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algext.cc
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1 /****************************************
2 * Computer Algebra System SINGULAR *
3 ****************************************/
4 /**
5  * ABSTRACT: numbers in an algebraic extension field K[a] / < f(a) >
6  * Assuming that we have a coeffs object cf, then these numbers
7  * are polynomials in the polynomial ring K[a] represented by
8  * cf->extRing.
9  * IMPORTANT ASSUMPTIONS:
10  * 1.) So far we assume that cf->extRing is a valid polynomial
11  * ring in exactly one variable, i.e., K[a], where K is allowed
12  * to be any field (representable in SINGULAR and which may
13  * itself be some extension field, thus allowing for extension
14  * towers).
15  * 2.) Moreover, this implementation assumes that
16  * cf->extRing->qideal is not NULL but an ideal with at
17  * least one non-zero generator which may be accessed by
18  * cf->extRing->qideal->m[0] and which represents the minimal
19  * polynomial f(a) of the extension variable 'a' in K[a].
20  * 3.) As soon as an std method for polynomial rings becomes
21  * availabe, all reduction steps modulo f(a) should be replaced
22  * by a call to std. Moreover, in this situation one can finally
23  * move from K[a] / < f(a) > to
24  * K[a_1, ..., a_s] / I, with I some zero-dimensional ideal
25  * in K[a_1, ..., a_s] given by a lex
26  * Gröbner basis.
27  * The code in algext.h and algext.cc is then capable of
28  * computing in K[a_1, ..., a_s] / I.
29  **/
30 
31 #include "misc/auxiliary.h"
32 
33 #include "reporter/reporter.h"
34 
35 #include "coeffs/coeffs.h"
36 #include "coeffs/numbers.h"
37 #include "coeffs/longrat.h"
38 
39 #include "polys/monomials/ring.h"
41 #include "polys/simpleideals.h"
42 #include "polys/PolyEnumerator.h"
43 
44 #include "factory/factory.h"
45 #include "polys/clapconv.h"
46 #include "polys/clapsing.h"
47 #include "polys/prCopy.h"
48 
50 #define TRANSEXT_PRIVATES 1
52 
53 #ifdef LDEBUG
54 #define naTest(a) naDBTest(a,__FILE__,__LINE__,cf)
55 BOOLEAN naDBTest(number a, const char *f, const int l, const coeffs r);
56 #else
57 #define naTest(a) do {} while (0)
58 #endif
59 
60 /* polynomial ring in which our numbers live */
61 #define naRing cf->extRing
62 
63 /* coeffs object in which the coefficients of our numbers live;
64  * methods attached to naCoeffs may be used to compute with the
65  * coefficients of our numbers, e.g., use naCoeffs->nAdd to add
66  * coefficients of our numbers */
67 #define naCoeffs cf->extRing->cf
68 
69 /* minimal polynomial */
70 #define naMinpoly naRing->qideal->m[0]
71 
72 /// forward declarations
73 BOOLEAN naGreaterZero(number a, const coeffs cf);
74 BOOLEAN naGreater(number a, number b, const coeffs cf);
75 BOOLEAN naEqual(number a, number b, const coeffs cf);
76 BOOLEAN naIsOne(number a, const coeffs cf);
77 BOOLEAN naIsMOne(number a, const coeffs cf);
78 number naInit(long i, const coeffs cf);
79 number naNeg(number a, const coeffs cf);
80 number naInvers(number a, const coeffs cf);
81 number naAdd(number a, number b, const coeffs cf);
82 number naSub(number a, number b, const coeffs cf);
83 number naMult(number a, number b, const coeffs cf);
84 number naDiv(number a, number b, const coeffs cf);
85 void naPower(number a, int exp, number *b, const coeffs cf);
86 number naCopy(number a, const coeffs cf);
87 void naWriteLong(number a, const coeffs cf);
88 void naWriteShort(number a, const coeffs cf);
89 number naGetDenom(number &a, const coeffs cf);
90 number naGetNumerator(number &a, const coeffs cf);
91 number naGcd(number a, number b, const coeffs cf);
92 void naDelete(number *a, const coeffs cf);
93 void naCoeffWrite(const coeffs cf, BOOLEAN details);
94 //number naIntDiv(number a, number b, const coeffs cf);
95 const char * naRead(const char *s, number *a, const coeffs cf);
96 
97 static BOOLEAN naCoeffIsEqual(const coeffs cf, n_coeffType n, void * param);
98 
99 
100 /// returns NULL if p == NULL, otherwise makes p monic by dividing
101 /// by its leading coefficient (only done if this is not already 1);
102 /// this assumes that we are over a ground field so that division
103 /// is well-defined;
104 /// modifies p
105 // void p_Monic(poly p, const ring r);
106 
107 /// assumes that p and q are univariate polynomials in r,
108 /// mentioning the same variable;
109 /// assumes a global monomial ordering in r;
110 /// assumes that not both p and q are NULL;
111 /// returns the gcd of p and q;
112 /// leaves p and q unmodified
113 // poly p_Gcd(const poly p, const poly q, const ring r);
114 
115 /* returns NULL if p == NULL, otherwise makes p monic by dividing
116  by its leading coefficient (only done if this is not already 1);
117  this assumes that we are over a ground field so that division
118  is well-defined;
119  modifies p */
120 static inline void p_Monic(poly p, const ring r)
121 {
122  if (p == NULL) return;
123  number n = n_Init(1, r->cf);
124  if (p->next==NULL) { p_SetCoeff(p,n,r); return; }
125  poly pp = p;
126  number lc = p_GetCoeff(p, r);
127  if (n_IsOne(lc, r->cf)) return;
128  number lcInverse = n_Invers(lc, r->cf);
129  p_SetCoeff(p, n, r); // destroys old leading coefficient!
