Functions
cfModResultant.h File Reference

modular resultant algorithm as described by G. More...

#include "canonicalform.h"

Go to the source code of this file.

Functions

CanonicalForm resultantFp (const CanonicalForm &A, const CanonicalForm &B, const Variable &x, bool prob=true)
 modular resultant algorihtm over Fp More...
 
CanonicalForm resultantZ (const CanonicalForm &A, const CanonicalForm &B, const Variable &x, bool prob=true)
 modular resultant algorihtm over Z More...
 

Detailed Description

modular resultant algorithm as described by G.

E. Collins in "The Calculation of multivariate polynomial resultants"

Author
Martin Lee

Definition in file cfModResultant.h.

Function Documentation

◆ resultantFp()

CanonicalForm resultantFp ( const CanonicalForm A,
const CanonicalForm B,
const Variable x,
bool  prob = true 
)

modular resultant algorihtm over Fp

Returns
resultantFp returns the resultant of A and B wrt. x
Parameters
[in]Asome poly
[in]Bsome poly
[in]xsome polynomial variable
[in]probif true use probabilistic algorithm

Definition at line 349 of file cfModResultant.cc.

351 {
352  ASSERT (getCharacteristic() > 0, "characteristic > 0 expected");
353 
354  if (A.isZero() || B.isZero())
355  return 0;
356 
357  int degAx= degree (A, x);
358  int degBx= degree (B, x);
359  if (A.level() < x.level())
360  return power (A, degBx);
361  if (B.level() < x.level())
362  return power (B, degAx);
363 
364  if (degAx == 0)
365  return power (A, degBx);
366  else if (degBx == 0)
367  return power (B, degAx);
368 
369  if (A.isUnivariate() && B.isUnivariate() && A.level() == B.level())
370  return uniResultant (A, B);
371 
372  CanonicalForm F= A;
373  CanonicalForm G= B;
374 
375  CFMap M, N;
376  myCompress (F, G, M, N, x);
377 
378  F= M (F);
379  G= M (G);
380 
381  Variable y= Variable (2);
382 
383  CanonicalForm GEval, FEval, recResult, H;
384  CanonicalForm newtonPoly= 1;
385  CanonicalForm modResult= 0;
386 
387  Variable z= Variable (1);
388  int bound= degAx*degree (G, 2) + degree (F, 2)*degBx;
389 
390  int p= getCharacteristic();
391  CanonicalForm minpoly;
392  Variable alpha= Variable (tmax (F.level(), G.level()) + 1);
393  bool algExt= hasFirstAlgVar (F, alpha) || hasFirstAlgVar (G, alpha);
394  CFGenerator * gen;
395  bool extOfExt= false;
396  Variable v= alpha;
397  CanonicalForm primElemAlpha, imPrimElemAlpha;
398  CFList source,dest;
399  if (!algExt && (p < (1 << 28)))
400  {
401  // pass to an extension of size at least 2^29
402  // for very very large input that is maybe too small though
403  int deg= ceil (29.0*((double) log (2)/log (p)))+1;
404  minpoly= randomIrredpoly (deg, z);
405  alpha= rootOf (minpoly);
406  AlgExtGenerator AlgExtGen (alpha);
407  gen= AlgExtGen.clone();
408  for (int i= 0; i < p; i++) // skip values from the prime field
409  (*gen).next();
410  }
411  else if (!algExt)
412  {
413  FFGenerator FFGen;
414  gen= FFGen.clone();
415  }
416  else
417  {
418  int deg= ceil (29.0*((double) log (2)/log (p)));
419  if (degree (getMipo (alpha)) < deg)
420  {
421  mpz_t field_size;
422  mpz_init (field_size);
423  mpz_ui_pow_ui (field_size, p,
424  deg + degree (getMipo (alpha)) - deg%degree (getMipo (alpha)));
425 
426  // field_size needs to fit in an int because of mapUp, mapDown, length of lists etc.
427  if (mpz_fits_sint_p (field_size))
428  {
429  minpoly= randomIrredpoly (deg + degree (getMipo (alpha))
