Factorization over algebraic function fields. More...

#include "canonicalform.h"

Functions
CanonicalForm	alg_gcd (const CanonicalForm &, const CanonicalForm &, const CFList &)

CFFList	facAlgFunc2 (const CanonicalForm &f, const CFList &as)
	factorize a polynomial f that is irreducible over the ground field modulo an extension given by an irreducible characteristic set as, f is assumed to be integral, i.e. $f\in K[x_1,\ldots,x_n]/(as)$ , and each element of as is assumed to be integral as well. must be either or . More...

CFFList	facAlgFunc (const CanonicalForm &f, const CFList &as)
	factorize a polynomial f modulo an extension given by an irreducible characteristic set as, f is assumed to be integral, i.e. $f\in K[x_1,\ldots,x_n]/(as)$ , and each element of as is assumed to be integral as well. must be either or . More...

Detailed Description

Factorization over algebraic function fields.

Note: some of the code is code from libfac or derived from code from libfac. Libfac is written by M. Messollen. See also COPYING for license information and README for general information on characteristic sets.

Author: Martin Lee

Definition in file facAlgFunc.h.

Function Documentation

◆ alg_gcd()

CanonicalForm alg_gcd	(	const CanonicalForm &	fff,
		const CanonicalForm &	ggg,
		const CFList &	as
	)

Definition at line 61 of file facAlgFunc.cc.

{
  if (fff.inCoeffDomain() || ggg.inCoeffDomain())
    return 1;
  CanonicalForm f=fff;
  CanonicalForm g=ggg;
  f=Prem(f,as);
  g=Prem(g,as);
  if ( f.isZero() )
  {
    if ( g.lc().sign() < 0 ) return -g;
    else                     return g;
  }
  else  if ( g.isZero() )
  {
    if ( f.lc().sign() < 0 ) return -f;
    else                     return f;
  }
 
  int v= as.getLast().level();
  if (f.level() <= v || g.level() <= v)
    return 1;
 
  CanonicalForm res;
 
  // does as appear in f and g ?
  bool has_alg_var=false;
  for ( CFListIterator j=as;j.hasItem(); j++ )
  {
    Variable v=j.getItem().mvar();
    if (hasVar (f, v))
      has_alg_var=true;
    if (hasVar (g, v))
      has_alg_var=true;
  }
  if (!has_alg_var)
  {
    /*if ((hasAlgVar(f))
    || (hasAlgVar(g)))
    {
      Varlist ord;
      for ( CFListIterator j=as;j.hasItem(); j++ )
        ord.append(j.getItem().mvar());
      res=algcd(f,g,as,ord);
    }
    else*/
    if (!hasAlgVar (f) && !hasAlgVar (g))
      return res=gcd(f,g);
  }
 
  int mvf=f.level();
  int mvg=g.level();
  if (mvg > mvf)
  {
    CanonicalForm tmp=f; f=g; g=tmp;
    int tmp2=mvf; mvf=mvg; mvg=tmp2;
  }
  if (g.inBaseDomain() || f.inBaseDomain())
    return CanonicalForm(1);
 
  CanonicalForm c_f= alg_content (f, as);
 
  if (mvf != mvg)
  {
    res= alg_gcd (g, c_f, as);
    return res;
  }
  Variable x= f.mvar();
 
  // now: mvf==mvg, f.level()==g.level()
  CanonicalForm c_g= alg_content (g, as);
 
  int delta= degree (f) - degree (g);
 
  f= divide (f, c_f, as);
  g= divide (g, c_g, as);
 
  // gcd of contents
  CanonicalForm c_gcd= alg_gcd (c_f, c_g, as);
  CanonicalForm tmp;
 
  if (delta < 0)
  {
    tmp= f;
    f= g;
    g= tmp;
    delta= -delta;
  }
 
  CanonicalForm r= 1;
 
  while (degree (g, x) > 0)
  {
    r= Prem (f, g);
    r= Prem (r, as);
    if (!r.isZero())
    {
      r= divide (r, alg_content (r,as), as);
      r /= vcontent (r,Variable (v+1));
    }
    f= g;
    g= r;
  }
 
  if (degree (g, x) == 0)
    return c_gcd;
 
  c_f= alg_content (f, as);
 
  f= divide (f, c_f, as);
 
  f *= c_gcd;
  f /= vcontent (f, Variable (v+1));
 
  return f;
}

◆ facAlgFunc()

CFFList facAlgFunc	(	const CanonicalForm &	f,
		const CFList &	as
	)

factorize a polynomial f modulo an extension given by an irreducible characteristic set as, f is assumed to be integral, i.e. $f\in K[x_1,\ldots,x_n]/(as)$ , and each element of as is assumed to be integral as well. must be either or .

Returns: the returned factors are not necessarily monic but only primitive and the product of the factors equals f up to a unit.

factorize a polynomial f modulo an extension given by an irreducible characteristic set as, f is assumed to be integral, i.e. $f\in K[x_1,\ldots,x_n]/(as)$ , and each element of as is assumed to be integral as well. must be either or .

Parameters

[in]	f	univariate poly
[in]	as	irreducible characteristic set

Definition at line 1043 of file facAlgFunc.cc.

{
  bool isRat= isOn (SW_RATIONAL);
  if (!isRat && getCharacteristic() == 0)
    On (SW_RATIONAL);
  CFFList Output, output, Factors= factorize(f);
  if (Factors.getFirst().factor().inCoeffDomain())
    Factors.removeFirst();
 
  if (as.length() == 0)
  {
    if (!isRat && getCharacteristic() == 0)
      Off (SW_RATIONAL);
    return Factors;
  }
  if (f.level() <= as.getLast().level())
  {
    if (!isRat && getCharacteristic() == 0)
      Off (SW_RATIONAL);
    return Factors;
  }
 
  for (CFFListIterator i=Factors; i.hasItem(); i++)
  {
    if (i.getItem().factor().level() > as.getLast().level())
    {
      output= facAlgFunc2 (i.getItem().factor(), as);
      for (CFFListIterator j= output; j.hasItem(); j++)
        Output= append (Output, CFFactor (j.getItem().factor(),
                                          j.getItem().exp()*i.getItem().exp()));
    }
  }
 
  if (!isRat && getCharacteristic() == 0)
    Off (SW_RATIONAL);
  return Output;
}

◆ facAlgFunc2()

CFFList facAlgFunc2	(	const CanonicalForm &	f,
		const CFList &	as
	)

factorize a polynomial f that is irreducible over the ground field modulo an extension given by an irreducible characteristic set as, f is assumed to be integral, i.e. $f\in K[x_1,\ldots,x_n]/(as)$ , and each element of as is assumed to be integral as well. must be either or .

Returns: the returned factors are not necessarily monic but only primitive and the product of the factors equals f up to a unit.