130  pIter(p);
131  while (p != NULL)
132  {
133  number n = n_Mult(p_GetCoeff(p, r), lcInverse, r->cf);
134  n_Normalize(n,r->cf);
135  p_SetCoeff(p, n, r); // destroys old leading coefficient!
136  pIter(p);
137  }
138  n_Delete(&lcInverse, r->cf);
139  p = pp;
140 }
141 
142 /// see p_Gcd;
143 /// additional assumption: deg(p) >= deg(q);
144 /// must destroy p and q (unless one of them is returned)
145 static inline poly p_GcdHelper(poly &p, poly &q, const ring r)
146 {
147  while (q != NULL)
148  {
149  p_PolyDiv(p, q, FALSE, r);
150  // swap p and q:
151  poly& t = q;
152  q = p;
153  p = t;
154 
155  }
156  return p;
157 }
158 
159 /* assumes that p and q are univariate polynomials in r,
160  mentioning the same variable;
161  assumes a global monomial ordering in r;
162  assumes that not both p and q are NULL;
163  returns the gcd of p and q;
164  leaves p and q unmodified */
165 static inline poly p_Gcd(const poly p, const poly q, const ring r)
166 {
167  assume((p != NULL) || (q != NULL));
168 
169  poly a = p; poly b = q;
170  if (p_Deg(a, r) < p_Deg(b, r)) { a = q; b = p; }
171  a = p_Copy(a, r); b = p_Copy(b, r);
172 
173  /* We have to make p monic before we return it, so that if the
174  gcd is a unit in the ground field, we will actually return 1. */
175  a = p_GcdHelper(a, b, r);
176  p_Monic(a, r);
177  return a;
178 }
179 
180 /* see p_ExtGcd;
181  additional assumption: deg(p) >= deg(q);
182  must destroy p and q (unless one of them is returned) */
183 static inline poly p_ExtGcdHelper(poly &p, poly &pFactor, poly &q, poly &qFactor,
184  ring r)
185 {
186  if (q == NULL)
187  {
188  qFactor = NULL;
189  pFactor = p_ISet(1, r);
190  p_SetCoeff(pFactor, n_Invers(p_GetCoeff(p, r), r->cf), r);
191  p_Monic(p, r);
192  return p;
193  }
194  else
195  {
196  poly pDivQ = p_PolyDiv(p, q, TRUE, r);
197  poly ppFactor = NULL; poly qqFactor = NULL;
198  poly theGcd = p_ExtGcdHelper(q, qqFactor, p, ppFactor, r);
199  pFactor = ppFactor;
200  qFactor = p_Add_q(qqFactor,
201  p_Neg(p_Mult_q(pDivQ, p_Copy(ppFactor, r), r), r),
202  r);
203  return theGcd;
204  }
205 }
206 
207 
208 /* assumes that p and q are univariate polynomials in r,
209  mentioning the same variable;
210  assumes a global monomial ordering in r;
211  assumes that not both p and q are NULL;
212  returns the gcd of p and q;
213  moreover, afterwards pFactor and qFactor contain appropriate
214  factors such that gcd(p, q) = p * pFactor + q * qFactor;
215  leaves p and q unmodified */
216 poly p_ExtGcd(poly p, poly &pFactor, poly q, poly &qFactor, ring r)
217 {
218  assume((p != NULL) || (q != NULL));
219  poly a = p; poly b = q; BOOLEAN aCorrespondsToP = TRUE;
220  if (p_Deg(a, r) < p_Deg(b, r))
221  { a = q; b = p; aCorrespondsToP = FALSE; }
222  a = p_Copy(a, r); b = p_Copy(b, r);
223  poly aFactor = NULL; poly bFactor = NULL;
224  poly theGcd = p_ExtGcdHelper(a, aFactor, b, bFactor, r);
225  if (aCorrespondsToP) { pFactor = aFactor; qFactor = bFactor; }
226  else { pFactor = bFactor; qFactor = aFactor; }
227  return theGcd;
228 }
229 
230 
231 
232 #ifdef LDEBUG
233 BOOLEAN naDBTest(number a, const char *f, const int l, const coeffs cf)
234 {
235  if (a == NULL) return TRUE;
