| |lp1|)
         (LET |res|
           (|append| |res| (|coefficientSet| |p|))))
 (LET |res|
   (|append| |res| (|discriminantSet| |lsqfrb|)))
 (|exit| 1 (|append| |res| (|resultantSet| |lsqfrb|))))
****** level 8  ******
$x:= 2
$m:= (List (Polynomial TheField))
$f:=
((((|#2| #) (|#1| #) (|f| #) (|lp1| #) ...)
  ((#:G798 #) (|c| #) (|p| # #) (|res| # #) ...)))

   >> Apparent user error:
   no mode found for
   #1

)lib )dir .

   Zero is now explicitly exposed in frame initial 
   Zero will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/ZERO.NRLIB/ZERO
   yyy is now explicitly exposed in frame initial 
   yyy will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/YYY.NRLIB/YYY
   xxx is now explicitly exposed in frame initial 
   xxx will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/XXX.NRLIB/XXX
   XPrimitiveArray is now explicitly exposed in frame initial 
   XPrimitiveArray will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/XPRIMARR.NRLIB/XPRIMARR
   Word is now explicitly exposed in frame initial 
   Word will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/WORD.NRLIB/WORD
   VectorField is now explicitly exposed in frame initial 
   VectorField will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/VF.NRLIB/VF
   TaylorSolve is now explicitly exposed in frame initial 
   TaylorSolve will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/UTSSOL.NRLIB/UTSSOL
   Units is now explicitly exposed in frame initial 
   Units will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/UNITS.NRLIB/UNITS
   UnivariateFormalPowerSeriesFunctions is now explicitly exposed in 
      frame initial 
   UnivariateFormalPowerSeriesFunctions will be automatically loaded 
      when needed from /var/aw/var/LatexWiki/UFPS1.NRLIB/UFPS1
   UnivariateFormalPowerSeries is now explicitly exposed in frame 
      initial 
   UnivariateFormalPowerSeries will be automatically loaded when needed
      from /var/aw/var/LatexWiki/UFPS.NRLIB/UFPS
   testtype is now explicitly exposed in frame initial 
   testtype will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TT.NRLIB/TT
   TryConditions is now explicitly exposed in frame initial 
   TryConditions will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TRYCOND.NRLIB/TRYCOND
   TryCondition2 is now explicitly exposed in frame initial 
   TryCondition2 will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TRYCON2.NRLIB/TRYCON2
   TranscendentalManipulations2 is now explicitly exposed in frame 
      initial 
   TranscendentalManipulations2 will be automatically loaded when 
      needed from /var/aw/var/LatexWiki/TRMANIP2.NRLIB/TRMANIP2
   TensorProduct2 is now explicitly exposed in frame initial 
   TensorProduct2 will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TPROD2.NRLIB/TPROD2
   TensorProduct is now explicitly exposed in frame initial 
   TensorProduct will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TPROD.NRLIB/TPROD
   SimplicialComplex is now explicitly exposed in frame initial 
   SimplicialComplex will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TOPAZ.NRLIB/TOPAZ
   TList is now explicitly exposed in frame initial 
   TList will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TLIST.NRLIB/TLIST
   test10 is now explicitly exposed in frame initial 
   test10 will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TESTTEN.NRLIB/TESTTEN
   test6 is now explicitly exposed in frame initial 
   test6 will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TESTSIX.NRLIB/TESTSIX
   TestPackage is now explicitly exposed in frame initial 
   TestPackage will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TESTP.NRLIB/TESTP
   test9 is now explicitly exposed in frame initial 
   test9 will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TESTNINE.NRLIB/TESTNINE
   test8 is now explicitly exposed in frame initial 
   test8 will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TESTEGT.NRLIB/TESTEGT
   Test is now explicitly exposed in frame initial 
   Test will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TEST.NRLIB/TEST
   TensorProductProperty is now explicitly exposed in frame initial 
   TensorProductProperty will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TENSORP.NRLIB/TENSORP
   TensorProductCategory is now explicitly exposed in frame initial 
   TensorProductCategory will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TENSORC.NRLIB/TENSORC
   TemplateUtilities is now explicitly exposed in frame initial 
   TemplateUtilities will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TEMUTL.NRLIB/TEMUTL
   tee2 is now explicitly exposed in frame initial 
   tee2 will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TEETWO.NRLIB/TEETWO
   tee1 is now explicitly exposed in frame initial 
   tee1 will be automatically loaded when needed from 
      /var/aw/var/LatexWiki/TEEONE.NRLIB/TEEONE
   T is now explicitly exposed in frame initial 

   >> System error:
   Lock on package COMMON-LISP violated when setting the symbol-function of T
while in package BOOT.
See also:
  The SBCL Manual, Node "Package Locks"
  The ANSI Standard, Section 11.1.2.1.2