source: git/factory/canonicalform.cc @ 77aa42

/* emacs edit mode for this file is -*- C++ -*- */
/* $Id: canonicalform.cc,v 1.25 1998-01-22 10:56:12 schmidt Exp $ */

#include <config.h>

#include "assert.h"

#include "cf_defs.h"
#include "cf_globals.h"
#include "canonicalform.h"
#include "cf_iter.h"
#include "int_cf.h"
#include "cf_factory.h"
#include "imm.h"
#include "gfops.h"
#include "cf_binom.h"
#if defined (USE_MEMUTIL) && ! defined (USE_OLD_MEMMAN)
#include "memman.h"
#endif

#ifndef NOSTREAMIO
CanonicalForm readCF( istream& );
#endif /* NOSTREAMIO */

//{{{ initialization
int initializeGMP();
int initializeCharacteristic();
#ifdef SINGULAR
int mmInit(void);
#endif

int
initCanonicalForm( void )
{
    static bool initialized = false;
    if ( ! initialized ) {
#if (defined (USE_MEMUTIL) && ! defined (USE_OLD_MEMMAN)) || defined (SINGULAR)
        (void)mmInit();
#endif

        (void)initializeCharacteristic();
        (void)initializeGMP();
        initPT();
        initialized = true;
    }
    return 1;
}
//}}}

//{{{ constructors, destructors, selectors
CanonicalForm::CanonicalForm() : value( CFFactory::basic( (int)0 ) )
{
}

CanonicalForm::CanonicalForm( const int i ) : value( CFFactory::basic( i ) )
{
}

CanonicalForm::CanonicalForm( const CanonicalForm & cf ) : value( is_imm( cf.value ) ? cf.value : cf.value->copyObject() )
{
}

CanonicalForm::CanonicalForm( InternalCF * cf ) : value( cf )
{
}

CanonicalForm::CanonicalForm( const Variable & v ) : value( CFFactory::poly( v ) )
{
}

CanonicalForm::CanonicalForm( const Variable & v, int e ) : value( CFFactory::poly( v, e ) )
{
}

CanonicalForm::CanonicalForm( const char * str ) : value( CFFactory::basic( str ) )
{
}

CanonicalForm::~CanonicalForm()
{
    if ( (! is_imm( value )) && value->deleteObject() )
        delete value;
}

InternalCF*
CanonicalForm::getval() const
{
    if ( is_imm( value ) )
        return value;
    else
        return value->copyObject();
}

CanonicalForm
CanonicalForm::deepCopy() const
{
    if ( is_imm( value ) )
        return *this;
    else
        return CanonicalForm( value->deepCopyObject() );
}
//}}}

//{{{ predicates
bool
CanonicalForm::isOne() const
{
    if ( is_imm( value ) == FFMARK )
        return imm_isone_p( value );
    else  if ( is_imm( value ) == GFMARK )
        return imm_isone_gf( value );
    else  if ( is_imm( value ) )
        return imm_isone( value );
    else
        return value->isOne();
}

bool
CanonicalForm::isZero() const
{
    if ( is_imm( value ) == FFMARK )
        return imm_iszero_p( value );
    else  if ( is_imm( value ) == GFMARK )
        return imm_iszero_gf( value );
    else  if ( is_imm( value ) )
        return imm_iszero( value );
    else
        return value->isZero();
}

bool
CanonicalForm::isImm() const
{
    return is_imm( value );
}

bool
CanonicalForm::inZ() const
{
    if ( is_imm( value ) == INTMARK )
        return true;
    else if ( is_imm( value ) )
        return false;
    else
        return value->levelcoeff() == IntegerDomain;
}

bool
CanonicalForm::inQ() const
{
    if ( is_imm( value ) == INTMARK )
        return true;
    else if ( is_imm( value ) )
        return false;
    else
        return value->levelcoeff() == IntegerDomain ||
            value->levelcoeff() == RationalDomain;
}

bool
CanonicalForm::inFF() const
{
    return is_imm( value ) == FFMARK;
}

bool
CanonicalForm::inGF() const
{
    return is_imm( value ) == GFMARK;
}

bool
CanonicalForm::inPP() const
{
    return ! is_imm( value ) && ( value->levelcoeff() == PrimePowerDomain );
}

bool
CanonicalForm::inBaseDomain() const
{
    if ( is_imm( value ) )
        return true;
    else
        return value->inBaseDomain();
}

bool
CanonicalForm::inExtension() const
{
    if ( is_imm( value ) )
        return false;
    else
        return value->inExtension();
}

bool
CanonicalForm::inCoeffDomain() const
{
    if ( is_imm( value ) )
        return true;
    else
        return value->inCoeffDomain();
}

bool
CanonicalForm::inPolyDomain() const
{
    if ( is_imm( value ) )
        return false;
    else
        return value->inPolyDomain();
}

bool
CanonicalForm::inQuotDomain() const
{
    if ( is_imm( value ) )
        return false;
    else
        return value->inQuotDomain();
}

bool
CanonicalForm::isFFinGF() const
{
    return is_imm( value ) == GFMARK && gf_isff( imm2int( value ) );
}

bool
CanonicalForm::isUnivariate() const
{
    if ( is_imm( value ) )
        return false;
    else
        return value->isUnivariate();
}
//}}}

//{{{ conversion functions
int
CanonicalForm::intval() const
{
    if ( is_imm( value ) )
        return imm_intval( value );
    else
        return value->intval();
}

CanonicalForm
CanonicalForm::mapinto () const
{
    ASSERT( is_imm( value ) ||  ! value->inExtension(), "cannot map into different Extension" );
    if ( is_imm( value ) )
        if ( getCharacteristic() == 0 )
            if ( is_imm( value ) == FFMARK )
                return CanonicalForm( int2imm( ff_symmetric( imm2int( value ) ) ) );
            else  if ( is_imm( value ) == GFMARK )
                return CanonicalForm( int2imm( ff_symmetric( gf_gf2ff( imm2int( value ) ) ) ) );
            else
                return *this;
        else  if ( CFFactory::gettype() == PrimePowerDomain )
            return CanonicalForm( CFFactory::basic( imm2int( value ) ) );
        else  if ( getGFDegree() == 1 )
            return CanonicalForm( int2imm_p( ff_norm( imm2int( value ) ) ) );
        else
            return CanonicalForm( int2imm_gf( gf_int2gf( imm2int( value ) ) ) );
    else  if ( value->inBaseDomain() )
        if ( getCharacteristic() == 0 )
            if ( value->levelcoeff() == PrimePowerDomain )
                return CFFactory::basic( getmpi( value, true ) );
            else
                return *this;
        else  if ( CFFactory::gettype() == PrimePowerDomain ) {
            ASSERT( value->levelcoeff() == PrimePowerDomain || value->levelcoeff() == IntegerDomain, "no proper map defined" );
            if ( value->levelcoeff() == PrimePowerDomain )
                return *this;
            else
                return CFFactory::basic( getmpi( value ) );
        }
        else {
            int val;
            if ( value->levelcoeff() == IntegerDomain )
                val = value->intmod( ff_prime );
            else  if ( value->levelcoeff() == RationalDomain )
                return num().mapinto() / den().mapinto();
            else {
                ASSERT( 0, "illegal domain" );
                return 0;
            }
            if ( getGFDegree() > 1 )
                return CanonicalForm( int2imm_gf( gf_int2gf( val ) ) );
            else
                return CanonicalForm( int2imm_p( val ) );
        }
    else {
        Variable x = value->variable();
        CanonicalForm result;
        for ( CFIterator i = *this; i.hasTerms(); i++ )
            result += power( x, i.exp() ) * i.coeff().mapinto();
        return result;
    }
}
//}}}