430  - deg%degree (getMipo (alpha)), z);
431  v= rootOf (minpoly);
432  Variable V_buf2;
433  bool primFail= false;
434  extOfExt= true;
435  primElemAlpha= primitiveElement (alpha, V_buf2, primFail);
436  ASSERT (!primFail, "failure in integer factorizer");
437  if (primFail)
438  ; //ERROR
439  else
440  imPrimElemAlpha= mapPrimElem (primElemAlpha, alpha, v);
441  F= mapUp (F, alpha, v, primElemAlpha, imPrimElemAlpha, source, dest);
442  G= mapUp (G, alpha, v, primElemAlpha, imPrimElemAlpha, source, dest);
443  }
444  else
445  {
446  deg= deg - deg % degree (getMipo (alpha));
447  mpz_ui_pow_ui (field_size, p, deg);
448  while (deg / degree (getMipo (alpha)) >= 2 && !mpz_fits_sint_p (field_size))
449  {
450  deg -= degree (getMipo (alpha));
451  mpz_ui_pow_ui (field_size, p, deg);
452  }
453  if (deg != degree (getMipo (alpha)))
454  {
455  minpoly= randomIrredpoly (deg, z);
456  v= rootOf (minpoly);
457  Variable V_buf2;
458  bool primFail= false;
459  extOfExt= true;
460  primElemAlpha= primitiveElement (alpha, V_buf2, primFail);
461  ASSERT (!primFail, "failure in integer factorizer");
462  if (primFail)
463  ; //ERROR
464  else
465  imPrimElemAlpha= mapPrimElem (primElemAlpha, alpha, v);
466  F= mapUp (F, alpha, v, primElemAlpha, imPrimElemAlpha, source, dest);
467  G= mapUp (G, alpha, v, primElemAlpha, imPrimElemAlpha, source, dest);
468  }
469  }
470  mpz_clear (field_size);
471  }
472  AlgExtGenerator AlgExtGen (v);
473  gen= AlgExtGen.clone();
474  for (int i= 0; i < p; i++)
475  (*gen).next();
476  }
477  int count= 0;
478  int equalCount= 0;
479  CanonicalForm point;
480  do
481  {
482  evalPoint (F, G, FEval, GEval, *gen);
483 
484  recResult= resultantFp (FEval, GEval, z, prob);
485 
486  H= newtonInterp ((*gen).item(), recResult, newtonPoly, modResult, y);
487 
488  if (H == modResult)
489  equalCount++;
490  else
491  equalCount= 0;
492 
493  count++;
494  if (count > bound || (prob && equalCount == 2 && !H.inCoeffDomain()))
495  {
496  if (!algExt && degree (H, alpha) <= 0)
497  break;
498  else if (algExt)
499  {
500  if (extOfExt && !isInExtension (H, imPrimElemAlpha, 1, primElemAlpha,
501  dest, source))
502  {
503  H= mapDown (H, primElemAlpha, imPrimElemAlpha, alpha, dest, source);
504  prune (v);
505  break;
506  }
507  else if (!extOfExt)
508  break;
509  }
510  }
511 
512  modResult= H;
513  newtonPoly *= (y - (*gen).item());
514  if ((*gen).hasItems())
515  (*gen).next();
516  else
517  STICKYASSERT (0, "out of evaluation points");
518  } while (1);
519 
520  delete gen;
521 
522  return N (H);
523 }
int status int void size_t count
Definition: si_signals.h:59
CanonicalForm power(const CanonicalForm &f, int n)
exponentiation
static CanonicalForm mapDown(const CanonicalForm &F, const Variable &alpha, const CanonicalForm &G, CFList &source, CFList &dest)
the CanonicalForm G is the output of map_up, returns F considered as an element over ...
Definition: cf_map_ext.cc:90
generate all elements in F_p(alpha) starting from 0
Definition: cf_generator.h:93
const CanonicalForm int const CFList const Variable & y
Definition: facAbsFact.cc:57
static CanonicalForm bound(const CFMatrix &M)
Definition: cf_linsys.cc:460
static CanonicalForm newtonInterp(const CanonicalForm &alpha, const CanonicalForm &u, const CanonicalForm &newtonPoly, const CanonicalForm &oldInterPoly, const Variable &x)
template CanonicalForm tmax(const CanonicalForm &, const CanonicalForm &)