236  p_Test((poly)a, naRing);
237  if (getCoeffType(cf)==n_algExt)
238  {
239  if((((poly)a)!=naMinpoly)
241  && (p_Totaldegree((poly)a, naRing)> 1)) // allow to output par(1)
242  {
243  dReportError("deg >= deg(minpoly) in %s:%d\n",f,l);
244  return FALSE;
245  }
246  }
247  return TRUE;
248 }
249 #endif
250 
251 void heuristicReduce(poly &p, poly reducer, const coeffs cf);
252 void definiteReduce(poly &p, poly reducer, const coeffs cf);
253 
254 /* returns the bottom field in this field extension tower; if the tower
255  is flat, i.e., if there is no extension, then r itself is returned;
256  as a side-effect, the counter 'height' is filled with the height of
257  the extension tower (in case the tower is flat, 'height' is zero) */
258 static coeffs nCoeff_bottom(const coeffs r, int &height)
259 {
260  assume(r != NULL);
261  coeffs cf = r;
262  height = 0;
263  while (nCoeff_is_Extension(cf))
264  {
265  assume(cf->extRing != NULL); assume(cf->extRing->cf != NULL);
266  cf = cf->extRing->cf;
267  height++;
268  }
269  return cf;
270 }
271 
272 BOOLEAN naIsZero(number a, const coeffs cf)
273 {
274  naTest(a);
275  return (a == NULL);
276 }
277 
278 void naDelete(number * a, const coeffs cf)
279 {
280  if (*a == NULL) return;
281  if (((poly)*a)==naMinpoly) { *a=NULL;return;}
282  poly aAsPoly = (poly)(*a);
283  p_Delete(&aAsPoly, naRing);
284  *a = NULL;
285 }
286 
287 BOOLEAN naEqual(number a, number b, const coeffs cf)
288 {
289  naTest(a); naTest(b);
290  /// simple tests
291  if (a == NULL) return (b == NULL);
292  if (b == NULL) return (a == NULL);
293  return p_EqualPolys((poly)a,(poly)b,naRing);
294 }
295 
296 number naCopy(number a, const coeffs cf)
297 {
298  naTest(a);
299  if (a == NULL) return NULL;
300  if (((poly)a)==naMinpoly) return a;
301  return (number)p_Copy((poly)a, naRing);
302 }
303 
304 number naGetNumerator(number &a, const coeffs cf)
305 {
306  return naCopy(a, cf);
307 }
308 
309 number naGetDenom(number &a, const coeffs cf)
310 {
311  naTest(a);
312  return naInit(1, cf);
313 }
314 
315 BOOLEAN naIsOne(number a, const coeffs cf)
316 {
317  naTest(a);
318  poly aAsPoly = (poly)a;
319  if ((a==NULL) || (!p_IsConstant(aAsPoly, naRing))) return FALSE;
320  return n_IsOne(p_GetCoeff(aAsPoly, naRing), naCoeffs);
321 }
322 
323 BOOLEAN naIsMOne(number a, const coeffs cf)
324 {
325  naTest(a);
326  poly aAsPoly = (poly)a;
327  if ((a==NULL) || (!p_IsConstant(aAsPoly, naRing))) return FALSE;
328  return n_IsMOne(p_GetCoeff(aAsPoly, naRing), naCoeffs);
329 }
330 
331 /// this is in-place, modifies a
332 number naNeg(number a, const coeffs cf)
333 {
334  naTest(a);
335  if (a != NULL) a = (number)p_Neg((poly)a, naRing);
336  return a;
337 }
338 
339 number naInit(long i, const coeffs cf)
340 {
341  if (i == 0) return NULL;
342  else return (number)p_ISet(i, naRing);
343 }
344 
345 long naInt(number &a, const coeffs cf)
346 {
347  naTest(a);
348  poly aAsPoly = (poly)a;
349  if(aAsPoly == NULL)
350  return 0;
351  if (!p_IsConstant(aAsPoly, naRing))
352  return 0;
353  assume( aAsPoly != NULL );
354  return n_Int(p_GetCoeff(aAsPoly, naRing), naCoeffs);
355 }
356 
357 /* TRUE iff (a != 0 and (b == 0 or deg(a) > deg(b) or (deg(a)==deg(b) && lc(a)>lc(b))) */