//{{{ CanonicalForm CanonicalForm::lc (), Lc (), LC (), LC ( v ) const
//{{{ docu
//
// lc(), Lc(), LC() - leading coefficient functions.
//
// All methods return CO if CO is in a base domain.
//
// lc() returns the leading coefficient of CO with respect to
// lexicographic ordering.  Elements in an algebraic extension
// are considered polynomials so lc() always returns a leading
// coefficient in a base domain.  This method is useful to get
// the base domain over which CO is defined.
//
// Lc() returns the leading coefficient of CO with respect to
// lexicographic ordering.  In contrast to lc() elements in an
// algebraic extension are considered coefficients so Lc() always
// returns a leading coefficient in a coefficient domain.
// 
// LC() returns the leading coefficient of CO where CO is
// considered a univariate polynomial in its main variable.  An
// element of an algebraic extension is considered an univariate
// polynomial, too.
//
// LC( v ) returns the leading coefficient of CO where CO is
// considered an univariate polynomial in the polynomial variable
// v.
// Note: If v is less than the main variable of CO we have to
// swap variables which may be quite expensive.
//
// Examples:
// Let x < y be polynomial variables, a an algebraic variable.
//
// (3*a*x*y^2+y+x).lc() = 3
// (3*a*x*y^2+y+x).Lc() = 3*a
// (3*a*x*y^2+y+x).LC() = 3*a*x
// (3*a*x*y^2+y+x).LC( x ) = 3*a*y^2+1
//
// (3*a^2+4*a).lc() = 3
// (3*a^2+4*a).Lc() = 3*a^2+4*a
// (3*a^2+4*a).LC() = 3
// (3*a^2+4*a).LC( x ) = 3*a^2+4*a
//
// See also: InternalCF::lc(), InternalCF::Lc(), InternalCF::LC(),
// InternalPoly::lc(), InternalPoly::Lc(), InternalPoly::LC(),
// ::lc(), ::Lc(), ::LC(), ::LC( v )
//
//}}}
CanonicalForm
CanonicalForm::lc () const
{
    if ( is_imm( value ) )
        return *this;
    else
        return value->lc();
}

CanonicalForm
CanonicalForm::Lc () const
{
    if ( is_imm( value ) || value->inCoeffDomain() )
        return *this;
    else
        return value->Lc();
}

CanonicalForm
CanonicalForm::LC () const
{
    if ( is_imm( value ) )
        return *this;
    else
        return value->LC();
}

CanonicalForm
CanonicalForm::LC ( const Variable & v ) const
{
    if ( is_imm( value ) || value->inCoeffDomain() )
        return *this;

    Variable x = value->variable();
    if ( v > x )
        return *this;
    else if ( v == x )
        return value->LC();
    else {
        CanonicalForm f = swapvar( *this, v, x );
        if ( f.mvar() == x )
            return swapvar( f.value->LC(), v, x );
        else
            // v did not occur in f
            return *this;
    }
}
//}}}

//{{{ int CanonicalForm::degree (), degree ( v ) const
//{{{ docu
//
// degree() - degree methods.
//
// Both methods returns -1 for the zero polynomial and 0 if
// CO is in a base domain.
//
// degree() returns the degree of CO in its main variable.
// Elements in an algebraic extension are considered polynomials.
//
// degree( v ) returns the degree of CO with respect to v.
// Elements in an algebraic extension are considered polynomials,
// and v may be algebraic.
//
// See also: InternalCf::degree(), InternalPoly::degree(),
// ::degree(), ::degree( v )
//
//}}}
int
CanonicalForm::degree() const
{
    int what = is_imm( value );
    if ( what )
        if ( what == FFMARK )
            return imm_iszero_p( value ) ? -1 : 0;
        else if ( what == INTMARK )
            return imm_iszero( value ) ? -1 : 0;
        else
            return imm_iszero_gf( value ) ? -1 : 0;
    else
        return value->degree();
}

int
CanonicalForm::degree( const Variable & v ) const
{
    int what = is_imm( value );
    if ( what )
        if ( what == FFMARK )
            return imm_iszero_p( value ) ? -1 : 0;
        else if ( what == INTMARK )
            return imm_iszero( value ) ? -1 : 0;
        else
            return imm_iszero_gf( value ) ? -1 : 0;
    else if ( value->inBaseDomain() )
        return value->degree();

    Variable x = value->variable();
    if ( v == x )
        return value->degree();
    else  if ( v > x )
        // relatively to v, f is in a coefficient ring
        return 0;
    else {
        int coeffdeg, result = 0;
        // search for maximum of coefficient degree
        for ( CFIterator i = *this; i.hasTerms(); i++ ) {
            coeffdeg = i.coeff().degree( v );
            if ( coeffdeg > result )
                result = coeffdeg;
        }
        return result;
    }
}
//}}}

//{{{ CanonicalForm CanonicalForm::tailcoeff (), int CanonicalForm::taildegree () const
//{{{ docu
//
// tailcoeff(), taildegree() - return least coefficient and
//   degree, resp.
//
// tailcoeff() returns the coefficient of the term with the least
// degree in CO where CO is considered an univariate polynomial
// in its main variable.  Elements in an algebraic extension are
// considered coefficients.
//
// taildegree() returns -1 for the zero polynomial, 0 if CO is in
// a base domain, otherwise the least degree of CO where CO is
// considered a univariate polynomial in its main variable.  In
// contrast to tailcoeff(), elements in an algebraic extension
// are considered polynomials, not coefficients, and such may
// have a taildegree larger than zero.
//
// See also: InternalCF::tailcoeff(), InternalCF::tailcoeff(),
// InternalPoly::tailcoeff(), InternalPoly::taildegree,
// ::tailcoeff(), ::taildegree()
//
//}}}
CanonicalForm
CanonicalForm::tailcoeff () const
{
    if ( is_imm( value ) || value->inCoeffDomain() )
        return *this;
    else
        return value->tailcoeff();
}

int
CanonicalForm::taildegree () const
{
    int what = is_imm( value );
    if ( what )
        if ( what == FFMARK )
            return imm_iszero_p( value ) ? -1 : 0;
        else if ( what == INTMARK )
            return imm_iszero( value ) ? -1 : 0;
        else
            return imm_iszero_gf( value ) ? -1 : 0;
    else
        return value->taildegree();
}
//}}}

