polywrap.cc

Go to the documentation of this file.

 
/* #[ License : */
/*
 *   Copyright (C) 1984-2026 J.A.M. Vermaseren
 *   When using this file you are requested to refer to the publication
 *   J.A.M.Vermaseren "New features of FORM" math-ph/0010025
 *   This is considered a matter of courtesy as the development was paid
 *   for by FOM the Dutch physics granting agency and we would like to
 *   be able to track its scientific use to convince FOM of its value
 *   for the community.
 *
 *   This file is part of FORM.
 *
 *   FORM is free software: you can redistribute it and/or modify it under the
 *   terms of the GNU General Public License as published by the Free Software
 *   Foundation, either version 3 of the License, or (at your option) any later
 *   version.
 *
 *   FORM is distributed in the hope that it will be useful, but WITHOUT ANY
 *   WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
 *   FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more
 *   details.
 *
 *   You should have received a copy of the GNU General Public License along
 *   with FORM.  If not, see <http://www.gnu.org/licenses/>.
 */
/* #] License : */ 
 
#include "poly.h"
#include "polygcd.h"
#include "polyfact.h"
 
#include <iostream>
#include <vector>
#include <map>
#include <climits>
#include <cassert>
 
//#define DEBUG
 
#ifdef DEBUG
#include "mytime.h"
#endif
 
// denompower is increased by this factor when divmod_heap fails
const int POLYWRAP_DENOMPOWER_INCREASE_FACTOR = 2;
 
using namespace std;
 
/*
    #[ poly_determine_modulus :
*/
 
WORD poly_determine_modulus (PHEAD bool multi_error, bool is_fun_arg, string message) {
    
    if (AC.ncmod==0) return 0;
 
    if (!is_fun_arg || (AC.modmode & ALSOFUNARGS)) {
        
        if (ABS(AC.ncmod)>1) {
            MLOCK(ErrorMessageLock);
            MesPrint ((char*)"ERROR: %s with modulus > WORDSIZE not implemented",message.c_str());
            MUNLOCK(ErrorMessageLock);
            Terminate(-1);
        }
            
        if (multi_error && AN.poly_num_vars>1) {
            MLOCK(ErrorMessageLock);
            MesPrint ((char*)"ERROR: multivariate %s with modulus not implemented",message.c_str());
            MUNLOCK(ErrorMessageLock);
            Terminate(-1);
        }
            
        return *AC.cmod;
    }
 
    AN.ncmod = 0;
    return 0;
}
 
/*
    #] poly_determine_modulus : 
    #[ poly_gcd :
*/
 
WORD *poly_gcd(PHEAD WORD *a, WORD *b, WORD fit) {
 
#ifdef WITHFLINT
    if ( AC.FlintPolyFlag && AC.ncmod==0 ) {
        WORD *ret = flint_gcd(BHEAD a, b, fit);
        return ret;
    }
#endif
 
#ifdef DEBUG
    cout << "*** [" << thetime() << "]  CALL : poly_gcd" << endl;
#endif
 
//
//MesPrint("Calling poly_gcd with:");
//{
//  WORD *at = a;
//  MesPrint("      a:");
//  while ( *at ) {
//      MesPrint("   %a",*at,at);
//      at += *at;
//  }
//  MesPrint("      b:");
//  at = b;
//  while ( *at ) {
//      MesPrint("   %a",*at,at);
//      at += *at;
//  }
//}
    
    // Extract variables
    vector<WORD *> e;
    e.reserve(2);
    e.push_back(a);
    e.push_back(b);
    poly::get_variables(BHEAD e, false, true);
 
    // Check for modulus calculus
    WORD modp=poly_determine_modulus(BHEAD true, true, "polynomial GCD");
 
    // Convert to polynomials
    poly pa(poly::argument_to_poly(BHEAD a, false, true), modp, 1);
    poly pb(poly::argument_to_poly(BHEAD b, false, true), modp, 1);
 
    // Calculate gcd
    poly gcd(polygcd::gcd(pa,pb));
    
    // Allocate new memory and convert to Form notation
    int newsize = (gcd.size_of_form_notation()+1);
    WORD *res;
    if ( fit ) {
        if ( newsize*sizeof(WORD) >= (size_t)(AM.MaxTer) ) {
            MLOCK(ErrorMessageLock);
            MesPrint("poly_gcd: Term too complex (%d words). Maybe increasing MaxTermSize (%d words) can help",
                newsize, AM.MaxTer/sizeof(WORD));
            MUNLOCK(ErrorMessageLock);
            Terminate(-1);
        }
        res = TermMalloc("poly_gcd");
    }
    else {
        res = (WORD *)Malloc1(newsize*sizeof(WORD), "poly_gcd");
    }
    poly::poly_to_argument(gcd, res, false);
 
    poly_free_poly_vars(BHEAD "AN.poly_vars_qcd");
 
    // reset modulo calculation
    AN.ncmod = AC.ncmod;
    return res;
}
 
/*
    #] poly_gcd : 
    #[ poly_divmod :
 
    if fit == 1 the answer must fit inside a term.
*/
 
WORD *poly_divmod(PHEAD WORD *a, WORD *b, int divmod, WORD fit) {
 
#ifdef DEBUG
    cout << "*** [" << thetime() << "]  CALL : poly_divmod" << endl;
#endif
 
    // check for modulus calculus
    WORD modp=poly_determine_modulus(BHEAD false, true, "polynomial division");
 
    // get variables
    vector<WORD *> e;
    e.reserve(2);
    e.push_back(a);
    e.push_back(b);
    poly::get_variables(BHEAD e, false, false);
 
    // add extra variables to keep track of denominators
    const int DENOMSYMBOL = MAXPOSITIVE;
    
//  WORD *new_poly_vars = (WORD *)Malloc1((AN.poly_num_vars+1)*sizeof(WORD), "AN.poly_vars");
//  WCOPY(new_poly_vars, AN.poly_vars, AN.poly_num_vars);
//  new_poly_vars[AN.poly_num_vars] = DENOMSYMBOL;
//  if (AN.poly_num_vars > 0)
//      M_free(AN.poly_vars, "AN.poly_vars");
//  AN.poly_num_vars++;
//  AN.poly_vars = new_poly_vars;
    AN.poly_vars[AN.poly_num_vars++] = DENOMSYMBOL;
    
    // convert to polynomials
    poly dena(BHEAD 0);
    poly denb(BHEAD 0);
    poly pa(poly::argument_to_poly(BHEAD a, false, true, &dena), modp, 1);
    poly pb(poly::argument_to_poly(BHEAD b, false, true, &denb), modp, 1);
    
    // remove contents
    poly numres(polygcd::integer_content(pa));
    poly denres(polygcd::integer_content(pb));
    pa /= numres;
    pb /= denres;
 
    if (divmod==0) {
        numres *= denb;
        denres *= dena;
    }
    else {
        denres = dena;
    }
 
    poly gcdres(polygcd::integer_gcd(numres,denres));
    numres /= gcdres;
    denres /= gcdres;                           
 
    // determine lcoeff(b)
    poly lcoeffb(pb.integer_lcoeff());
 
    poly pres(BHEAD 0);
 
    if (!lcoeffb.is_one()) {
 
        if (AN.poly_num_vars > 2) {
            // the original polynomial is multivariate (one dummy variable has
            // been added), so it is not trivial to determine which power of
            // lcoeff(b) can be in the answer
 
            int denompower = 0, prevdenompower = 0;
            poly pq(BHEAD 0);
            poly pr(BHEAD 0);
 
            // try denompower = 0, if this fails increase denompower until division succeeds
            bool div_fail = true;
            do
            {
                if(denompower < prevdenompower)
                {
                    // denompower increased beyond INT_MAX
/* INTERNAL_ERROR_EXCL_START */
                    MLOCK(ErrorMessageLock);
                    MesPrint ((char*)"!>ERROR: pseudo-division failed in poly_divmod (denompower > INT_MAX)");
                    MUNLOCK(ErrorMessageLock);
                    Terminate(1);
/* INTERNAL_ERROR_EXCL_STOP */
                }
 
                if(denompower != 0)
                {
                    // multiply a by lcoeffb^(denompower-prevdenompower)
                    WORD n = lcoeffb[lcoeffb[1]];
                    RaisPow(BHEAD (UWORD *)&lcoeffb[2+AN.poly_num_vars], &n, denompower-prevdenompower);
                    lcoeffb[1] = 2 + AN.poly_num_vars + ABS(n);
                    lcoeffb[0] = 1 + lcoeffb[1];
                    lcoeffb[lcoeffb[1]] = n;
 
                    pa *= lcoeffb;
                    denres *= lcoeffb;
                }
 
                // try division
                poly ppow(BHEAD 0);
                poly::divmod_heap(pa,pb,pq,pr,false,true,div_fail); // sets div_fail
 