CanonicalForm primitiveElement(const Variable &alpha, Variable &beta, bool &fail)
determine a primitive element of , is a primitive element of a field which is isomorphic to ...
Definition: cf_map_ext.cc:310
factory&#39;s class for variables
Definition: factory.h:117
virtual class for generators
Definition: cf_generator.h:21
CF_NO_INLINE bool isZero() const
Definition: cf_inline.cc:372
bool isInExtension(const CanonicalForm &F, const CanonicalForm &gamma, const int k, const CanonicalForm &delta, CFList &source, CFList &dest)
tests if F is not contained in a subfield defined by gamma (Fq case) or k (GF case) ...
generate all elements in F_p starting from 0
Definition: cf_generator.h:55
gmp_float log(const gmp_float &a)
Definition: mpr_complex.cc:343
factory&#39;s main class
Definition: canonicalform.h:77
int myCompress(const CanonicalForm &F, const CanonicalForm &G, CFMap &M, CFMap &N, bool topLevel)
compressing two polynomials F and G, M is used for compressing, N to reverse the compression ...
Definition: cfModGcd.cc:93
Variable alpha
Definition: facAbsBiFact.cc:52
CanonicalForm getMipo(const Variable &alpha, const Variable &x)
Definition: variable.cc:207
int getCharacteristic()
Definition: cf_char.cc:51
void prune(Variable &alpha)
Definition: variable.cc:261
Variable rootOf(const CanonicalForm &, char name='@')
returns a symbolic root of polynomial with name name Use it to define algebraic variables ...
Definition: variable.cc:162
static CanonicalForm mapUp(const Variable &alpha, const Variable &beta)
and is a primitive element, returns the image of
Definition: cf_map_ext.cc:67
const signed long ceil(const ampf< Precision > &x)
Definition: amp.h:788
int level() const
Definition: factory.h:134
bool isUnivariate() const
const CanonicalForm CFMap CFMap & N
#define A
Definition: sirandom.c:23
CanonicalForm resultantFp(const CanonicalForm &A, const CanonicalForm &B, const Variable &x, bool prob)
modular resultant algorihtm over Fp
bool hasFirstAlgVar(const CanonicalForm &f, Variable &a)
check if poly f contains an algebraic variable a
Definition: cf_ops.cc:665
const CanonicalForm CFMap & M
int i
Definition: cfEzgcd.cc:125
CanonicalForm H
Definition: facAbsFact.cc:64
class CFMap
Definition: cf_map.h:84
const Variable & v
< [in] a sqrfree bivariate poly
Definition: facBivar.h:37
#define STICKYASSERT(expression, message)
Definition: cf_assert.h:64
const CanonicalForm & G
static CanonicalForm uniResultant(const CanonicalForm &F, const CanonicalForm &G)
b *CanonicalForm B
Definition: facBivar.cc:52
virtual CFGenerator * clone() const
Definition: cf_generator.h:30
int level() const
level() returns the level of CO.
CanonicalForm randomIrredpoly(int i, const Variable &x)
computes a random monic irreducible univariate polynomial in x over Fp of degree i via NTL ...
Definition: cf_irred.cc:42
#define ASSERT(expression, message)
Definition: cf_assert.h:99
int p
Definition: cfModGcd.cc:4019
int degree(const CanonicalForm &f)
CanonicalForm mapPrimElem(const CanonicalForm &primElem, const Variable &alpha, const Variable &beta)
compute the image of a primitive element of in . We assume .
Definition: cf_map_ext.cc:377
virtual void next()
Definition: cf_generator.h:29
CFGenerator * clone() const
Definition: cf_generator.cc:52
static void evalPoint(const CanonicalForm &F, const CanonicalForm &G, CanonicalForm &FEval, CanonicalForm &GEval, CFGenerator &evalPoint)
bool inCoeffDomain() const