//{{{ int CanonicalForm::level (), Variable CanonicalForm::mvar () const
//{{{ docu
//
// level(), mvar() - return level and main variable of CO.
//
// level() returns the level of CO.  For a list of the levels and
// their meanings, see cf_defs.h.
//
// mvar() returns the main variable of CO or Variable() if CO is
// in a base domain.
//
// See also: InternalCF::level(), InternalCF::variable(),
// InternalPoly::level(), InternalPoly::variable(), ::level(),
// ::mvar()
// 
//}}}
int
CanonicalForm::level () const
{
    if ( is_imm( value ) )
        return LEVELBASE;
    else
        return value->level();
}

Variable
CanonicalForm::mvar () const
{
    if ( is_imm( value ) )
        return Variable();
    else
        return value->variable();
}
//}}}

//{{{ CanonicalForm CanonicalForm::num (), den () const
//{{{ docu
//
// num(), den() - return numinator and denominator of CO.
//
// num() returns the numinator of CO if CO is a rational number,
// CO itself otherwise.
//
// den() returns the denominator of CO if CO is a rational
// number, 1 (from the current domain!) otherwise.
//
// See also: InternalCF::num(), InternalCF::den(),
// InternalRational::num(), InternalRational::den(), ::num(),
// ::den()
//
//}}}
CanonicalForm
CanonicalForm::num () const
{
    if ( is_imm( value ) )
        return *this;
    else
        return CanonicalForm( value->num() );
}

CanonicalForm
CanonicalForm::den () const
{
    if ( is_imm( value ) )
        return CanonicalForm( 1 );
    else
        return CanonicalForm( value->den() );
}
//}}}

//{{{ assignment operators
CanonicalForm &
CanonicalForm::operator = ( const CanonicalForm & cf )
{
    if ( this != &cf ) {
        if ( (! is_imm( value )) && value->deleteObject() )
            delete value;
        value = (is_imm( cf.value )) ? cf.value : cf.value->copyObject();
    }
    return *this;
}

CanonicalForm &
CanonicalForm::operator = ( const int cf )
{
    if ( (! is_imm( value )) && value->deleteObject() )
        delete value;
    value = CFFactory::basic( cf );
    return *this;
}

CanonicalForm &
CanonicalForm::operator += ( const CanonicalForm & cf )
{
    int what = is_imm( value );
    if ( what ) {
        ASSERT ( ! is_imm( cf.value ) || (what==is_imm( cf.value )), "illegal base coefficients" );
        if ( (what = is_imm( cf.value )) == FFMARK )
            value = imm_add_p( value, cf.value );
        else  if ( what == GFMARK )
            value = imm_add_gf( value, cf.value );
        else  if ( what )
            value = imm_add( value, cf.value );
        else {
            InternalCF * dummy = cf.value->copyObject();
            value = dummy->addcoeff( value );
        }
    }
    else  if ( is_imm( cf.value ) )
        value = value->addcoeff( cf.value );
    else  if ( value->level() == cf.value->level() ) {
        if ( value->levelcoeff() == cf.value->levelcoeff() )
            value = value->addsame( cf.value );
        else  if ( value->levelcoeff() > cf.value->levelcoeff() )
            value = value->addcoeff( cf.value );
        else {
            InternalCF * dummy = cf.value->copyObject();
            dummy = dummy->addcoeff( value );
            if ( value->deleteObject() ) delete value;
            value = dummy;
        }
    }
    else  if ( level() > cf.level() )
        value = value->addcoeff( cf.value );
    else {
        InternalCF * dummy = cf.value->copyObject();
        dummy = dummy->addcoeff( value );
        if ( value->deleteObject() ) delete value;
        value = dummy;
    }
    return *this;
}

CanonicalForm &
CanonicalForm::operator -= ( const CanonicalForm & cf )
{
    int what = is_imm( value );
    if ( what ) {
        ASSERT ( ! is_imm( cf.value ) || (what==is_imm( cf.value )), "illegal base coefficients" );
        if ( (what = is_imm( cf.value )) == FFMARK )
            value = imm_sub_p( value, cf.value );
        else  if ( what == GFMARK )
            value = imm_sub_gf( value, cf.value );
        else  if ( what )
            value = imm_sub( value, cf.value );
        else {
            InternalCF * dummy = cf.value->copyObject();
            value = dummy->subcoeff( value, true );
        }
    }
    else  if ( is_imm( cf.value ) )
        value = value->subcoeff( cf.value, false );
    else  if ( value->level() == cf.value->level() ) {
        if ( value->levelcoeff() == cf.value->levelcoeff() )
            value = value->subsame( cf.value );
        else  if ( value->levelcoeff() > cf.value->levelcoeff() )
            value = value->subcoeff( cf.value, false );
        else {
            InternalCF * dummy = cf.value->copyObject();
            dummy = dummy->subcoeff( value, true );
            if ( value->deleteObject() ) delete value;
            value = dummy;
        }
    }
    else  if ( level() > cf.level() )
        value = value->subcoeff( cf.value, false );
    else {
        InternalCF * dummy = cf.value->copyObject();
        dummy = dummy->subcoeff( value, true );
        if ( value->deleteObject() ) delete value;
        value = dummy;
    }
    return *this;
}

CanonicalForm &
CanonicalForm::operator *= ( const CanonicalForm & cf )
{
    int what = is_imm( value );
    if ( what ) {
        ASSERT ( ! is_imm( cf.value ) || (what==is_imm( cf.value )), "illegal base coefficients" );
        if ( (what = is_imm( cf.value )) == FFMARK )
            value = imm_mul_p( value, cf.value );
        else  if ( what == GFMARK )
            value = imm_mul_gf( value, cf.value );
        else  if ( what )
            value = imm_mul( value, cf.value );
        else {
            InternalCF * dummy = cf.value->copyObject();
            value = dummy->mulcoeff( value );
        }
    }
    else  if ( is_imm( cf.value ) )
        value = value->mulcoeff( cf.value );
    else  if ( value->level() == cf.value->level() ) {
        if ( value->levelcoeff() == cf.value->levelcoeff() )
            value = value->mulsame( cf.value );
        else  if ( value->levelcoeff() > cf.value->levelcoeff() )
            value = value->mulcoeff( cf.value );
        else {
            InternalCF * dummy = cf.value->copyObject();
            dummy = dummy->mulcoeff( value );
            if ( value->deleteObject() ) delete value;
            value = dummy;
        }
    }
    else  if ( level() > cf.level() )
        value = value->mulcoeff( cf.value );
    else {
        InternalCF * dummy = cf.value->copyObject();
        dummy = dummy->mulcoeff( value );
        if ( value->deleteObject() ) delete value;
        value = dummy;
    }
    return *this;
}