                // increase denompower for next iteration
                prevdenompower = denompower;
                denompower = (denompower==0 ? 1 : denompower*POLYWRAP_DENOMPOWER_INCREASE_FACTOR+1 ); // generates 2^n-1 when POLYWRAP_DENOMPOWER_INCREASE_FACTOR = 2
            }
            while(div_fail);
 
            pres = (divmod==0 ? pq : pr);
 
        }
        else {
            // one variable, so the power is the difference of the degrees
 
            int denompower = MaX(0, pa.degree(0) - pb.degree(0) + 1);
 
            // multiply a by that power
            WORD n = lcoeffb[lcoeffb[1]];
            RaisPow(BHEAD (UWORD *)&lcoeffb[2+AN.poly_num_vars], &n, denompower);
            lcoeffb[1] = 2 + AN.poly_num_vars + ABS(n);
            lcoeffb[0] = 1 + lcoeffb[1];
            lcoeffb[lcoeffb[1]] = n;
 
            pa *= lcoeffb;
            denres *= lcoeffb;
 
            pres = (divmod==0 ? pa/pb : pa%pb);
 
        }
 
    }
    else {
 
        pres = (divmod==0 ? pa/pb : pa%pb);
 
    }
 
    // convert to Form notation
    // NOTE: this part can be rewritten with poly::size_of_form_notation_with_den()
    // and poly::poly_to_argument_with_den().
    WORD *res;
 
    // special case: a=0
    if (pres[0]==1) {
        if ( fit ) {
            res = TermMalloc("poly_divmod");
        }
        else {
            res = (WORD *)Malloc1(sizeof(WORD), "poly_divmod");
        }
        res[0] = 0;
    }
    else {
        pres *= numres;
        
        WORD nden;
        UWORD *den = (UWORD *)NumberMalloc("poly_divmod");
 
        // allocate the memory; note that this overestimates the size,
        // since the estimated denominators are too large
        int ressize = pres.size_of_form_notation() + pres.number_of_terms()*2*ABS(denres[denres[1]]) + 1;
 
        if ( fit ) {
            if ( ressize*sizeof(WORD) > (size_t)(AM.MaxTer) ) {
                MLOCK(ErrorMessageLock);
                MesPrint("poly_divmod: Term too complex (%d words). Maybe increasing MaxTermSize (%d words) can help",
                    ressize, AM.MaxTer/sizeof(WORD));
                MUNLOCK(ErrorMessageLock);
                Terminate(-1);
            }
            res = TermMalloc("poly_divmod");
        }
        else {
            res = (WORD *)Malloc1(ressize*sizeof(WORD), "poly_divmod");
        }
        int L=0;
 
        for (int i=1; i!=pres[0]; i+=pres[i]) {
 
            res[L]=1; // length
            bool first = true;
            
            for (int j=0; j<AN.poly_num_vars; j++)
                if (pres[i+1+j] > 0) {
                    if (first) {
                        first = false;
                        res[L+1] = 1; // symbols
                        res[L+2] = 2; // length
                    }
                    res[L+1+res[L+2]++] = AN.poly_vars[j]; // symbol
                    res[L+1+res[L+2]++] = pres[i+1+j];     // power
                }
            
            if (!first) res[L] += res[L+2]; // fix length
 
            // numerator            
            WORD nnum = pres[i+pres[i]-1];
            WCOPY(&res[L+res[L]], &pres[i+pres[i]-1-ABS(nnum)], ABS(nnum));
 
            // calculate denominator
            nden = denres[denres[1]];
            WCOPY(den, &denres[2+AN.poly_num_vars], ABS(nden));
 
            if (nden!=1 || den[0]!=1)
                Simplify(BHEAD (UWORD *)&res[L+res[L]], &nnum, den, &nden); // gcd(num,den)
            Pack((UWORD *)&res[L+res[L]], &nnum, den, nden);              // format
            res[L] += 2*ABS(nnum)+1;                                      // fix length
            res[L+res[L]-1] = SGN(nnum)*(2*ABS(nnum)+1);                  // length of coefficient
            L += res[L];                                                  // fix length
        }
        
        res[L] = 0;
        
        NumberFree(den,"poly_divmod");  
    }
 
    // clean up
    poly_free_poly_vars(BHEAD "AN.poly_vars_divmod");
 
    // reset modulo calculation
    AN.ncmod = AC.ncmod;
 
    return res;
}
 
/*
    #] poly_divmod : 
    #[ poly_div :
 
    Routine divides the expression in arg1 by the expression in arg2.
    We did not take out special cases.
    The arguments are zero terminated sequences of term(s).
    The action is to divide arg1 by arg2: [arg1/arg2].
    The answer should be a buffer (allocated by Malloc1) with a zero
    terminated sequence of terms (or just zero).
*/
WORD *poly_div(PHEAD WORD *a, WORD *b, WORD fit) {
 
#ifdef WITHFLINT
    if ( AC.FlintPolyFlag && AC.ncmod==0 ) {
        WORD *ret = flint_div(BHEAD a, b, fit);
        return ret;
    }
#endif
 
#ifdef DEBUG
    cout << "*** [" << thetime() << "]  CALL : poly_div" << endl;
#endif
    
    return poly_divmod(BHEAD a, b, 0, fit);
}
 
/*
    #] poly_div : 
    #[ poly_rem :
 
    Routine divides the expression in arg1 by the expression in arg2
    and takes the remainder.
    We did not take out special cases.
    The arguments are zero terminated sequences of term(s).
    The action is to divide arg1 by arg2 and take the remainder: [arg1%arg2].
    The answer should be a buffer (allocated by Malloc1) with a zero
    terminated sequence of terms (or just zero).
*/
WORD *poly_rem(PHEAD WORD *a, WORD *b, WORD fit) {
 
#ifdef WITHFLINT
    if ( AC.FlintPolyFlag && AC.ncmod==0 ) {
        WORD *ret = flint_rem(BHEAD a, b, fit);
        return ret;
    }
#endif
 
#ifdef DEBUG
    cout << "*** [" << thetime() << "]  CALL : poly_rem" << endl;
#endif
 
    return poly_divmod(BHEAD a, b, 1, fit);
}
 
/*
    #] poly_rem : 
    #[ poly_ratfun_read :
*/
 
void poly_ratfun_read (WORD *a, poly &num, poly &den) {
 
#ifdef DEBUG
    cout << "*** [" << thetime() << "]  CALL : poly_ratfun_read" << endl;
#endif
 
    POLY_GETIDENTITY(num);
 
    int modp = num.modp;
    
    WORD *astop = a+a[1];
 
    bool clean = (a[2] & MUSTCLEANPRF) == 0;
 
    a += FUNHEAD;
    if (a >= astop) {
        MLOCK(ErrorMessageLock);
        MesPrint ((char*)"ERROR: PolyRatFun cannot have zero arguments");
        MUNLOCK(ErrorMessageLock);
        Terminate(-1);
    }
 
    poly den_num(BHEAD 1),den_den(BHEAD 1);
    
    num = poly::argument_to_poly(BHEAD a, true, !clean, &den_num);
    num.setmod(modp,1);
    NEXTARG(a);
    
    if (a < astop) {
        den = poly::argument_to_poly(BHEAD a, true, !clean, &den_den);
        den.setmod(modp,1);
        NEXTARG(a);
    }
    else {
        den = poly(BHEAD 1, modp, 1);
    }
 
    if (a < astop) {
        MLOCK(ErrorMessageLock);
        MesPrint ((char*)"ERROR: PolyRatFun cannot have more than two arguments");
        MUNLOCK(ErrorMessageLock);
        Terminate(-1);
    }
 