358 BOOLEAN naGreater(number a, number b, const coeffs cf)
359 {
360  naTest(a); naTest(b);
361  if (naIsZero(a, cf))
362  {
363  if (naIsZero(b, cf)) return FALSE;
364  return !n_GreaterZero(pGetCoeff((poly)b),naCoeffs);
365  }
366  if (naIsZero(b, cf))
367  {
368  return n_GreaterZero(pGetCoeff((poly)a),naCoeffs);
369  }
370  int aDeg = p_Totaldegree((poly)a, naRing);
371  int bDeg = p_Totaldegree((poly)b, naRing);
372  if (aDeg>bDeg) return TRUE;
373  if (aDeg<bDeg) return FALSE;
374  return n_Greater(pGetCoeff((poly)a),pGetCoeff((poly)b),naCoeffs);
375 }
376 
377 /* TRUE iff a != 0 and (LC(a) > 0 or deg(a) > 0) */
378 BOOLEAN naGreaterZero(number a, const coeffs cf)
379 {
380  naTest(a);
381  if (a == NULL) return FALSE;
382  if (n_GreaterZero(p_GetCoeff((poly)a, naRing), naCoeffs)) return TRUE;
383  if (p_Totaldegree((poly)a, naRing) > 0) return TRUE;
384  return FALSE;
385 }
386 
387 void naCoeffWrite(const coeffs cf, BOOLEAN details)
388 {
389  assume( cf != NULL );
390 
391  const ring A = cf->extRing;
392 
393  assume( A != NULL );
394  assume( A->cf != NULL );
395 
396  n_CoeffWrite(A->cf, details);
397 
398 // rWrite(A);
399 
400  const int P = rVar(A);
401  assume( P > 0 );
402 
403  PrintS("[");
404 
405  for (int nop=0; nop < P; nop ++)
406  {
407  Print("%s", rRingVar(nop, A));
408  if (nop!=P-1) PrintS(", ");
409  }
410 
411  PrintS("]/(");
412 
413  const ideal I = A->qideal;
414 
415  assume( I != NULL );
416  assume( IDELEMS(I) == 1 );
417 
418 
419  if ( details )
420  {
421  p_Write0( I->m[0], A);
422  PrintS(")");
423  }
424  else
425  PrintS("...)");
426 
427 /*
428  char *x = rRingVar(0, A);
429 
430  Print("// Coefficients live in the extension field K[%s]/<f(%s)>\n", x, x);
431  Print("// with the minimal polynomial f(%s) = %s\n", x,
432  p_String(A->qideal->m[0], A));
433  PrintS("// and K: ");
434 */
435 }
436 
437 number naAdd(number a, number b, const coeffs cf)
438 {
439  naTest(a); naTest(b);
440  if (a == NULL) return naCopy(b, cf);
441  if (b == NULL) return naCopy(a, cf);
442  poly aPlusB = p_Add_q(p_Copy((poly)a, naRing),
443  p_Copy((poly)b, naRing), naRing);
444  //definiteReduce(aPlusB, naMinpoly, cf);
445  return (number)aPlusB;
446 }
447 
448 number naSub(number a, number b, const coeffs cf)
449 {
450  naTest(a); naTest(b);
451  if (b == NULL) return naCopy(a, cf);
452  poly minusB = p_Neg(p_Copy((poly)b, naRing), naRing);
453  if (a == NULL) return (number)minusB;
454  poly aMinusB = p_Add_q(p_Copy((poly)a, naRing), minusB, naRing);
455  //definiteReduce(aMinusB, naMinpoly, cf);
456  return (number)aMinusB;
457 }
458 
459 number naMult(number a, number b, const coeffs cf)
460 {
461  naTest(a); naTest(b);
462  if ((a == NULL)||(b == NULL)) return NULL;
463  poly aTimesB = pp_Mult_qq((poly)a, (poly)b, naRing);
464  definiteReduce(aTimesB, naMinpoly, cf);
465  p_Normalize(aTimesB,naRing);
466  return (number)aTimesB;
467 }
468 
469 number naDiv(number a, number b, const coeffs cf)
470 {
471  naTest(a); naTest(b);
472  if (b == NULL) WerrorS(nDivBy0);
473  if (a == NULL) return NULL;
474  poly bInverse = (poly)naInvers(b, cf);
475  if(bInverse != NULL) // b is non-zero divisor!