CanonicalForm &
CanonicalForm::operator /= ( const CanonicalForm & cf )
{
    int what = is_imm( value );
    if ( what ) {
        ASSERT ( ! is_imm( cf.value ) || (what==is_imm( cf.value )), "illegal base coefficients" );
        if ( (what = is_imm( cf.value )) == FFMARK )
            value = imm_div_p( value, cf.value );
        else  if ( what == GFMARK )
            value = imm_div_gf( value, cf.value );
        else  if ( what )
            value = imm_divrat( value, cf.value );
        else {
            InternalCF * dummy = cf.value->copyObject();
            value = dummy->dividecoeff( value, true );
        }
    }
    else  if ( is_imm( cf.value ) )
        value = value->dividecoeff( cf.value, false );
    else  if ( value->level() == cf.value->level() ) {
        if ( value->levelcoeff() == cf.value->levelcoeff() )
            value = value->dividesame( cf.value );
        else  if ( value->levelcoeff() > cf.value->levelcoeff() )
            value = value->dividecoeff( cf.value, false );
        else {
            InternalCF * dummy = cf.value->copyObject();
            dummy = dummy->dividecoeff( value, true );
            if ( value->deleteObject() ) delete value;
            value = dummy;
        }
    }
    else  if ( level() > cf.level() )
        value = value->dividecoeff( cf.value, false );
    else {
        InternalCF * dummy = cf.value->copyObject();
        dummy = dummy->dividecoeff( value, true );
        if ( value->deleteObject() ) delete value;
        value = dummy;
    }
    return *this;
}

CanonicalForm &
CanonicalForm::div ( const CanonicalForm & cf )
{
    int what = is_imm( value );
    if ( what ) {
        ASSERT ( ! is_imm( cf.value ) || (what==is_imm( cf.value )), "illegal base coefficients" );
        if ( (what = is_imm( cf.value )) == FFMARK )
            value = imm_div_p( value, cf.value );
        else  if ( what == GFMARK )
            value = imm_div_gf( value, cf.value );
        else  if ( what )
            value = imm_div( value, cf.value );
        else {
            InternalCF * dummy = cf.value->copyObject();
            value = dummy->divcoeff( value, true );
        }
    }
    else  if ( is_imm( cf.value ) )
        value = value->divcoeff( cf.value, false );
    else  if ( value->level() == cf.value->level() ) {
        if ( value->levelcoeff() == cf.value->levelcoeff() )
            value = value->divsame( cf.value );
        else  if ( value->levelcoeff() > cf.value->levelcoeff() )
            value = value->divcoeff( cf.value, false );
        else {
            InternalCF * dummy = cf.value->copyObject();
            dummy = dummy->divcoeff( value, true );
            if ( value->deleteObject() ) delete value;
            value = dummy;
        }
    }
    else  if ( level() > cf.level() )
        value = value->divcoeff( cf.value, false );
    else {
        InternalCF * dummy = cf.value->copyObject();
        dummy = dummy->divcoeff( value, true );
        if ( value->deleteObject() ) delete value;
        value = dummy;
    }
    return *this;
}

CanonicalForm &
CanonicalForm::operator %= ( const CanonicalForm & cf )
{
    int what = is_imm( value );
    if ( what ) {
        ASSERT ( ! is_imm( cf.value ) || (what==is_imm( cf.value )), "illegal base coefficients" );
        if ( (what = is_imm( cf.value )) == FFMARK )
            value = imm_mod_p( value, cf.value );
        else  if ( what == GFMARK )
            value = imm_mod_gf( value, cf.value );
        else  if ( what )
            value = imm_mod( value, cf.value );
        else {
            InternalCF * dummy = cf.value->copyObject();
            value = dummy->modulocoeff( value, true );
        }
    }
    else  if ( is_imm( cf.value ) )
        value = value->modulocoeff( cf.value, false );
    else  if ( value->level() == cf.value->level() ) {
        if ( value->levelcoeff() == cf.value->levelcoeff() )
            value = value->modulosame( cf.value );
        else  if ( value->levelcoeff() > cf.value->levelcoeff() )
            value = value->modulocoeff( cf.value, false );
        else {
            InternalCF * dummy = cf.value->copyObject();
            dummy = dummy->modulocoeff( value, true );
            if ( value->deleteObject() ) delete value;
            value = dummy;
        }
    }
    else  if ( level() > cf.level() )
        value = value->modulocoeff( cf.value, false );
    else {
        InternalCF * dummy = cf.value->copyObject();
        dummy = dummy->modulocoeff( value, true );
        if ( value->deleteObject() ) delete value;
        value = dummy;
    }
    return *this;
}

CanonicalForm &
CanonicalForm::mod ( const CanonicalForm & cf )
{
    int what = is_imm( value );
    if ( what ) {
        ASSERT ( ! is_imm( cf.value ) || (what==is_imm( cf.value )), "illegal base coefficients" );
        if ( (what = is_imm( cf.value )) == FFMARK )
            value = imm_mod_p( value, cf.value );
        else  if ( what == GFMARK )
            value = imm_mod_gf( value, cf.value );
        else  if ( what )
            value = imm_mod( value, cf.value );
        else {
            InternalCF * dummy = cf.value->copyObject();
            value = dummy->modcoeff( value, true );
        }
    }
    else  if ( is_imm( cf.value ) )
        value = value->modcoeff( cf.value, false );
    else  if ( value->level() == cf.value->level() ) {
        if ( value->levelcoeff() == cf.value->levelcoeff() )
            value = value->modsame( cf.value );
        else  if ( value->levelcoeff() > cf.value->levelcoeff() )
            value = value->modcoeff( cf.value, false );
        else {
            InternalCF * dummy = cf.value->copyObject();
            dummy = dummy->modcoeff( value, true );
            if ( value->deleteObject() ) delete value;
            value = dummy;
        }
    }
    else  if ( level() > cf.level() )
        value = value->modcoeff( cf.value, false );
    else {
        InternalCF * dummy = cf.value->copyObject();
        dummy = dummy->modcoeff( value, true );
        if ( value->deleteObject() ) delete value;
        value = dummy;
    }
    return *this;
}

void
divrem ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & q, CanonicalForm & r )
{
    InternalCF * qq = 0, * rr = 0;
    int what = is_imm( f.value );
    if ( what )
        if ( is_imm( g.value ) ) {
            if ( what == FFMARK )
                imm_divrem_p( f.value, g.value, qq, rr );
            else  if ( what == GFMARK )
                imm_divrem_gf( f.value, g.value, qq, rr );
            else
                imm_divrem( f.value, g.value, qq, rr );
        }
        else
            g.value->divremcoeff( f.value, qq, rr, true );
    else  if ( (what=is_imm( g.value )) )
        f.value->divremcoeff( g.value, qq, rr, false );
    else  if ( f.value->level() == g.value->level() )
        if ( f.value->levelcoeff() == g.value->levelcoeff() )
            f.value->divremsame( g.value, qq, rr );
        else  if ( f.value->levelcoeff() > g.value->levelcoeff() )
            f.value->divremcoeff( g.value, qq, rr, false );
        else
            g.value->divremcoeff( f.value, qq, rr, true );
    else  if ( f.value->level() > g.value->level() )
        f.value->divremcoeff( g.value, qq, rr, false );
    else
        g.value->divremcoeff( f.value, qq, rr, true );
    ASSERT( qq != 0 && rr != 0, "error in divrem" );
    q = CanonicalForm( qq );
    r = CanonicalForm( rr );
}