    // JD: At this point, num and den are certainly sorted into the correct order by
    // poly::argument_to_poly, but we can't rely on the clean flag to know if there
    // are any negative powers. Check for them, and set clean = false if there are any.
    vector<WORD> minpower(AN.poly_num_vars, MAXPOSITIVE);
 
    for (int i=1; i<num[0]; i+=num[i]) {
        for (int j=0; j<AN.poly_num_vars; j++) {
            minpower[j] = MiN(minpower[j], num[i+1+j]);
        }
    }
    for (int i=1; i<den[0]; i+=den[i]) {
        for (int j=0; j<AN.poly_num_vars; j++) {
            minpower[j] = MiN(minpower[j], den[i+1+j]);
            if ( minpower[j] < 0 ) clean = false;
        }
    }
 
    if (!clean) {
        for (int i=1; i<num[0]; i+=num[i])
            for (int j=0; j<AN.poly_num_vars; j++)
                num[i+1+j] -= minpower[j];
        for (int i=1; i<den[0]; i+=den[i])
            for (int j=0; j<AN.poly_num_vars; j++)
                den[i+1+j] -= minpower[j];
 
        num *= den_den;
        den *= den_num;
 
        poly gcd = polygcd::gcd(num,den);
        num /= gcd;
        den /= gcd;
    }
}
 
/*
    #] poly_ratfun_read : 
    #[ poly_sort :
*/
 
void poly_sort(PHEAD WORD *a) {
 
#ifdef DEBUG
    cout << "*** [" << thetime() << "]  CALL : poly_sort" << endl;
#endif
    if (NewSort(BHEAD0)) { Terminate(-1); }
    AR.CompareRoutine = (COMPAREDUMMY)(&CompareSymbols);
    
    for (int i=ARGHEAD; i<a[0]; i+=a[i]) {
        if (SymbolNormalize(a+i)<0 || StoreTerm(BHEAD a+i)) {
            AR.CompareRoutine = (COMPAREDUMMY)(&Compare1);
            LowerSortLevel();
            Terminate(-1);
        }
    }
    
    if (EndSort(BHEAD a+ARGHEAD,1) < 0) {
        AR.CompareRoutine = (COMPAREDUMMY)(&Compare1);
        Terminate(-1);
    }
    
    AR.CompareRoutine = (COMPAREDUMMY)(&Compare1);
    a[1] = 0; // set dirty flag to zero
}
 
/*
    #] poly_sort : 
    #[ poly_ratfun_add :
*/
 
WORD *poly_ratfun_add (PHEAD WORD *t1, WORD *t2) {
 
#ifdef WITHFLINT
    if ( AC.FlintPolyFlag && AC.ncmod==0 ) {
        WORD *ret = flint_ratfun_add(BHEAD t1, t2);
        return ret;
    }
#endif
 
    if ( AR.PolyFunExp == 1 ) return PolyRatFunSpecial(BHEAD t1, t2);
 
#ifdef DEBUG
    cout << "*** [" << thetime() << "]  CALL : poly_ratfun_add" << endl;
#endif
 
    WORD *oldworkpointer = AT.WorkPointer;
    
    // Extract variables
    vector<WORD *> e;
    e.reserve(4);
    
    for (WORD *t=t1+FUNHEAD; t<t1+t1[1];) {
        e.push_back(t);
        NEXTARG(t);
    }
    for (WORD *t=t2+FUNHEAD; t<t2+t2[1];) {
        e.push_back(t);
        NEXTARG(t);
    }
 
    assert(e.size() == 4);
 
    poly::get_variables(BHEAD e, true, true);
    
    // Check for modulus calculus
    WORD modp=poly_determine_modulus(BHEAD true, true, "PolyRatFun");
    
    // Find numerators / denominators
    poly num1(BHEAD 0,modp,1), den1(BHEAD 0,modp,1), num2(BHEAD 0,modp,1), den2(BHEAD 0,modp,1);
 
    poly_ratfun_read(t1, num1, den1);
    poly_ratfun_read(t2, num2, den2);
 
    poly num(BHEAD 0),den(BHEAD 0),gcd(BHEAD 0);
 
    // Calculate result
    if (den1 != den2) {
        gcd = polygcd::gcd(den1,den2);
#ifdef OLDADDITION
        num = num1*(den2/gcd) + num2*(den1/gcd);
        den = (den1/gcd)*den2;
        gcd = polygcd::gcd(num,den);
#else
        den = den1/gcd;
        num = num1*(den2/gcd) + num2*den;
        den = den*den2;
        gcd = polygcd::gcd(num,gcd);
#endif
    }
    else {
        num = num1 + num2;
        den = den1;
        gcd = polygcd::gcd(num,den);
    }
 
    num /= gcd;
    den /= gcd;
    
    // Fix sign
    if (den.sign() == -1) { num*=poly(BHEAD -1); den*=poly(BHEAD -1); }
 
    // Check size: include FUNHEAD for the prf itself, an ARGHEAD each for num and den,
    // and 3 for the final coeff "1/1". We don't know here what the rest of the term looks like,
    // but it certainly has at least its total size (so +1):
    if ((num.size_of_form_notation() + den.size_of_form_notation() + FUNHEAD + 2*ARGHEAD + 3 + 1)
        > AM.MaxTer/(int)sizeof(WORD)) {
 
        MLOCK(ErrorMessageLock);
        MesPrint ("ERROR: PolyRatFun doesn't fit in a term");
        MesPrint ("(1) num size = %d, den size = %d, rest = %d, MaxTermSize = %d words",
                num.size_of_form_notation()+ARGHEAD,
                den.size_of_form_notation()+ARGHEAD,
                FUNHEAD + 3 + 1,
                AM.MaxTer/sizeof(WORD));
        MUNLOCK(ErrorMessageLock);
        Terminate(-1);
    }
 
    // Format result in Form notation
    WORD *t = oldworkpointer;
 
    *t++ = AR.PolyFun;                   // function 
    *t++ = 0;                            // length (to be determined)
//  *t++ &= ~MUSTCLEANPRF;               // clean polyratfun
    *t++ = 0;
    FILLFUN3(t);                         // header
    poly::poly_to_argument(num,t, true); // argument 1 (numerator)
    if (*t>0 && t[1]==DIRTYFLAG)          // to Form order
        poly_sort(BHEAD t);        
    t += (*t>0 ? *t : 2);
    poly::poly_to_argument(den,t, true); // argument 2 (denominator)
    if (*t>0 && t[1]==DIRTYFLAG)          // to Form order
        poly_sort(BHEAD t);        
    t += (*t>0 ? *t : 2);
 
    oldworkpointer[1] = t - oldworkpointer; // length
    AT.WorkPointer = t;
 
    poly_free_poly_vars(BHEAD "AN.poly_vars_ratfun_add");
 
    // reset modulo calculation
    AN.ncmod = AC.ncmod;
    
    return oldworkpointer;
}
 
/*
    #] poly_ratfun_add : 
    #[ poly_ratfun_normalize :
*/
 
int poly_ratfun_normalize (PHEAD WORD *term) {
 
#ifdef WITHFLINT
    if ( AC.FlintPolyFlag && AC.ncmod==0 ) {
        flint_ratfun_normalize(BHEAD term);
        return 0;
    }
#endif
 
#ifdef DEBUG
    cout << "*** [" << thetime() << "]  CALL : poly_ratfun_normalize" << endl;
#endif
    
    // Strip coefficient
    WORD *tstop = term + *term;
    int ncoeff = tstop[-1];
    tstop -= ABS(ncoeff);
 
    // if only one clean polyratfun, return immediately
    int num_polyratfun = 0;
 
    for (WORD *t=term+1; t<tstop; t+=t[1]) 
        if (*t == AR.PolyFun) {
            num_polyratfun++;
            if ((t[2] & MUSTCLEANPRF) != 0)
                num_polyratfun = INT_MAX;
            if (num_polyratfun > 1) break;
        }
        
    if (num_polyratfun <= 1) return 0;
 
    WORD oldsorttype = AR.SortType;
    AR.SortType = SORTHIGHFIRST;
 
/*
    When there are polyratfun's with only one variable: rename them
    temporarily to TMPPOLYFUN.
*/
    for (WORD *t=term+1; t<tstop; t+=t[1]) {
        if (*t == AR.PolyFun && (t[1] == FUNHEAD+t[FUNHEAD]
            || t[1] == FUNHEAD+2 ) ) { *t = TMPPOLYFUN; }
    }
 
    
    // Extract all variables in the polyfuns
    vector<WORD *> e;
    
    for (WORD *t=term+1; t<tstop; t+=t[1]) {
        if (*t == AR.PolyFun) 
            for (WORD *t2 = t+FUNHEAD; t2<t+t[1];) {
                e.push_back(t2);
                NEXTARG(t2);
            }       
    }
    poly::get_variables(BHEAD e, true, true);
 