476  {
477  poly aDivB = p_Mult_q(p_Copy((poly)a, naRing), bInverse, naRing);
478  definiteReduce(aDivB, naMinpoly, cf);
479  p_Normalize(aDivB,naRing);
480  return (number)aDivB;
481  }
482  return NULL;
483 }
484 
485 /* 0^0 = 0;
486  for |exp| <= 7 compute power by a simple multiplication loop;
487  for |exp| >= 8 compute power along binary presentation of |exp|, e.g.
488  p^13 = p^1 * p^4 * p^8, where we utilise that
489  p^(2^(k+1)) = p^(2^k) * p^(2^k);
490  intermediate reduction modulo the minimal polynomial is controlled by
491  the in-place method heuristicReduce(poly, poly, coeffs); see there.
492 */
493 void naPower(number a, int exp, number *b, const coeffs cf)
494 {
495  naTest(a);
496 
497  /* special cases first */
498  if (a == NULL)
499  {
500  if (exp >= 0) *b = NULL;
501  else WerrorS(nDivBy0);
502  return;
503  }
504  else if (exp == 0) { *b = naInit(1, cf); return; }
505  else if (exp == 1) { *b = naCopy(a, cf); return; }
506  else if (exp == -1) { *b = naInvers(a, cf); return; }
507 
508  int expAbs = exp; if (expAbs < 0) expAbs = -expAbs;
509 
510  /* now compute a^expAbs */
511  poly pow; poly aAsPoly = (poly)a;
512  if (expAbs <= 7)
513  {
514  pow = p_Copy(aAsPoly, naRing);
515  for (int i = 2; i <= expAbs; i++)
516  {
517  pow = p_Mult_q(pow, p_Copy(aAsPoly, naRing), naRing);
519  }
521  }
522  else
523  {
524  pow = p_ISet(1, naRing);
525  poly factor = p_Copy(aAsPoly, naRing);
526  while (expAbs != 0)
527  {
528  if (expAbs & 1)
529  {
532  }
533  expAbs = expAbs / 2;
534  if (expAbs != 0)
535  {
538  }
539  }
542  }
543 
544  /* invert if original exponent was negative */
545  number n = (number)pow;
546  if (exp < 0)
547  {
548  number m = naInvers(n, cf);
549  naDelete(&n, cf);
550  n = m;
551  }
552  *b = n;
553 }
554 
555 /* may reduce p modulo the reducer by calling definiteReduce;
556  the decision is made based on the following heuristic
557  (which should also only be changed here in this method):
558  if (deg(p) > 10*deg(reducer) then perform reduction;
559  modifies p */
560 void heuristicReduce(poly &p, poly reducer, const coeffs cf)
561 {
562  #ifdef LDEBUG
563  p_Test((poly)p, naRing);
564  p_Test((poly)reducer, naRing);
565  #endif
566  if (p_Totaldegree(p, naRing) > 10 * p_Totaldegree(reducer, naRing))
567  definiteReduce(p, reducer, cf);
568 }
569 
570 void naWriteLong(number a, const coeffs cf)
571 {
572  naTest(a);
573  if (a == NULL)
574  StringAppendS("0");
575  else
576  {
577  poly aAsPoly = (poly)a;
578  /* basically, just write aAsPoly using p_Write,
579  but use brackets around the output, if a is not
580  a constant living in naCoeffs = cf->extRing->cf */
581  BOOLEAN useBrackets = !(p_IsConstant(aAsPoly, naRing));
582  if (useBrackets) StringAppendS("(");
583  p_String0Long(aAsPoly, naRing, naRing);
584  if (useBrackets) StringAppendS(")");
585  }
586 }
587 
588 void naWriteShort(number a, const coeffs cf)
589 {
590  naTest(a);
591  if (a == NULL)
592  StringAppendS("0");
593  else
594  {
595  poly aAsPoly = (poly)a;
596  /* basically, just write aAsPoly using p_Write,
597  but use brackets around the output, if a is not
598  a constant living in naCoeffs = cf->extRing->cf */