bool
divremt ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & q, CanonicalForm & r )
{
    InternalCF * qq = 0, * rr = 0;
    int what = is_imm( f.value );
    bool result = true;
    if ( what )
        if ( is_imm( g.value ) ) {
            if ( what == FFMARK )
                imm_divrem_p( f.value, g.value, qq, rr );
            else  if ( what == GFMARK )
                imm_divrem_gf( f.value, g.value, qq, rr );
            else
                imm_divrem( f.value, g.value, qq, rr );
        }
        else
            result = g.value->divremcoefft( f.value, qq, rr, true );
    else  if ( (what=is_imm( g.value )) )
        result = f.value->divremcoefft( g.value, qq, rr, false );
    else  if ( f.value->level() == g.value->level() )
        if ( f.value->levelcoeff() == g.value->levelcoeff() )
            result = f.value->divremsamet( g.value, qq, rr );
        else  if ( f.value->levelcoeff() > g.value->levelcoeff() )
            result = f.value->divremcoefft( g.value, qq, rr, false );
        else
            result = g.value->divremcoefft( f.value, qq, rr, true );
    else  if ( f.value->level() > g.value->level() )
        result = f.value->divremcoefft( g.value, qq, rr, false );
    else
        result = g.value->divremcoefft( f.value, qq, rr, true );
    if ( result ) {
        ASSERT( qq != 0 && rr != 0, "error in divrem" );
        q = CanonicalForm( qq );
        r = CanonicalForm( rr );
    }
    else {
        q = 0; r = 0;
    }
    return result;
}
//}}}

//{{{ CanonicalForm CanonicalForm::operator () ( f ), operator () ( f, v ) const
//{{{ docu
//
// operator ()() - evaluation operator.
//
// Both operators return CO if CO is in a base domain.
//
// operator () ( f ) returns CO with f inserted for the main
// variable.  Elements in an algebraic extension are considered
// polynomials.
//
// operator () ( f, v ) returns CO with f inserted for v.
// Elements in an algebraic extension are considered polynomials
// and v may be an algebraic variable.
//
//}}}
CanonicalForm
CanonicalForm::operator () ( const CanonicalForm & f ) const
{
    if ( is_imm( value ) || value->inBaseDomain() )
        return *this;
    else {
        CFIterator i = *this;
        int lastExp = i.exp();
        CanonicalForm result = i.coeff();
        i++;
        while ( i.hasTerms() ) {
            if ( (lastExp - i.exp()) == 1 )
                result *= f;
            else
                result *= power( f, lastExp - i.exp() );
            result += i.coeff();
            lastExp = i.exp();
            i++;
        }
        if ( lastExp != 0 )
            result *= power( f, lastExp );
        return result;
    }
}

CanonicalForm
CanonicalForm::operator () ( const CanonicalForm & f, const Variable & v ) const
{
    if ( is_imm( value ) || value->inBaseDomain() )
        return *this;

    Variable x = value->variable();
    if ( v > x )
        return *this;
    else  if ( v == x )
        return (*this)( f );
    else {
        // v is less than main variable of f
        CanonicalForm result = 0;
        for ( CFIterator i = *this; i.hasTerms(); i++ )
            result += i.coeff()( f, v ) * power( x, i.exp() );
        return result;
    }
}
//}}}

//{{{ CanonicalForm CanonicalForm::operator [] ( int i ) const
//{{{ docu
//
// operator []() - return i'th coefficient from CO.
//
// Returns CO if CO is in a base domain and i equals zero.
// Returns zero (from the current domain) if CO is in a base
// domain and i is larger than zero.  Otherwise, returns the
// coefficient to x^i in CO (if x denotes the main variable of
// CO) or zero if CO does not contain x^i.  Elements in an
// algebraic extension are considered polynomials.  i should be
// larger or equal zero.
//
// Note: Never use a loop like
//
// for ( int i = degree( f ); i >= 0; i-- )
//     foo( i, f[ i ] );
//
// which is much slower than
//
// for ( int i = degree( f ), CFIterator I = f; I.hasTerms(); I++ ) {
//     // fill gap with zeroes
//     for ( ; i > I.exp(); i-- )
//         foo( i, 0 );
//     // at this point, i == I.exp()
//     foo( i, i.coeff() );
//     i--;
// }
// // work through trailing zeroes
// for ( ; i >= 0; i-- )
//     foo( i, 0 );
//
//}}}
CanonicalForm
CanonicalForm::operator [] ( int i ) const
{
    ASSERT( i >= 0, "index to operator [] less than zero" );
    if ( is_imm( value ) )
        if ( i == 0 )
            return *this;
        else
            return CanonicalForm( 0 );
    else
        return value->coeff( i );
}
//}}}

//{{{ CanonicalForm CanonicalForm::deriv (), deriv ( x )
//{{{ docu
//
// deriv() - return the formal derivation of CO.
//
// deriv() derives CO with respect to its main variable.  Returns
// zero from the current domain if f is in a coefficient domain.
//
// deriv( x ) derives CO with respect to x.  x should be a
// polynomial variable.  Returns zero from the current domain if
// f is in a coefficient domain.
//
// See also: ::deriv()
//
//}}}
CanonicalForm
CanonicalForm::deriv () const
{
    if ( is_imm( value ) || value->inCoeffDomain() )
        return CanonicalForm( 0 );
    else {
        CanonicalForm result = 0;
        Variable x = value->variable();
        for ( CFIterator i = *this; i.hasTerms(); i++ )
            if ( i.exp() > 0 )
                result += power( x, i.exp()-1 ) * i.coeff() * i.exp();
        return result;
    }
}

CanonicalForm
CanonicalForm::deriv ( const Variable & x ) const
{
    ASSERT( x.level() > 0, "cannot derive with respect to algebraic variables" );
    if ( is_imm( value ) || value->inCoeffDomain() )
        return CanonicalForm( 0 );

    Variable y = value->variable();
    if ( x > y )
        return CanonicalForm( 0 );
    else if ( x == y )
        return deriv();
    else {
        CanonicalForm result = 0;
        for ( CFIterator i = *this; i.hasTerms(); i++ )
            result += i.coeff().deriv( x ) * power( y, i.exp() );
        return result;
    }
}
//}}}

//{{{ int CanonicalForm::sign () const
//{{{ docu
//
// sign() - return sign of CO.
//
// If CO is an integer or a rational number, the sign is defined
// as usual.  If CO is an element of a prime power domain or of
// FF(p) and SW_SYMMETRIC_FF is on, the sign of CO is the sign of
// the symmetric representation of CO.  If CO is in GF(q) or in
// FF(p) and SW_SYMMETRIC_FF is off, the sign of CO is zero iff
// CO is zero, otherwise the sign is one.
//
// If CO is a polynomial or in an extension of one of the base
// domains, the sign of CO is the sign of its leading
// coefficient.
//
// See also: InternalCF::sign(), InternalInteger::sign(),
// InternalPrimePower::sign(), InternalRational::sign(),
// InternalPoly::sign(), imm_sign(), gf_sign()
//
//}}}
int
CanonicalForm::sign () const
{
    if ( is_imm( value ) )
        return imm_sign( value );
    else
        return value->sign();
}
//}}}