    // Check for modulus calculus
    WORD modp=poly_determine_modulus(BHEAD true, true, "PolyRatFun");
    
    // Accumulate total denominator/numerator and copy the remaining terms
    // We start with 'trivial' polynomials
    poly num1(BHEAD (UWORD *)tstop, ncoeff/2, modp, 1);
    poly den1(BHEAD (UWORD *)tstop+ABS(ncoeff/2), ABS(ncoeff)/2, modp, 1);
 
    WORD *s = term+1;
 
    for (WORD *t=term+1; t<tstop;) 
        if (*t == AR.PolyFun) {
 
            poly num2(BHEAD 0,modp,1);
            poly den2(BHEAD 0,modp,1);
            poly_ratfun_read(t,num2,den2);
 
            if ((t[2] & MUSTCLEANPRF) != 0) { // first normalize
                poly gcd1(polygcd::gcd(num2,den2));
                num2 = num2/gcd1;
                den2 = den2/gcd1;
            }
            t += t[1];
            poly gcd1(polygcd::gcd(num1,den2));
            poly gcd2(polygcd::gcd(num2,den1));
 
            num1 = (num1 / gcd1) * (num2 / gcd2);
            den1 = (den1 / gcd2) * (den2 / gcd1);
        }
        else {
            int i = t[1];
            if ( s != t ) { NCOPY(s,t,i) }
            else { t += i; s += i; }
        }           
 
    // Fix sign
    if (den1.sign() == -1) { num1*=poly(BHEAD -1); den1*=poly(BHEAD -1); }
 
    // Check size: include FUNHEAD for the prf itself, an ARGHEAD each for num and den,
    // s-term for the copied term so far, and 3 for final coeff "1/1"
    if ((num1.size_of_form_notation() + den1.size_of_form_notation() + FUNHEAD + 2*ARGHEAD
        + s-term + 3) > AM.MaxTer/(int)sizeof(WORD)) {
 
        MLOCK(ErrorMessageLock);
        MesPrint ("ERROR: PolyRatFun doesn't fit in a term");
        MesPrint ("(2) num size = %d, den size = %d, rest = %d, MaxTermSize = %d words",
                num1.size_of_form_notation()+ARGHEAD,
                den1.size_of_form_notation()+ARGHEAD,
                FUNHEAD + s-term + 3,
                AM.MaxTer/sizeof(WORD));
        MUNLOCK(ErrorMessageLock);
        Terminate(-1);
    }
 
    // Format result in Form notation
    WORD *t = s;
    *t++ = AR.PolyFun;                   // function
    *t++ = 0;                            // size (to be determined)
    *t++ &= ~MUSTCLEANPRF;                   // clean polyratfun
    FILLFUN3(t);                         // header
    poly::poly_to_argument(num1,t,true); // argument 1 (numerator)
    if (*t>0 && t[1]==DIRTYFLAG)         // to Form order
        poly_sort(BHEAD t);
    t += (*t>0 ? *t : 2);
    poly::poly_to_argument(den1,t,true); // argument 2 (denominator)
    if (*t>0 && t[1]==DIRTYFLAG)         // to Form order
        poly_sort(BHEAD t);        
    t += (*t>0 ? *t : 2);
 
    s[1] = t - s;                        // function length
 
    *t++ = 1;                            // term coefficient
    *t++ = 1;
    *t++ = 3;
    
    term[0] = t-term;                    // term length
 
    poly_free_poly_vars(BHEAD "AN.poly_vars_ratfun_normalize");
 
    // reset modulo calculation
    AN.ncmod = AC.ncmod;
 
    tstop = term + *term; tstop -= ABS(tstop[-1]);
    for (WORD *t=term+1; t<tstop; t+=t[1]) {
        if (*t == TMPPOLYFUN ) *t = AR.PolyFun;
    }
 
    AR.SortType = oldsorttype;
    return 0;
}
 
/*
    #] poly_ratfun_normalize : 
    #[ poly_fix_minus_signs :
*/
 
void poly_fix_minus_signs (factorized_poly &a) {
 
    if ( a.factor.empty() ) return;
 
    POLY_GETIDENTITY(a.factor[0]);
    
    int overall_sign = +1;
 
    // find term with maximum power of highest symbol
    for (int i=0; i<(int)a.factor.size(); i++) {
 
        int maxvar = -1;
        int maxpow = -1;
        int sign = +1;
 
        WORD *tstop = a.factor[i].terms; tstop += *tstop;
        for (WORD *t=a.factor[i].terms+1; t<tstop; t+=*t) 
            for (int j=0; j<AN.poly_num_vars; j++) {
                int var = AN.poly_vars[j];
                int pow = t[1+j];
                if (pow>0 && (var>maxvar || (var==maxvar && pow>maxpow))) {
                    maxvar = var;
                    maxpow = pow;
                    sign = SGN(*(t+*t-1));
                }
            }
 
        // if negative coefficient, multiply by -1
        if (sign==-1) {
            a.factor[i] *= poly(BHEAD sign);
            if (a.power[i] % 2 == 1) overall_sign*=-1;
        }
    }
 
    // if overall minus sign
    if (overall_sign == -1) {
        // look at constant factor and multiply by -1
        for (int i=0; i<(int)a.factor.size(); i++)
            if (a.factor[i].is_integer()) {
                a.factor[i] *= poly(BHEAD -1);
                return;
            }
 
        // otherwise, add a factor of -1
        a.add_factor(poly(BHEAD -1), 1);
    }
}
 
/*
    #] poly_fix_minus_signs : 
    #[ poly_factorize :
*/
 
WORD *poly_factorize (PHEAD WORD *argin, WORD *argout, bool with_arghead, bool is_fun_arg) {
 
#ifdef DEBUG
    cout << "*** [" << thetime() << "]  CALL : poly_factorize" << endl;
#endif
    
    poly::get_variables(BHEAD vector<WORD*>(1,argin), with_arghead, true);
    poly den(BHEAD 0);
    poly a(poly::argument_to_poly(BHEAD argin, with_arghead, true, &den));
 
    // check for modulus calculus
    WORD modp=poly_determine_modulus(BHEAD true, is_fun_arg, "polynomial factorization");
    a.setmod(modp,1);
 
    // factorize
    factorized_poly f(polyfact::factorize(a));
    poly_fix_minus_signs(f);
 
    poly num(BHEAD 1);
    for (int i=0; i<(int)f.factor.size(); i++)
        if (f.factor[i].is_integer())
            num = f.factor[i];  
    
    // determine size
    int len = with_arghead ? ARGHEAD : 0;
 
    if (!num.is_one() || !den.is_one()) {
        len++;
        len += MaX(ABS(num[num[1]]), den[den[1]])*2+1;
        len += with_arghead ? ARGHEAD : 1;
    }
    
    for (int i=0; i<(int)f.factor.size(); i++) {
        if (!f.factor[i].is_integer()) {
            len += f.power[i] * f.factor[i].size_of_form_notation();
            len += f.power[i] * (with_arghead ? ARGHEAD : 1);
        }
    }
    
    len++;
    
    if (argout != NULL) {
        // check size
        if (len >= AM.MaxTer) {
            MLOCK(ErrorMessageLock);
            MesPrint ("ERROR: factorization doesn't fit in a term (len = %d, MaxTermSize = %d words)",
                len/sizeof(WORD), AM.MaxTer/sizeof(WORD));
            MUNLOCK(ErrorMessageLock);
            Terminate(-1);
        }
    }
    else {
        // allocate size
        argout = (WORD*) Malloc1(len*sizeof(WORD), "poly_factorize");
    }
 
    WORD *old_argout = argout;
 
    // constant factor
    if (!num.is_one() || !den.is_one()) {
        int n = max(ABS(num[num[1]]), ABS(den[den[1]]));
 
        if (with_arghead) {
            *argout++ = ARGHEAD + 2 + 2*n;
            for (int i=1; i<ARGHEAD; i++)
                *argout++ = 0;
        }
        
        *argout++ = 2 + 2*n;
        
        for (int i=0; i<n; i++) 
            *argout++ = i<ABS(num[num[1]]) ? num[2+AN.poly_num_vars+i] : 0;
        for (int i=0; i<n; i++) 
            *argout++ = i<ABS(den[den[1]]) ? den[2+AN.poly_num_vars+i] : 0;
 