599  BOOLEAN useBrackets = !(p_IsConstant(aAsPoly, naRing));
600  if (useBrackets) StringAppendS("(");
601  p_String0Short(aAsPoly, naRing, naRing);
602  if (useBrackets) StringAppendS(")");
603  }
604 }
605 
606 const char * naRead(const char *s, number *a, const coeffs cf)
607 {
608  poly aAsPoly;
609  const char * result = p_Read(s, aAsPoly, naRing);
610  if (aAsPoly!=NULL) definiteReduce(aAsPoly, naMinpoly, cf);
611  *a = (number)aAsPoly;
612  return result;
613 }
614 
615 #if 0
616 /* implemented by the rule lcm(a, b) = a * b / gcd(a, b) */
617 number naLcm(number a, number b, const coeffs cf)
618 {
619  naTest(a); naTest(b);
620  if (a == NULL) return NULL;
621  if (b == NULL) return NULL;
622  number theProduct = (number)pp_Mult_qq((poly)a, (poly)b, naRing);
623  /* note that theProduct needs not be reduced w.r.t. naMinpoly;
624  but the final division will take care of the necessary reduction */
625  number theGcd = naGcd(a, b, cf);
626  return naDiv(theProduct, theGcd, cf);
627 }
628 #endif
629 number napNormalizeHelper(number b, const coeffs cf)
630 {
631  number h=n_Init(1,naRing->cf);
632  poly bb=(poly)b;
633  number d;
634  while(bb!=NULL)
635  {
636  d=n_NormalizeHelper(h,pGetCoeff(bb), naRing->cf);
637  n_Delete(&h,naRing->cf);
638  h=d;
639  pIter(bb);
640  }
641  return h;
642 }
643 number naLcmContent(number a, number b, const coeffs cf)
644 {
645  if (nCoeff_is_Zp(naRing->cf)) return naCopy(a,cf);
646 #if 0
647  else {
648  number g = ndGcd(a, b, cf);
649  return g;
650  }
651 #else
652  {
653  a=(number)p_Copy((poly)a,naRing);
654  number t=napNormalizeHelper(b,cf);
655  if(!n_IsOne(t,naRing->cf))
656  {
657  number bt, rr;
658  poly xx=(poly)a;
659  while (xx!=NULL)
660  {
661  bt = n_SubringGcd(t, pGetCoeff(xx), naRing->cf);
662  rr = n_Mult(t, pGetCoeff(xx), naRing->cf);
663  n_Delete(&pGetCoeff(xx),naRing->cf);
664  pGetCoeff(xx) = n_Div(rr, bt, naRing->cf);
665  n_Normalize(pGetCoeff(xx),naRing->cf);
666  n_Delete(&bt,naRing->cf);
667  n_Delete(&rr,naRing->cf);
668  pIter(xx);
669  }
670  }
671  n_Delete(&t,naRing->cf);
672  return (number) a;
673  }
674 #endif
675 }
676 
677 /* expects *param to be castable to AlgExtInfo */
678 static BOOLEAN naCoeffIsEqual(const coeffs cf, n_coeffType n, void * param)
679 {
680  if (n_algExt != n) return FALSE;
681  AlgExtInfo *e = (AlgExtInfo *)param;
682  /* for extension coefficient fields we expect the underlying
683  polynomial rings to be IDENTICAL, i.e. the SAME OBJECT;
684  this expectation is based on the assumption that we have properly
685  registered cf and perform reference counting rather than creating
686  multiple copies of the same coefficient field/domain/ring */
687  if (naRing == e->r)
688  return TRUE;
689  /* (Note that then also the minimal ideals will necessarily be
690  the same, as they are attached to the ring.) */
691 
692  // NOTE: Q(a)[x] && Q(a)[y] should better share the _same_ Q(a)...
693  if( rEqual(naRing, e->r, TRUE) ) // also checks the equality of qideals
694  {
695  const ideal mi = naRing->qideal;
696  assume( IDELEMS(mi) == 1 );
697  const ideal ii = e->r->qideal;
698  assume( IDELEMS(ii) == 1 );
699 
700  // TODO: the following should be extended for 2 *equal* rings...