◆ resultantZ()

CanonicalForm resultantZ ( const CanonicalForm A,
const CanonicalForm B,
const Variable x,
bool  prob = true 
)

modular resultant algorihtm over Z

Returns
resultantZ returns the resultant of A and B wrt. x
Parameters
[in]Asome poly
[in]Bsome poly
[in]xsome polynomial variable
[in]probif true use probabilistic algorithm

Definition at line 560 of file cfModResultant.cc.

562 {
563  ASSERT (getCharacteristic() == 0, "characteristic > 0 expected");
564 #ifndef NOASSERT
565  bool isRat= isOn (SW_RATIONAL);
566  On (SW_RATIONAL);
567  ASSERT (bCommonDen (A).isOne(), "input A is rational");
568  ASSERT (bCommonDen (B).isOne(), "input B is rational");
569  if (!isRat)
570  Off (SW_RATIONAL);
571 #endif
572 
573  int degAx= degree (A, x);
574  int degBx= degree (B, x);
575  if (A.level() < x.level())
576  return power (A, degBx);
577  if (B.level() < x.level())
578  return power (B, degAx);
579 
580  if (degAx == 0)
581  return power (A, degBx);
582  else if (degBx == 0)
583  return power (B, degAx);
584 
585  CanonicalForm F= A;
586  CanonicalForm G= B;
587 
588  Variable X= x;
589  if (F.level() != x.level() || G.level() != x.level())
590  {
591  if (F.level() > G.level())
592  X= F.mvar();
593  else
594  X= G.mvar();
595  F= swapvar (F, X, x);
596  G= swapvar (G, X, x);
597  }
598 
599  // now X is the main variable
600 
601  CanonicalForm d= 0;
602  CanonicalForm dd= 0;
604  for (CFIterator i= F; i.hasTerms(); i++)
605  {
606  buf= oneNorm (i.coeff());
607  d= (buf > d) ? buf : d;
608  }
609  CanonicalForm e= 0, ee= 0;
610  for (CFIterator i= G; i.hasTerms(); i++)
611  {
612  buf= oneNorm (i.coeff());
613  e= (buf > e) ? buf : e;
614  }
615  d= power (d, degBx);
616  e= power (e, degAx);
617  CanonicalForm bound= 1;
618  for (int i= degBx + degAx; i > 1; i--)
619  bound *= i;
620  bound *= d*e;
621  bound *= 2;
622 
623  bool onRational= isOn (SW_RATIONAL);
624  if (onRational)
625  Off (SW_RATIONAL);
626  int i = cf_getNumBigPrimes() - 1;
627  int p;
628  CanonicalForm l= lc (F)*lc(G);
629  CanonicalForm resultModP, q (0), newResult, newQ;
631  int equalCount= 0;
632  CanonicalForm test, newTest;
633  int count= 0;
634  do
635  {
636  p = cf_getBigPrime( i );
637  i--;
638  while ( i >= 0 && mod( l, p ) == 0)
639  {
640  p = cf_getBigPrime( i );
641  i--;
642  }
643 
644  if (i <= 0)
645  return resultant (A, B, x);
646 
647  setCharacteristic (p);
648 
649  TIMING_START (fac_resultant_p);
650  resultModP= resultantFp (mapinto (F), mapinto (G), X, prob);
651  TIMING_END_AND_PRINT (fac_resultant_p, "time to compute resultant mod p: ");
652 
653  setCharacteristic (0);
654 
655  count++;
656  if ( q.isZero() )
657  {
658  result= mapinto(resultModP);
659  q= p;
660  }
661  else
662  {
663  chineseRemainder( result, q, mapinto (resultModP), p, newResult, newQ );
664  q= newQ;
665  result= newResult;
666  test= symmetricRemainder (result,q);
667  if (test != newTest)
668  {
669  newTest= test;
670  equalCount= 0;
671  }
672  else
673  equalCount++;