        *argout++ = SGN(num[num[1]]) * (2*n+1);
 
        if (!with_arghead)
            *argout++ = 0;
        else {
            if (ToFast(old_argout, old_argout))
                argout = old_argout+2;
        }
    }
 
    // non-constant factors
    for (int i=0; i<(int)f.factor.size(); i++)
        if (!f.factor[i].is_integer()) 
            for (int j=0; j<f.power[i]; j++) {          
                poly::poly_to_argument(f.factor[i],argout,with_arghead);
                
                if (with_arghead)
                    argout += *argout > 0 ? *argout : 2;
                else {
                    while (*argout!=0) argout+=*argout;
                    argout++;
                }
            }
    
    *argout=0;
 
    poly_free_poly_vars(BHEAD "AN.poly_vars_factorize");
 
    // reset modulo calculation
    AN.ncmod = AC.ncmod;
    
    return old_argout;
}
 
/*
    #] poly_factorize : 
    #[ poly_factorize_argument :
*/
 
int poly_factorize_argument(PHEAD WORD *argin, WORD *argout) {
 
#ifdef DEBUG
    cout << "*** [" << thetime() << "]  CALL : poly_factorize_argument" << endl;
#endif
 
#ifdef WITHFLINT
    if ( AC.FlintPolyFlag && AC.ncmod==0 ) {
        flint_factorize_argument(BHEAD argin, argout);
        return 0;
    }
#endif
 
    poly_factorize(BHEAD argin,argout,true,true);
    return 0;
}
 
/*
    #] poly_factorize_argument : 
    #[ poly_factorize_dollar :
*/
 
WORD *poly_factorize_dollar (PHEAD WORD *argin) {
 
#ifdef DEBUG
    cout << "*** [" << thetime() << "]  CALL : poly_factorize_dollar" << endl;
#endif
 
#ifdef WITHFLINT
    if ( AC.FlintPolyFlag && AC.ncmod==0 ) {
        return flint_factorize_dollar(BHEAD argin);
    }
#endif
 
    return poly_factorize(BHEAD argin,NULL,false,false);
}
 
/*
    #] poly_factorize_dollar : 
    #[ poly_factorize_expression :
*/
 
int poly_factorize_expression(EXPRESSIONS expr) {
 
#ifdef DEBUG
    cout << "*** [" << thetime() << "]  CALL : poly_factorize_expression" << endl;
#endif
 
    GETIDENTITY;
 
    if (AT.WorkPointer + AM.MaxTer > AT.WorkTop ) {
        MLOCK(ErrorMessageLock);
        MesWork();
        MUNLOCK(ErrorMessageLock);
        Terminate(-1);
    }
    
    WORD *term = AT.WorkPointer;
    WORD startebuf = cbuf[AT.ebufnum].numrhs;
    FILEHANDLE *file;
    POSITION pos;
 
    FILEHANDLE *oldinfile = AR.infile;
    FILEHANDLE *oldoutfile = AR.outfile;
    WORD oldBracketOn = AR.BracketOn;
    WORD *oldBrackBuf = AT.BrackBuf;
    WORD oldbracketindexflag = AT.bracketindexflag;
    char oldCommercial[COMMERCIALSIZE+2];
 
    strcpy(oldCommercial, (char*)AC.Commercial);
    strcpy((char*)AC.Commercial, "factorize");
 
    // locate is the input  
    if (expr->status == HIDDENGEXPRESSION || expr->status == HIDDENLEXPRESSION ||
            expr->status == INTOHIDEGEXPRESSION || expr->status == INTOHIDELEXPRESSION) {
        AR.InHiBuf = 0; file = AR.hidefile; AR.GetFile = 2;
    }
    else {
        AR.InInBuf = 0; file = AR.outfile; AR.GetFile = 0;
    }
 
    // read and write to expression file
    AR.infile = AR.outfile = file;
    
    // dummy indices are not allowed
    if (expr->numdummies > 0) {
        MesPrint("ERROR: factorization with dummy indices not implemented");
        Terminate(-1);
    }
 
    // determine whether the expression in on file or in memory
    if (file->handle >= 0) {
        pos = expr->onfile;
        SeekFile(file->handle,&pos,SEEK_SET);
        if (ISNOTEQUALPOS(pos,expr->onfile)) {
/* INTERNAL_ERROR_EXCL_START */
            MesPrint("!>ERROR: something wrong in scratch file [poly_factorize_expression]");
            Terminate(-1);
/* INTERNAL_ERROR_EXCL_STOP */
        }
        file->POposition = expr->onfile;
        file->POfull = file->PObuffer;
        if (expr->status == HIDDENGEXPRESSION)
            AR.InHiBuf = 0;
        else
            AR.InInBuf = 0;
    }
    else {
        file->POfill = (WORD *)((UBYTE *)(file->PObuffer)+BASEPOSITION(expr->onfile));
    }
    
    SetScratch(AR.infile, &(expr->onfile));
 
    // read the first header term
    WORD size = GetTerm(BHEAD term);
    if (size <= 0) {
/* INTERNAL_ERROR_EXCL_START */
        MesPrint ("!>ERROR: something wrong with expression [poly_factorize_expression]");
        Terminate(-1);
/* INTERNAL_ERROR_EXCL_STOP */
    }
 
    // store position: this is where the output will go
    pos = expr->onfile;
    ADDPOS(pos, size*sizeof(WORD));
    
    // use polynomial as buffer, because it is easy to extend
    poly buffer(BHEAD 0);
    int bufpos = 0;
    int sumcommu = 0;
 
    // read all terms
    while (GetTerm(BHEAD term)) {
        // substitute non-symbols by extra symbols
        sumcommu += DoesCommu(term);
        if ( sumcommu > 1 ) {
            MesPrint("ERROR: Cannot factorize an expression with more than one noncommuting object");
            Terminate(-1);
        }
        buffer.check_memory(bufpos);        
        if (LocalConvertToPoly(BHEAD term, buffer.terms + bufpos, startebuf,0) < 0) {
/* INTERNAL_ERROR_EXCL_START */
            MesPrint("!>ERROR: in LocalConvertToPoly [factorize_expression]");
            Terminate(-1);
/* INTERNAL_ERROR_EXCL_STOP */
        }
        bufpos += *(buffer.terms + bufpos);
    }
    buffer[bufpos] = 0;
 
    // parse the polynomial
             
    AN.poly_num_vars = 0;
    poly::get_variables(BHEAD vector<WORD*>(1,buffer.terms), false, true);
    poly den(BHEAD 0);
    poly a(poly::argument_to_poly(BHEAD buffer.terms, false, true, &den));
 
    // check for modulus calculus
    WORD modp=poly_determine_modulus(BHEAD true, false, "polynomial factorization");
    a.setmod(modp,1);
 
    // create output
    SetScratch(file, &pos);
    NewSort(BHEAD0);    
    
    CBUF *C = cbuf+AC.cbufnum;
    CBUF *CC = cbuf+AT.ebufnum;
    
    // turn brackets on. We force the existence of a bracket index.
    WORD nexpr = expr - Expressions;
    AR.BracketOn = 1;
    AT.BrackBuf = AM.BracketFactors;
    AT.bracketindexflag = 1;
    ClearBracketIndex(-nexpr-2); // Clears the index made during primary generation
    OpenBracketIndex(nexpr);     // Set up a new index
        
    if (a.is_zero()) {
        expr->numfactors = 1;
    }
    else if (a.is_one() && den.is_one()) {
        expr->numfactors = 1;
        
        term[0] = 8;
        term[1] = SYMBOL;
        term[2] = 4;
        term[3] = FACTORSYMBOL;
        term[4] = 1;
        term[5] = 1;
        term[6] = 1;
        term[7] = 3;
 
        AT.WorkPointer += *term;
        Generator(BHEAD term, C->numlhs);
        AT.WorkPointer = term;
    }
    else {
        factorized_poly fac;
        bool iszero = false;
        
        if (!(expr->vflags & ISFACTORIZED)) {
            // factorize the polynomial
            fac = polyfact::factorize(a);
            poly_fix_minus_signs(fac);
        }
        else {
            int factorsymbol=-1;
            for (int i=0; i<AN.poly_num_vars; i++)
                if (AN.poly_vars[i] == FACTORSYMBOL)
                    factorsymbol = i;
 
            poly denpow(BHEAD 1);
            