//{{{ CanonicalForm CanonicalForm::sqrt () const
//{{{ docu
//
// sqrt() - calculate integer square root.
//
// CO has to be an integer greater or equal zero.  Returns the
// largest integer less or equal sqrt(CO).
//
// In the immediate case, we use the newton method to find the
// root.  The algorithm is from H. Cohen - 'A Course in
// Computational Algebraic Number Theory', ch. 1.7.1.
//
// See also: InternalCF::sqrt(), InternalInteger::sqrt(), ::sqrt()
//
//}}}
CanonicalForm
CanonicalForm::sqrt () const
{
    if ( is_imm( value ) ) {
        ASSERT( is_imm( value ) == INTMARK, "sqrt() not implemented" );
        int n = imm2int( value );
        ASSERT( n >= 0, "arg to sqrt() less than zero" );
        if ( n == 0 || n == 1 )
            return CanonicalForm( n );
        else {
            int x, y = n;
            do {
                x = y;
                // the intermediate result may not fit into an
                // integer, but the result does
                y = (unsigned int)(x + n/x)/2;
            } while ( y < x );
            return CanonicalForm( x );
        }
    }
    else
        return CanonicalForm( value->sqrt() );
}
//}}}

//{{{ int CanonicalForm::ilog2 () const
//{{{ docu
//
// ilog2() - integer logarithm to base 2.
//
// Returns the largest integer less or equal logarithm of CO to
// base 2.  CO should be a positive integer.
//
// See also: InternalCF::ilog2(), InternalInteger::ilog2(), ::ilog2()
//
//}}}
int
CanonicalForm::ilog2 () const
{
    if ( is_imm( value ) ) {
        ASSERT( is_imm( value ) == INTMARK, "ilog2() not implemented" );
        int a = imm2int( value );
        ASSERT( a > 0, "arg to ilog2() less or equal zero" );
        int n = -1;
        while ( a != 0 ) {
            n++;
            a /= 2;
        }
        return n;
    }
    else
        return value->ilog2();
}
//}}}

//{{{ bool operator ==, operator != ( const CanonicalForm & lhs, const CanonicalForm & rhs )
//{{{ docu
//
// operator ==(), operator !=() - compare canonical forms on
//   (in)equality.
//
// operator ==() returns true iff lhs equals rhs.
// operator !=() returns true iff lhs does not equal rhs.
//
// This is the point in factory where we essentially use that
// CanonicalForms in fact are canonical.  There must not be two
// different representations of the same mathematical object,
// otherwise, such (in)equality will not be recognized by these
// operators.  In other word, we rely on the fact that structural
// different factory objects in any case represent different
// mathematical objects.
//
// So we use the following procedure to test on equality (and
// analogously on inequality).  First, we check whether lhs.value
// equals rhs.value.  If so we are ready and return true.
// Second, if one of the operands is immediate, but the other one
// not, we return false.  Third, if the operand's levels differ
// we return false.  Fourth, if the operand's levelcoeffs differ
// we return false.  At last, we call the corresponding internal
// method to compare both operands.
//
// Both operands should have coefficients from the same base domain.
//
// Note: To compare with the zero or the unit of the current domain,
// you better use the methods `CanonicalForm::isZero()' or
// `CanonicalForm::isOne()', resp., than something like `f == 0',
// since the latter is quite a lot slower.
//
// See also: InternalCF::comparesame(),
// InternalInteger::comparesame(), InternalRational::comparesame(),
// InternalPrimePower::comparesame(), InternalPoly::comparesame()
//
//}}}
bool
operator == ( const CanonicalForm & lhs, const CanonicalForm & rhs )
{
    if ( lhs.value == rhs.value )
        return true;
    else if ( is_imm( rhs.value ) || is_imm( lhs.value ) ) {
        ASSERT( ! is_imm( rhs.value ) ||
                ! is_imm( lhs.value ) ||
                is_imm( rhs.value ) == is_imm( lhs.value ),
                "incompatible operands" );
        return false;
    }
    else  if ( lhs.value->level() != rhs.value->level() )
        return false;
    else  if ( lhs.value->levelcoeff() != rhs.value->levelcoeff() )
        return false;
    else
        return rhs.value->comparesame( lhs.value ) == 0;
}

bool
operator != ( const CanonicalForm & lhs, const CanonicalForm & rhs )
{
    if ( lhs.value == rhs.value )
        return false;
    else if ( is_imm( rhs.value ) || is_imm( lhs.value ) ) {
        ASSERT( ! is_imm( rhs.value ) ||
                ! is_imm( lhs.value ) ||
                is_imm( rhs.value ) == is_imm( lhs.value ),
                "incompatible operands" );
        return true;
    }
    else  if ( lhs.value->level() != rhs.value->level() )
        return true;
    else  if ( lhs.value->levelcoeff() != rhs.value->levelcoeff() )
        return true;
    else        return rhs.value->comparesame( lhs.value ) != 0;
}
//}}}

//{{{ bool operator >, operator < ( const CanonicalForm & lhs, const CanonicalForm & rhs )
//{{{ docu
//
// operator >(), operator <() - compare canonical forms. on size or
//   level.
//
// The most common and most useful application of these operators
// is to compare two integers or rationals, of course.  However,
// these operators are defined on all other base domains and on
// polynomials, too.  From a mathematical point of view this may
// seem meaningless, since there is no ordering on finite fields
// or on polynomials respecting the algebraic structure.
// Nevertheless, from a programmer's point of view it may be
// sensible to order these objects, e.g. to sort them.
//
// Therefore, the ordering defined by these operators in any case
// is a total ordering which fulfills the law of trichotomy.
//
// It is clear how this is done in the case of the integers and
// the rationals.  For finite fields, all you can say is that
// zero is the minimal element w.r.t. the ordering, the other
// elements are ordered in an arbitrary (but total!)  way.  For
// polynomials, you have an ordering derived from the
// lexicographical ordering of monomials.  E.g. if lm(f) < lm(g)
// w.r.t. lexicographic ordering, then f < g.  For more details,
// refer to the documentation of `InternalPoly::operator <()'.
// 
// Both operands should have coefficients from the same base domain.
//
// The scheme how both operators are implemented is allmost the
// same as for the assignment operators (check for immediates,
// then check levels, then check levelcoeffs, then call the
// appropriate internal comparesame()/comparecoeff() method).
// For more information, confer to the overview for the
// arithmetic operators.
//
// See also: InternalCF::comparesame(),
// InternalInteger::comparesame(), InternalRational::comparesame(),
// InternalPrimePower::comparesame(), InternalPoly::comparesame(),
// InternalCF::comparecoeff(), InternalInteger::comparecoeff(),
// InternalRational::comparecoeff(),
// InternalPrimePower::comparecoeff(), InternalPoly::comparecoeff(),
// imm_cmp(), imm_cmp_p(), imm_cmp_gf()
//
//}}}
bool
operator > ( const CanonicalForm & lhs, const CanonicalForm & rhs )
{
    int what = is_imm( rhs.value );
    if ( is_imm( lhs.value ) ) {
        ASSERT( ! what || (what == is_imm( lhs.value )), "incompatible operands" );
        if ( what == 0 )
            return rhs.value->comparecoeff( lhs.value ) < 0;
        else if ( what == INTMARK )
            return imm_cmp( lhs.value, rhs.value ) > 0;
        else if ( what == FFMARK )
            return imm_cmp_p( lhs.value, rhs.value ) > 0;
        else
            return imm_cmp_gf( lhs.value, rhs.value ) > 0;
    }
    else  if ( what )
        return lhs.value->comparecoeff( rhs.value ) > 0;
    else  if ( lhs.value->level() == rhs.value->level() )
        if ( lhs.value->levelcoeff() == rhs.value->levelcoeff() )
            return lhs.value->comparesame( rhs.value ) > 0;
        else  if ( lhs.value->levelcoeff() > rhs.value->levelcoeff() )
            return lhs.value->comparecoeff( rhs.value ) > 0;
        else
            return rhs.value->comparecoeff( lhs.value ) < 0;
    else
        return lhs.value->level() > rhs.value->level();
}