701  assume( p_EqualPolys(mi->m[0], ii->m[0], naRing, e->r) );
702 
703  rDelete(e->r);
704 
705  return TRUE;
706  }
707 
708  return FALSE;
709 
710 }
711 
712 int naSize(number a, const coeffs cf)
713 {
714  if (a == NULL) return 0;
715  poly aAsPoly = (poly)a;
716  int theDegree = 0; int noOfTerms = 0;
717  while (aAsPoly != NULL)
718  {
719  noOfTerms++;
720  int d = p_GetExp(aAsPoly, 1, naRing);
721  if (d > theDegree) theDegree = d;
722  pIter(aAsPoly);
723  }
724  return (theDegree +1) * noOfTerms;
725 }
726 
727 /* performs polynomial division and overrides p by the remainder
728  of division of p by the reducer;
729  modifies p */
730 void definiteReduce(poly &p, poly reducer, const coeffs cf)
731 {
732  #ifdef LDEBUG
733  p_Test((poly)p, naRing);
734  p_Test((poly)reducer, naRing);
735  #endif
736  if ((p!=NULL) && (p_GetExp(p,1,naRing)>=p_GetExp(reducer,1,naRing)))
737  {
738  p_PolyDiv(p, reducer, FALSE, naRing);
739  }
740 }
741 
742 void naNormalize(number &a, const coeffs cf)
743 {
744  poly aa=(poly)a;
745  if (aa!=naMinpoly)
747  a=(number)aa;
748 }
749 
751 {
752  if (n.isZero()) return NULL;
753  poly p=convFactoryPSingP(n,naRing);
754  return (number)p;
755 }
756 CanonicalForm naConvSingNFactoryN( number n, BOOLEAN /*setChar*/, const coeffs cf )
757 {
758  naTest(n);
759  if (n==NULL) return CanonicalForm(0);
760 
761  return convSingPFactoryP((poly)n,naRing);
762 }
763 
764 /* IMPORTANT NOTE: Since an algebraic field extension is again a field,
765  the gcd of two elements is not very interesting. (It
766  is actually any unit in the field, i.e., any non-
767  zero element.) Note that the below method does not operate
768  in this strong sense but rather computes the gcd of
769  two given elements in the underlying polynomial ring. */
770 number naGcd(number a, number b, const coeffs cf)
771 {
772  if (a==NULL) return naCopy(b,cf);
773  if (b==NULL) return naCopy(a,cf);
774 
775  poly ax=(poly)a;
776  poly bx=(poly)b;
777  if (pNext(ax)!=NULL)
778  return (number)p_Copy(ax, naRing);
779  else
780  {
781  if(nCoeff_is_Zp(naRing->cf))
782  return naInit(1,cf);
783  else
784  {
785  number x = n_Copy(pGetCoeff((poly)a),naRing->cf);
786  if (n_IsOne(x,naRing->cf))
787  return (number)p_NSet(x,naRing);
788  while (pNext(ax)!=NULL)
789  {
790  pIter(ax);
791  number y = n_SubringGcd(x, pGetCoeff(ax), naRing->cf);
792  n_Delete(&x,naRing->cf);
793  x = y;
794  if (n_IsOne(x,naRing->cf))
795  return (number)p_NSet(x,naRing);
796  }
797  do
798  {
799  number y = n_SubringGcd(x, pGetCoeff(bx), naRing->cf);
800  n_Delete(&x,naRing->cf);
801  x = y;
802  if (n_IsOne(x,naRing->cf))
803  return (number)p_NSet(x,naRing);
804  pIter(bx);
805  }
806  while (bx!=NULL);
807  return (number)p_NSet(x,naRing);
808  }
809  }
810 #if 0
811  naTest(a); naTest(b);
812  const ring R = naRing;
813  return (number) singclap_gcd_r((poly)a, (poly)b, R);
814 #endif
815 // return (number)p_Gcd((poly)a, (poly)b, naRing);
816 }
817 
818 number naInvers(number a, const coeffs cf)
819 {
820  naTest(a);
821  if (a == NULL) WerrorS(nDivBy0);
822 
823  poly aFactor = NULL; poly mFactor = NULL; poly theGcd = NULL;
824 // singclap_extgcd!