            // already factorized, so factorize the factors
            for (int i=1; i<=expr->numfactors; i++) {
                poly origfac(a.coefficient(factorsymbol, i));
                factorized_poly fac2;
                if (origfac.is_zero())
                    iszero=true;
                else {
                    fac2 = polyfact::factorize(origfac);
                    poly_fix_minus_signs(fac2);
                    denpow *= den;
                }
                for (int j=0; j<(int)fac2.power.size(); j++)
                    fac.add_factor(fac2.factor[j], fac2.power[j]);
            }
 
            // update denominator, since each factor was scaled
            den=denpow;
        }
 
        expr->numfactors = 0;
        
        // coefficient
        poly num(BHEAD 1);      
        for (int i=0; i<(int)fac.factor.size(); i++) 
            if (fac.factor[i].is_integer())
                num *= fac.factor[i];
 
        poly gcd(polygcd::integer_gcd(num,den));
        den/=gcd;
        num/=gcd;
 
        if (iszero) 
            expr->numfactors++;
 
        if (!iszero || (expr->vflags & KEEPZERO)) {
            if (!num.is_one() || !den.is_one()) {
                expr->numfactors++;
                
                int n = max(ABS(num[num[1]]), ABS(den[den[1]]));
                
                term[0] = 6 + 2*n;
                term[1] = SYMBOL;
                term[2] = 4;
                term[3] = FACTORSYMBOL;
                term[4] = expr->numfactors;
                for (int i=0; i<n; i++) {
                    term[5+i]   = i<ABS(num[num[1]]) ? num[2+AN.poly_num_vars+i] : 0;
                    term[5+n+i] = i<ABS(den[den[1]]) ? den[2+AN.poly_num_vars+i] : 0;
                }
                term[5+2*n] = SGN(num[num[1]]) * (2*n+1);
                AT.WorkPointer += *term;
                Generator(BHEAD term, C->numlhs);
                AT.WorkPointer = term;
            }
 
            vector<poly> fac_arg(fac.factor.size(), poly(BHEAD 0));
            
            // convert the non-constant factors to Form-style arguments
            for (int i=0; i<(int)fac.factor.size(); i++)
                if (!fac.factor[i].is_integer()) {
                    buffer.check_memory(fac.factor[i].size_of_form_notation()+1);       
                    poly::poly_to_argument(fac.factor[i], buffer.terms, false);
                    NewSort(BHEAD0);
                    
                    for (WORD *t=buffer.terms; *t!=0; t+=*t) {
                        // substitute extra symbols
                        if (ConvertFromPoly(BHEAD t, term, numxsymbol, CC->numrhs-startebuf+numxsymbol,
                                                                startebuf-numxsymbol, 1) <= 0 ) {
/* INTERNAL_ERROR_EXCL_START */
                            MesPrint("!>ERROR: in ConvertFromPoly [factorize_expression]");
                            Terminate(-1);
                            return(-1);
/* INTERNAL_ERROR_EXCL_STOP */
                        }
                        
                        // store term
                        AT.WorkPointer += *term;
                        Generator(BHEAD term, C->numlhs);
                        AT.WorkPointer = term;
                    }
 
                    // sort and store in buffer
                    WORD *buffer;
                    if (EndSort(BHEAD (WORD *)((void *)(&buffer)),2) < 0) return -1;
                    
                    LONG bufsize=0;
                    for (WORD *t=buffer; *t!=0; t+=*t)
                        bufsize+=*t;
                    
                    fac_arg[i].check_memory(bufsize+ARGHEAD+1);
 
                    for (int j=0; j<ARGHEAD; j++)
                        fac_arg[i].terms[j] = 0;
                    fac_arg[i].terms[0] = ARGHEAD + bufsize;
                    WCOPY(fac_arg[i].terms+ARGHEAD, buffer, bufsize);
                    M_free(buffer, "polynomial factorization");
                }
        
            // compare and sort the factors in Form notation
            vector<int> order;
            vector<vector<int> > comp(fac.factor.size(), vector<int>(fac.factor.size(), 0));
            
            for (int i=0; i<(int)fac.factor.size(); i++)
                if (!fac.factor[i].is_integer()) {
                    order.push_back(i);
                    
                    for (int j=i+1; j<(int)fac.factor.size(); j++)
                        if (!fac.factor[j].is_integer()) {                      
                            comp[i][j] = CompArg(fac_arg[j].terms, fac_arg[i].terms);
                            comp[j][i] = -comp[i][j];
                        }
                }
            
            for (int i=0; i<(int)order.size(); i++) 
                for (int j=0; j+1<(int)order.size(); j++)
                    if (comp[order[i]][order[j]] == 1)
                        swap(order[i],order[j]);        
            
            // create the final expression
            for (int i=0; i<(int)order.size(); i++)
                for (int j=0; j<fac.power[order[i]]; j++) {
                    
                    expr->numfactors++;
                    
                    WORD *tstop = fac_arg[order[i]].terms + *fac_arg[order[i]].terms;
                    for (WORD *t=fac_arg[order[i]].terms+ARGHEAD; t<tstop; t+=*t) {
                        
                        WCOPY(term+4, t, *t);
                        
                        // add special symbol "factor_"
                        *term = *(term+4) + 4;
                        *(term+1) = SYMBOL;
                        *(term+2) = 4;
                        *(term+3) = FACTORSYMBOL;
                        *(term+4) = expr->numfactors;
                        
                        // store term
                        AT.WorkPointer += *term;
                        Generator(BHEAD term, C->numlhs);
                        AT.WorkPointer = term;
                    }               
                }
        }
    }
    
    // final sorting
    if (EndSort(BHEAD NULL,0) < 0) {
        LowerSortLevel();
        Terminate(-1);
    }
 
    // set factorized flag
    if (expr->numfactors > 0) 
        expr->vflags |= ISFACTORIZED;
 
    // clean up
    AR.infile = oldinfile;
    AR.outfile = oldoutfile;
    AR.BracketOn = oldBracketOn;
    AT.BrackBuf = oldBrackBuf;
    AT.bracketindexflag = oldbracketindexflag;
    strcpy((char*)AC.Commercial, oldCommercial);
    
    poly_free_poly_vars(BHEAD "AN.poly_vars_factorize_expression");
 
    return 0;
}
 
/*
    #] poly_factorize_expression : 
    #[ poly_unfactorize_expression :
*/
 
#if ( SUBEXPSIZE == 5 )
static WORD genericterm[] = {38,1,4,FACTORSYMBOL,0
    ,EXPRESSION,15,0,1,0,13,10,8,1,4,FACTORSYMBOL,0,1,1,3
    ,EXPRESSION,15,0,1,0,13,10,8,1,4,FACTORSYMBOL,0,1,1,3
    ,1,1,3,0};
static WORD genericterm2[] = {23,1,4,FACTORSYMBOL,0
    ,EXPRESSION,15,0,1,0,13,10,8,1,4,FACTORSYMBOL,0,1,1,3
    ,1,1,3,0};
#endif
 
int poly_unfactorize_expression(EXPRESSIONS expr)
{
    GETIDENTITY;
    int i, j, nfac = expr->numfactors, nfacp, nexpr = expr - Expressions;
    int expriszero = 0;
 
    FILEHANDLE *oldinfile = AR.infile;
    FILEHANDLE *oldoutfile = AR.outfile;
    char oldCommercial[COMMERCIALSIZE+2];
 
    WORD *oldworkpointer = AT.WorkPointer;  
    WORD *term = AT.WorkPointer, *t, *w, size;
    
    FILEHANDLE *file;
    POSITION pos, oldpos;
 