bool
operator < ( const CanonicalForm & lhs, const CanonicalForm & rhs )
{
    int what = is_imm( rhs.value );
    if ( is_imm( lhs.value ) ) {
        ASSERT( ! what || (what == is_imm( lhs.value )), "incompatible operands" );
        if ( what == 0 )
            return rhs.value->comparecoeff( lhs.value ) > 0;
        else if ( what == INTMARK )
            return imm_cmp( lhs.value, rhs.value ) < 0;
        else if ( what == FFMARK )
            return imm_cmp_p( lhs.value, rhs.value ) < 0;
        else
            return imm_cmp_gf( lhs.value, rhs.value ) < 0;
    }
    else  if ( what )
        return lhs.value->comparecoeff( rhs.value ) < 0;
    else  if ( lhs.value->level() == rhs.value->level() )
        if ( lhs.value->levelcoeff() == rhs.value->levelcoeff() )
            return lhs.value->comparesame( rhs.value ) < 0;
        else  if ( lhs.value->levelcoeff() > rhs.value->levelcoeff() )
            return lhs.value->comparecoeff( rhs.value ) < 0;
        else
            return rhs.value->comparecoeff( lhs.value ) > 0;
    else
        return lhs.value->level() < rhs.value->level();
}
//}}}

//{{{ arithmetic operators
CanonicalForm
operator - ( const CanonicalForm & cf )
{
    CanonicalForm result( cf );
    int what = is_imm( result.value );
    if ( what == FFMARK )
        result.value = imm_neg_p( result.value );
    else  if ( what == GFMARK )
        result.value = imm_neg_gf( result.value );
    else  if ( what )
        result.value = imm_neg( result.value );
    else
        result.value = result.value->neg();
    return result;
}

CanonicalForm
operator + ( const CanonicalForm &c1, const CanonicalForm &c2 )
{
    CanonicalForm result( c1 );
    result += c2;
    return result;
}

CanonicalForm
operator - ( const CanonicalForm &c1, const CanonicalForm &c2 )
{
    CanonicalForm result( c1 );
    result -= c2;
    return result;
}

CanonicalForm
operator * ( const CanonicalForm &c1, const CanonicalForm &c2 )
{
    CanonicalForm result( c1 );
    result *= c2;
    return result;
}

CanonicalForm
operator / ( const CanonicalForm &c1, const CanonicalForm &c2 )
{
    CanonicalForm result( c1 );
    result /= c2;
    return result;
}

CanonicalForm
div ( const CanonicalForm &c1, const CanonicalForm &c2 )
{
    CanonicalForm result( c1 );
    result.div( c2 );
    return result;
}

CanonicalForm
mod ( const CanonicalForm &c1, const CanonicalForm &c2 )
{
    CanonicalForm result( c1 );
    result.mod( c2 );
    return result;
}

CanonicalForm
operator % ( const CanonicalForm &c1, const CanonicalForm &c2 )
{
    CanonicalForm result( c1 );
    result %= c2;
    return result;
}
//}}}

//{{{ CanonicalForm bgcd ( const CanonicalForm & f, const CanonicalForm & g )
//{{{ docu
//
// bgcd() - return base coefficient gcd.
//
// If both f and g are integers and `SW_RATIONAL' is off the
// positive greatest common divisor of f and g is returned.
// Otherwise, if `SW_RATIONAL' is on or one of f and g is not an
// integer, the greatest common divisor is trivial: either zero
// if f and g equal zero or one (both from the current domain).
//
// f and g should come from one base domain which should be not
// the prime power domain.
//
// Implementation: 
//
// CanonicalForm::bgcd() handles the immediate case with a
//   standard euclidean algorithm.  For the non-immediate cases
//   `InternalCF::bgcdsame()' or `InternalCF::bgcdcoeff()', resp. are
//   called following the usual level/levelcoeff approach.
//
// InternalCF::bgcdsame() and
// InternalCF::bgcdcoeff() throw an assertion ("not implemented")
//
// InternalInteger::bgcdsame() is a wrapper around `mpz_gcd()'
//   which takes some care about immediate results and the sign
//   of the result
// InternalInteger::bgcdcoeff() is a wrapper around
//   `mpz_gcd_ui()' which takes some care about the sign
//   of the result
//
// InternalRational::bgcdsame() and
// InternalRational::bgcdcoeff() always return one
//
//}}}
CanonicalForm
bgcd ( const CanonicalForm & f, const CanonicalForm & g )
{
    // check immediate cases
    int what = is_imm( g.value );
    if ( is_imm( f.value ) ) {
        ASSERT( ! what || (what == is_imm( f.value )), "incompatible operands" );
        if ( what == 0 )
            return g.value->bgcdcoeff( f.value );
        else if ( what == INTMARK && ! cf_glob_switches.isOn( SW_RATIONAL ) ) {
            // calculate gcd using standard integer
            // arithmetic
            int fInt = imm2int( f.value );
            int gInt = imm2int( g.value );

            if ( fInt < 0 ) fInt = -fInt;
            if ( gInt < 0 ) gInt = -gInt;
            // swap fInt and gInt
            if ( gInt > fInt ) {
                int swap = gInt;
                gInt = fInt;
                fInt = swap;
            }

            // now, 0 <= gInt <= fInt.  Start the loop.
            while ( gInt ) {
                // calculate (fInt, gInt) = (gInt, fInt%gInt)
                int r = fInt % gInt;
                fInt = gInt;
                gInt = r;
            }

            return CanonicalForm( fInt );
        } else
            // we do not go for maximal speed for these stupid
            // special cases
            return CanonicalForm( f.isZero() && g.isZero ? 0 : 1 );
    }
    else if ( what )
        return f.value->bgcdcoeff( g.value );

    int fLevel = f.value->level();
    int gLevel = g.value->level();