674  if (newQ > bound || (prob && equalCount == 2))
675  {
676  result= test;
677  break;
678  }
679  }
680  } while (1);
681 
682  if (onRational)
683  On (SW_RATIONAL);
684  return swapvar (result, X, x);
685 }
TIMING_END_AND_PRINT(fac_alg_resultant, "time to compute resultant0: ")
int status int void size_t count
Definition: si_signals.h:59
CanonicalForm power(const CanonicalForm &f, int n)
exponentiation
int cf_getNumBigPrimes()
Definition: cf_primes.cc:45
static CanonicalForm bound(const CFMatrix &M)
Definition: cf_linsys.cc:460
CF_NO_INLINE CanonicalForm mod(const CanonicalForm &, const CanonicalForm &)
Definition: cf_inline.cc:564
void Off(int sw)
switches
TIMING_START(fac_alg_resultant)
const CanonicalForm CFMap CFMap const Variable & x
factory&#39;s class for variables
Definition: factory.h:117
factory&#39;s main class
Definition: canonicalform.h:77
static CanonicalForm symmetricRemainder(const CanonicalForm &f, const CanonicalForm &q)
void setCharacteristic(int c)
Definition: cf_char.cc:23
int getCharacteristic()
Definition: cf_char.cc:51
CanonicalForm lc(const CanonicalForm &f)
CanonicalForm swapvar(const CanonicalForm &, const Variable &, const Variable &)
swapvar() - swap variables x1 and x2 in f.
Definition: cf_ops.cc:168
int status int void * buf
Definition: si_signals.h:59
int level() const
Definition: factory.h:134
#define A
Definition: sirandom.c:23
CanonicalForm resultantFp(const CanonicalForm &A, const CanonicalForm &B, const Variable &x, bool prob)
modular resultant algorihtm over Fp
static const int SW_RATIONAL
set to 1 for computations over Q
Definition: cf_defs.h:28
bool isOn(int sw)
switches
void On(int sw)
switches
int i
Definition: cfEzgcd.cc:125
CanonicalForm mapinto(const CanonicalForm &f)
Variable mvar() const
mvar() returns the main variable of CO or Variable() if CO is in a base domain.
CanonicalForm bCommonDen(const CanonicalForm &f)
CanonicalForm bCommonDen ( const CanonicalForm & f )
class to iterate through CanonicalForm&#39;s
Definition: cf_iter.h:44
CanonicalForm test
Definition: cfModGcd.cc:4037
void chineseRemainder(const CanonicalForm &x1, const CanonicalForm &q1, const CanonicalForm &x2, const CanonicalForm &q2, CanonicalForm &xnew, CanonicalForm &qnew)
void chineseRemainder ( const CanonicalForm & x1, const CanonicalForm & q1, const CanonicalForm & x2...
Definition: cf_chinese.cc:52
const CanonicalForm & G
b *CanonicalForm B
Definition: facBivar.cc:52
int cf_getBigPrime(int i)
Definition: cf_primes.cc:39
int level() const
level() returns the level of CO.
#define ASSERT(expression, message)
Definition: cf_assert.h:99
int p
Definition: cfModGcd.cc:4019
int degree(const CanonicalForm &f)
CanonicalForm resultant(const CanonicalForm &f, const CanonicalForm &g, const Variable &x)
CanonicalForm resultant ( const CanonicalForm & f, const CanonicalForm & g, const Variable & x ) ...
static CanonicalForm oneNorm(const CanonicalForm &F)
return result
Definition: facAbsBiFact.cc:76
int l
Definition: cfEzgcd.cc:93