    WORD oldBracketOn = AR.BracketOn;
    WORD *oldBrackBuf = AT.BrackBuf;
    CBUF *C = cbuf+AC.cbufnum;
 
    if ( ( expr->vflags & ISFACTORIZED ) == 0 ) return(0);
 
    if ( AT.WorkPointer + AM.MaxTer > AT.WorkTop ) {
        MLOCK(ErrorMessageLock);
        MesWork();
        MUNLOCK(ErrorMessageLock);
        Terminate(-1);
    }
 
    oldpos = AS.OldOnFile[nexpr];
    AS.OldOnFile[nexpr] = expr->onfile;
 
    strcpy(oldCommercial, (char*)AC.Commercial);
    strcpy((char*)AC.Commercial, "unfactorize");
/*
    locate the input    
*/
    if ( expr->status == HIDDENGEXPRESSION || expr->status == HIDDENLEXPRESSION ||
            expr->status == INTOHIDEGEXPRESSION || expr->status == INTOHIDELEXPRESSION ) {
        AR.InHiBuf = 0; file = AR.hidefile; AR.GetFile = 2;
    }
    else {
        AR.InInBuf = 0; file = AR.outfile; AR.GetFile = 0;
    }
/*
    read and write to expression file
*/
    AR.infile = AR.outfile = file;
/*
    set the input file to the correct position  
*/
    if ( file->handle >= 0 ) {
        pos = expr->onfile;
        SeekFile(file->handle,&pos,SEEK_SET);
        if (ISNOTEQUALPOS(pos,expr->onfile)) {
/* INTERNAL_ERROR_EXCL_START */
            MesPrint("!>ERROR: something wrong in scratch file unfactorize_expression");
            Terminate(-1);
/* INTERNAL_ERROR_EXCL_STOP */
        }
        file->POposition = expr->onfile;
        file->POfull = file->PObuffer;
        if ( expr->status == HIDDENGEXPRESSION )
            AR.InHiBuf = 0;
        else
            AR.InInBuf = 0;
    }
    else {
        file->POfill = (WORD *)((UBYTE *)(file->PObuffer)+BASEPOSITION(expr->onfile));
    }
    SetScratch(AR.infile, &(expr->onfile));
/*
    Test for whether the first factor is zero.
*/
    if ( GetFirstBracket(term,nexpr) < 0 ) Terminate(-1);
    if ( term[4] != 1 || *term != 8 || term[1] != SYMBOL || term[3] != FACTORSYMBOL || term[4] != 1 ) {
        expriszero = 1;
    }
    SetScratch(AR.infile, &(expr->onfile));
/*
    Read the prototype. After this we have the file ready for the output at pos.
*/
    size = GetTerm(BHEAD term);
    if ( size <= 0 ) {
/* INTERNAL_ERROR_EXCL_START */
        MesPrint ("!>ERROR: something wrong with expression unfactorize_expression");
        Terminate(-1);
/* INTERNAL_ERROR_EXCL_STOP */
    }
    pos = expr->onfile;
    ADDPOS(pos, size*sizeof(WORD));
/*
    Set the brackets straight
*/
    AR.BracketOn = 1;
    AT.BrackBuf = AM.BracketFactors;
    AT.bracketinfo = 0;
    while ( nfac > 2 ) {
        nfacp = nfac - nfac%2;
/*
        Prepare the bracket index. We have:
            e->bracketinfo:    the old input bracket index
            e->newbracketinfo: the bracket index made for our current input
        We need to keep e->bracketinfo in case other workers need it (InParallel)
        Hence we work with AT.bracketinfo which takes priority.
        Note that in Processor we forced a newbracketinfo to be made.
*/
        if ( AT.bracketinfo != 0 ) ClearBracketIndex(-1);
        AT.bracketinfo = expr->newbracketinfo;
        OpenBracketIndex(nexpr);
/*
        Now emulate the terms:
            sum_(i,0,nfacp,2,factor_^(i/2+1)*F[factor_^(i+1)]*F[factor_^(i+2)])
            +factor_^(nfacp/2+1)*F[factor_^nfac]
*/
        NewSort(BHEAD0);
        if ( expriszero == 0 ) {
            for ( i = 0; i < nfacp; i += 2 ) {
                t = genericterm; w = term = oldworkpointer;
                j = *t; NCOPY(w,t,j);
                term[4] = i/2+1;
                term[7] = nexpr;
                term[16] = i+1;
                term[22] = nexpr;
                term[31] = i+2;
                AT.WorkPointer = term + *term;
                Generator(BHEAD term, C->numlhs);
            }
            if ( nfac > nfacp ) {
                t = genericterm2; w = term = oldworkpointer;
                j = *t; NCOPY(w,t,j);
                term[4] = i/2+1;
                term[7] = nexpr;
                term[16] = nfac;
                AT.WorkPointer = term + *term;
                Generator(BHEAD term, C->numlhs);
            }
        }
        if ( EndSort(BHEAD AM.S0->sBuffer,0) < 0 ) {
            LowerSortLevel();
            Terminate(-1);
        }
/*
        Set the file back into reading position
*/
        SetScratch(file, &pos);
        nfac = (nfac+1)/2;
        if ( expriszero ) { nfac = 1; }
    }
    if ( AT.bracketinfo != 0 ) ClearBracketIndex(-1);
    AT.bracketinfo = expr->newbracketinfo;
    expr->newbracketinfo = 0;
/*
    Reset the brackets to make them ready for the final pass
*/
    AR.BracketOn = oldBracketOn;
    AT.BrackBuf = oldBrackBuf;
    if ( AR.BracketOn ) OpenBracketIndex(nexpr);
/*
    We distinguish two cases: nfac == 2 and nfac == 1
    After preparing the term we skip the factor_ part.
*/
    NewSort(BHEAD0);
    if ( expriszero == 0 ) {
        if ( nfac == 1 ) {
            t = genericterm2; w = term = oldworkpointer;
            j = *t; NCOPY(w,t,j);
            term[7] = nexpr;
            term[16] = nfac;
        }
        else if ( nfac == 2 ) {
            t = genericterm; w = term = oldworkpointer;
            j = *t; NCOPY(w,t,j);
            term[7] = nexpr;
            term[16] = 1;
            term[22] = nexpr;
            term[31] = 2;
        }
        else {
            AS.OldOnFile[nexpr] = oldpos;
            return(-1);
        }
        term[4] = term[0]-4;
        term += 4;
        AT.WorkPointer = term + *term;
        Generator(BHEAD term, C->numlhs);
    }
    if ( EndSort(BHEAD AM.S0->sBuffer,0) < 0 ) {
        LowerSortLevel();
        Terminate(-1);
    }
/*
    Final Cleanup
*/
    expr->numfactors = 0;
    expr->vflags &= ~ISFACTORIZED;
    if ( AT.bracketinfo != 0 ) ClearBracketIndex(-1);
 
    AR.infile = oldinfile;
    AR.outfile = oldoutfile;
    strcpy((char*)AC.Commercial, oldCommercial);
    AT.WorkPointer = oldworkpointer;
    AS.OldOnFile[nexpr] = oldpos;
    
    return(0);
}
 
/*
    #] poly_unfactorize_expression : 
    #[ poly_inverse : 
*/
 
WORD *poly_inverse(PHEAD WORD *arga, WORD *argb) {
 
#ifdef WITHFLINT
    if ( AC.FlintPolyFlag && AC.ncmod==0 ) {
        return flint_inverse(BHEAD arga, argb);
    }
#endif
 
#ifdef DEBUG
    cout << "*** [" << thetime() << "]  CALL : poly_inverse" << endl;
#endif
 
    // Extract variables
    vector<WORD *> e;
    e.reserve(2);
    e.push_back(arga);
    e.push_back(argb);
    poly::get_variables(BHEAD e, false, true);
 
    if (AN.poly_num_vars > 1) {
        MLOCK(ErrorMessageLock);
        MesPrint ((char*)"ERROR: multivariate polynomial inverse is generally impossible");
        MUNLOCK(ErrorMessageLock);
        Terminate(-1);
    }
 
    poly finalden(BHEAD 1), finalres(BHEAD 1);
    int ressize = 0;
    WORD *res = NULL;
 
    // Convert to polynomials
    poly dena(BHEAD 0); // We need to keep the overall denominator of arga, to multiply the result
    poly a(poly::argument_to_poly(BHEAD arga, false, true, &dena));
    poly b(poly::argument_to_poly(BHEAD argb, false, true));
 
    // Divide out the integer content, FORM has not already done this.
    poly content_a(BHEAD 0), content_b(BHEAD 0);
    content_a = polygcd::integer_content(a);
    content_b = polygcd::integer_content(b);
    a /= content_a;
    b /= content_b;
 
    poly invamodp(BHEAD 0), invbmodp(BHEAD 0);
    WORD modp = 0;
 
    // Special cases:
    // Possibly strange that we give 1 for inverse_(x1,1) but here we take MMA's convention.
    if ((a.is_one() && b.is_one()) || a.is_one()) {
        finalres = poly(BHEAD 1);
    }
    else if (b.is_one()) {
        finalres = poly(BHEAD 0);
    }
    else {
        // Check for modulus calculus
        modp=poly_determine_modulus(BHEAD true, true, "polynomial inverse");
        const bool mod_calc = modp==0 ? false : true;
        a.setmod(modp,1);
        b.setmod(modp,1);
 