    // check levels
    if ( fLevel == gLevel ) {
        fLevel = f.value->levelcoeff();
        gLevel = g.value->levelcoeff();

        // check levelcoeffs
        if ( fLevel == gLevel )
            return f.value->bgcdsame( g.value );
        else if ( fLevel < gLevel )
            return g.value->bgcdcoeff( f.value );
        else
            return f.value->bgcdcoeff( g.value );
    }
    else if ( fLevel < gLevel )
        return g.value->bgcdcoeff( f.value );
    else
        return f.value->bgcdcoeff( g.value );
}
//}}}

//{{{ CanonicalForm bextgcd ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & a, CanonicalForm & b )
//{{{ docu
//
// bextgcd() - return base coefficient extended gcd.
//
//}}}
CanonicalForm
bextgcd ( const CanonicalForm & f, const CanonicalForm & g, CanonicalForm & a, CanonicalForm & b )
{
    // check immediate cases
    int what = is_imm( g.value );
    if ( is_imm( f.value ) ) {
        ASSERT( ! what || (what == is_imm( f.value )), "incompatible operands" );
        if ( what == 0 )
            return g.value->bextgcdcoeff( f.value, b, a );
        else if ( what == INTMARK && ! cf_glob_switches.isOn( SW_RATIONAL ) ) {
            // calculate extended gcd using standard integer
            // arithmetic
            int fInt = imm2int( f.value );
            int gInt = imm2int( g.value );

            // to avoid any system dpendencies with `%', we work
            // with positive numbers only.  To a pity, we have to
            // redo all the checks when assigning to a and b.
            if ( fInt < 0 ) fInt = -fInt;
            if ( gInt < 0 ) gInt = -gInt;
            // swap fInt and gInt
            if ( gInt > fInt ) {
                int swap = gInt;
                gInt = fInt;
                fInt = swap;
            }

            int u = 1; int v = 0;
            int uNext = 0; int vNext = 1;

            // at any step, we have:
            // fInt_0 * u + gInt_0 * v = fInt
            // fInt_0 * uNext + gInt_0 * vNext = gInt
            // where fInt_0 and gInt_0 denote the values of fint
            // and gInt, resp., at the beginning
            while ( gInt ) {
                int r = fInt % gInt;
                int q = fInt / gInt;
                int uSwap = u - q * uNext;
                int vSwap = v - q * vNext;

                // update variables
                fInt = gInt;
                gInt = r;
                u = uNext; v = vNext;
                uNext = uSwap; vNext = vSwap;
            }

            // now, assign to a and b
            int fTest = imm2int( f.value );
            int gTest = imm2int( g.value );
            if ( gTest > fTest ) {
                a = v; b = u;
            } else {
                a = u; b = v;
            }
            if ( fTest < 0 ) a = -a;
            if ( gTest < 0 ) b = -b;
            return CanonicalForm( fInt );
        } else
            // stupid special cases
            if ( ! f.isZero() ) {
                a = 1/f; b = 0; return CanonicalForm( 1 );
            } else if ( ! g.isZero() ) {
                a = 0; b = 1/g; return CanonicalForm( 1 );
            } else {
                a = 0; b = 0; return CanonicalForm( 0 );
            }
    }
    else if ( what )
        return f.value->bextgcdcoeff( g.value, a, b );

    int fLevel = f.value->level();
    int gLevel = g.value->level();

    // check levels
    if ( fLevel == gLevel ) {
        fLevel = f.value->levelcoeff();
        gLevel = g.value->levelcoeff();

        // check levelcoeffs
        if ( fLevel == gLevel )
            return f.value->bextgcdsame( g.value, a, b );
        else if ( fLevel < gLevel )
            return g.value->bextgcdcoeff( f.value, b, a );
        else
            return f.value->bextgcdcoeff( g.value, a, b );
    }
    else if ( fLevel < gLevel )
        return g.value->bextgcdcoeff( f.value, b, a );
    else
        return f.value->bextgcdcoeff( g.value, a, b );
}
//}}}

//{{{ input/output
#ifndef NOSTREAMIO
void
CanonicalForm::print( ostream & os, char * str ) const
{
    if ( is_imm( value ) )
        imm_print( os, value, str );
    else
        value->print( os, str );
}

ostream&
operator << ( ostream & os, const CanonicalForm & cf )
{
    cf.print( os, "" );
    return os;
}

istream&
operator >> ( istream & is, CanonicalForm & cf )
{
#ifndef SINGULAR
    cf = readCF( is );
    return is;
#else /* SINGULAR */
    return 0;
#endif /* SINGULAR */
}
#endif /* NOSTREAMIO */
//}}}

//{{{ genCoeff(), genOne(), genZero()
CanonicalForm
CanonicalForm::genCoeff( int type, int i )
{
    return CanonicalForm( CFFactory::basic( type, i ) );
}

CanonicalForm
CanonicalForm::genZero() const
{
    int what = is_imm( value );
    if ( what == FFMARK )
        return CanonicalForm( CFFactory::basic( FiniteFieldDomain, 0 ) );
    else  if ( what == GFMARK )
        return CanonicalForm( CFFactory::basic( GaloisFieldDomain, 0 ) );
    else  if ( what )
        return CanonicalForm( CFFactory::basic( IntegerDomain, 0 ) );
    else
        return CanonicalForm( value->genZero() );
}

CanonicalForm
CanonicalForm::genOne() const
{
    int what = is_imm( value );
    if ( what == FFMARK )
        return CanonicalForm( CFFactory::basic( FiniteFieldDomain, 1 ) );
    else  if ( what == GFMARK )
        return CanonicalForm( CFFactory::basic( GaloisFieldDomain, 1 ) );
    else  if ( what )
        return CanonicalForm( CFFactory::basic( IntegerDomain, 1 ) );
    else
        return CanonicalForm( value->genOne() );
}
//}}}

//{{{ exponentiation
CanonicalForm
power ( const CanonicalForm & f, int n )
{
    ASSERT( n >= 0, "illegal exponent" );
    if ( f == 0 )
        return 0;
    else  if ( f == 1 )
        return f;
    else  if ( f == -1 ) {
        if ( n % 2 == 0 )
            return 1;
        else
            return -1;
    }
    else  if ( n == 0 )
        return 1;
    else {
        CanonicalForm result = f;
        for ( int i = 1; i < n; i++ )
            result *= f;
        return result;
    }
}

CanonicalForm
power ( const Variable & v, int n )
{
    ASSERT( n >= 0, "illegal exponent" );
    if ( n == 0 )
        return 1;
    else  if ( n == 1 )
        return v;
    else  if ( v.level() < 0 ) {
        CanonicalForm result( v, n-1 );
        return result * v;
    }
    else
        return CanonicalForm( v, n );
}
//}}}

//{{{ switches
void
On( int sw )
{
    cf_glob_switches.On( sw );
}

void
Off( int sw )
{
    cf_glob_switches.Off( sw );
}

bool
isOn( int sw )
{
    return cf_glob_switches.isOn( sw );
}
//}}}