        // Check the gcd of a,b: if it is != 1, the inverse does not exist.
        poly gcd(polygcd::gcd(a,b));
        if (!gcd.is_one()) {
            MLOCK(ErrorMessageLock);
            if (mod_calc) {
                MesPrint ((char*)"ERROR: polynomial inverse does not exist (mod %d)", modp);
            }
            else {
                MesPrint ((char*)"ERROR: polynomial inverse does not exist");
            }
            MUNLOCK(ErrorMessageLock);
            Terminate(-1);
        }
 
        bool inv_exists = true;
        do {
            // If we are not using modulus calculus, find a suitable prime for xgcd:
            if (!mod_calc) {
                vector<int> x(1,0);
                modp = polyfact::choose_prime(a.integer_lcoeff()*b.integer_lcoeff(), x, modp);
            }
 
            poly amodp(a,modp,1);
            poly bmodp(b,modp,1);
 
            // Calculate gcd
            vector<poly> xgcd(polyfact::extended_gcd_Euclidean_lifted(amodp,bmodp));
            invamodp = poly(xgcd[0]);
            invbmodp = poly(xgcd[1]);
 
            inv_exists = ((invamodp * amodp) % bmodp).is_one();
            if (!inv_exists && mod_calc) {
                // Control should not reach here!
                MLOCK(ErrorMessageLock);
                MesPrint ((char*)"ERROR: polynomial inverse does not exist (mod %d) B", modp);
                MUNLOCK(ErrorMessageLock);
                Terminate(-1);
            }
 
        // If the inverse does not exist and we are not working in modulus calculus,
        // choose a new prime and try again. This loop should always terminate, as
        // we have already checked that gcd(a,b) == 1.
        } while (!inv_exists);
 
        // estimate of the size of the Form notation; might be extended later
        ressize = invamodp.size_of_form_notation()+1;
        res = (WORD *)Malloc1(ressize*sizeof(WORD), "poly_inverse");
 
        // initialize polynomials to store the result
        poly primepower(BHEAD modp);
        poly inva(invamodp,modp,1);
        poly invb(invbmodp,modp,1);
 
        while (true) {
            // convert to Form notation
            int j=0;
            WORD n=0;
            for (int i=1; i<inva[0]; i+=inva[i]) {
 
                // check whether res should be extended
                while (ressize < j + 2*ABS(inva[i+inva[i]-1]) + (inva[i+1]>0?4:0) + 3) {
                    int newressize = 2*ressize;
 
                    WORD *newres = (WORD *)Malloc1(newressize*sizeof(WORD), "poly_inverse");
                    WCOPY(newres, res, ressize);
                    M_free(res, "poly_inverse");
                    res = newres;
                    ressize = newressize;
                }
 
                res[j] = 1;
                if (inva[i+1]>0) {
                    res[j+res[j]++] = SYMBOL;
                    res[j+res[j]++] = 4;
                    res[j+res[j]++] = AN.poly_vars[0];
                    res[j+res[j]++] = inva[i+1];
                }
                MakeLongRational(BHEAD (UWORD *)&inva[i+2], inva[i+inva[i]-1],
                                                 (UWORD*)&primepower.terms[3], primepower.terms[primepower.terms[1]],
                                                 (UWORD *)&res[j+res[j]], &n);
                res[j] += 2*ABS(n);
                res[j+res[j]++] = SGN(n)*(2*ABS(n)+1);
                j += res[j];
            }
            res[j]=0;
 
            // if modulus calculus is set, this is the answer
            if (a.modp != 0) break;
 
            // otherwise check over integers
            poly den(BHEAD 0);
            poly check(poly::argument_to_poly(BHEAD res, false, true, &den));
            // Shortcut: if b's lcoeff doesn't divide the lcoeff of check*a,
            // b certainly doesn't divide check*a:
            if (poly::divides(b.integer_lcoeff(), check.integer_lcoeff()*a.integer_lcoeff())) {
                check = check*a - den;
                if (poly::divides(b, check)) break;
            }
 
            // if incorrect, lift with quadratic p-adic Newton's iteration.
            poly error((poly(BHEAD 1) - a*inva - b*invb) / primepower);
            poly errormodpp(error, modp, inva.modn);
 
            inva.modn *= 2;
            invb.modn *= 2;
 
            poly dinva((inva * errormodpp) % b);
            poly dinvb((invb * errormodpp) % a);
 
            inva += dinva * primepower;
            invb += dinvb * primepower;
 
            primepower *= primepower;
        }
 
        // One more round trip from form -> poly -> form, to multiply by dena in an easy way
        finalres = poly(poly::argument_to_poly(BHEAD res, false, true, &finalden));
    }
 
    finalres *= dena;
    // The overall denominator additionally needs to be multiplied by content_a:
    finalden *= content_a;
    const WORD finalden_size = finalden.terms[finalden.terms[1]];
    const int finalsize = finalres.size_of_form_notation_with_den(finalden_size)+1;
    if (ressize < finalsize) {
        if (res != NULL) {
            M_free(res, "poly_inverse");
        }
        res = (WORD *)Malloc1(finalsize*sizeof(WORD), "poly_inverse");
    }
    poly::poly_to_argument_with_den(finalres, finalden_size,
        (UWORD*)&(finalden.terms[finalden.terms[1] - ABS(finalden_size)]), res, false);
 
    // clean up and reset modulo calculation
    poly_free_poly_vars(BHEAD "AN.poly_vars_inverse");
    
    AN.ncmod = AC.ncmod;
    return res;
}
 
/*
    #] poly_inverse : 
    #[ poly_mul :
*/
 
WORD *poly_mul(PHEAD WORD *a, WORD *b) {
 
#ifdef DEBUG
    cout << "*** [" << thetime() << "]  CALL : poly_mul" << endl;
#endif
 
#ifdef WITHFLINT
    if ( AC.FlintPolyFlag && AC.ncmod==0 ) {
        return flint_mul(BHEAD a, b);
    }
#endif
 
    // Extract variables
    vector<WORD *> e;
    e.reserve(2);
    e.push_back(a);
    e.push_back(b);
    poly::get_variables(BHEAD e, false, false);  // TODO: any performance effect by sort_vars=true?
 
    // Convert to polynomials
    poly dena(BHEAD 0);
    poly denb(BHEAD 0);
    poly pa(poly::argument_to_poly(BHEAD a, false, true, &dena));
    poly pb(poly::argument_to_poly(BHEAD b, false, true, &denb));
 
    // Check for modulus calculus
    WORD modp = poly_determine_modulus(BHEAD true, true, "polynomial multiplication");
    pa.setmod(modp, 1);
 
    // NOTE: mul_ is currently implemented by translating negative powers of
    // symbols to extra symbols. For future improvement, it may be better to
    // compute
    //   (content(a) * content(b)) * (a/content(a)) * (b/content(b))
    // to avoid introducing extra symbols for "mixed" cases, e.g.,
    // (1+x) * (1/x) -> (1+x) * (1+Z1_).
    assert(dena.is_integer());
    assert(denb.is_integer());
    assert(modp == 0 || dena.is_one());
    assert(modp == 0 || denb.is_one());
 
    // multiplication
    pa *= pb;
 
    // convert to Form notation
    WORD *res;
    if (dena.is_one() && denb.is_one()) {
        res = (WORD *)Malloc1((pa.size_of_form_notation() + 1) * sizeof(WORD), "poly_mul");
        poly::poly_to_argument(pa, res, false);
    } else {
        dena *= denb;
        res = (WORD *)Malloc1((pa.size_of_form_notation_with_den(dena[dena[1]]) + 1) * sizeof(WORD), "poly_mul");
        poly::poly_to_argument_with_den(pa, dena[dena[1]], (const UWORD *)&dena[2+AN.poly_num_vars], res, false);
    }
 
    // clean up and reset modulo calculation
    poly_free_poly_vars(BHEAD "AN.poly_vars_mul");
    AN.ncmod = AC.ncmod;
 
    return res;
}
 
/*
    #] poly_mul : 
    #[ poly_free_poly_vars :
*/
 
void poly_free_poly_vars(PHEAD const char *text)
{
    if ( AN.poly_vars_type == 0 ) {
        TermFree(AN.poly_vars, text);
    }
    else {
        M_free(AN.poly_vars, text);
    }
    AN.poly_num_vars = 0;
    AN.poly_vars = 0;
}
 
/*
    #] poly_free_poly_vars : 
*/