825  const BOOLEAN ret = singclap_extgcd ((poly)a, naMinpoly, theGcd, aFactor, mFactor, naRing);
826 
827  assume( !ret );
828 
829 // if( ret ) theGcd = p_ExtGcd((poly)a, aFactor, naMinpoly, mFactor, naRing);
830 
831  naTest((number)theGcd); naTest((number)aFactor); naTest((number)mFactor);
832  p_Delete(&mFactor, naRing);
833 
834  // /* the gcd must be 1 since naMinpoly is irreducible and a != NULL: */
835  // assume(naIsOne((number)theGcd, cf));
836 
837  if( !naIsOne((number)theGcd, cf) )
838  {
839  WerrorS("zero divisor found - your minpoly is not irreducible");
840  p_Delete(&aFactor, naRing); aFactor = NULL;
841  }
842  p_Delete(&theGcd, naRing);
843 
844  return (number)(aFactor);
845 }
846 
847 /* assumes that src = Q or Z, dst = Q(a) */
848 number naMap00(number a, const coeffs src, const coeffs dst)
849 {
850  if (n_IsZero(a, src)) return NULL;
851  assume(src->rep == dst->extRing->cf->rep);
852  poly result = p_One(dst->extRing);
853  p_SetCoeff(result, n_Copy(a, src), dst->extRing);
854  return (number)result;
855 }
856 
857 /* assumes that src = Z, dst = K(a) */
858 number naMapZ0(number a, const coeffs src, const coeffs dst)
859 {
860  if (n_IsZero(a, src)) return NULL;
861  poly result = p_One(dst->extRing);
862  nMapFunc nMap=n_SetMap(src,dst->extRing->cf);
863  p_SetCoeff(result, nMap(a, src, dst->extRing->cf), dst->extRing);
864  if (n_IsZero(pGetCoeff(result),dst->extRing->cf))
865  p_Delete(&result,dst->extRing);
866  return (number)result;
867 }
868 
869 /* assumes that src = Z/p, dst = Q(a) */
870 number naMapP0(number a, const coeffs src, const coeffs dst)
871 {
872  if (n_IsZero(a, src)) return NULL;
873  /* mapping via intermediate int: */
874  int n = n_Int(a, src);
875  number q = n_Init(n, dst->extRing->cf);
876  poly result = p_One(dst->extRing);
877  p_SetCoeff(result, q, dst->extRing);
878  return (number)result;
879 }
880 
881 #if 0
882 /* assumes that either src = Q(a), dst = Q(a), or
883  src = Z/p(a), dst = Z/p(a) */
884 number naCopyMap(number a, const coeffs src, const coeffs dst)
885 {
886  return naCopy(a, dst);
887 }
888 #endif
889 
890 number naCopyTrans2AlgExt(number a, const coeffs src, const coeffs dst)
891 {
892  assume (nCoeff_is_transExt (src));
893  assume (nCoeff_is_algExt (dst));
894  fraction fa=(fraction)a;
895  poly p, q;
896  if (rSamePolyRep(src->extRing, dst->extRing))
897  {
898  p = p_Copy(NUM(fa),src->extRing);
899  if (!DENIS1(fa))
900  {
901  q = p_Copy(DEN(fa),src->extRing);
902  assume (q != NULL);
903  }
904  }
905  else
906  {
907  assume ((strcmp(rRingVar(0,src->extRing),rRingVar(0,dst->extRing))==0) && (rVar (src->extRing) == rVar (dst->extRing)));
908 
909  nMapFunc nMap= n_SetMap (src->extRing->cf, dst->extRing->cf);
910 
911  assume (nMap != NULL);
912  p= p_PermPoly (NUM (fa), NULL, src->extRing, dst->extRing,nMap, NULL,rVar (src->extRing));
913  if (!DENIS1(fa))
914  {
915  q= p_PermPoly (DEN (fa), NULL, src->extRing, dst->extRing,nMap, NULL,rVar (src->extRing));
916  assume (q != NULL);
917  }
918  }
919  definiteReduce(p, dst->extRing->qideal->m[0], dst);
920  p_Test (p, dst->extRing);
921  if (!DENIS1(fa))
922  {
923  definiteReduce(q, dst->extRing->qideal->m[0], dst);
924  p_Test (q, dst->extRing);
925  if (q != NULL)
926  {
927  number t= naDiv ((number)p,(number)q, dst);
928  p_Delete (&p, dst->extRing);
929  p_Delete (&q, dst->extRing);
930  return t;
931  }
932  WerrorS ("mapping denominator to zero");
933  }
934  return (number) p;
935 }
936 
937 /* assumes that src = Q, dst = Z/p(a) */
938 number naMap0P(number a, const coeffs src, const coeffs dst)
939 {
940  if (n_IsZero(a, src)) return NULL;
941  // int p = rChar(dst->extRing);
942 
943  number q = nlModP(a, src, dst->extRing->cf); // FIXME? TODO? // extern number nlModP(number q, const coeffs Q, const coeffs Zp); // Map q \in QQ \to pZ
944 
945  poly result = p_NSet(q, dst->extRing);
946 
947  return (number)result;
948 }
949 
950 /* assumes that src = Z/p, dst = Z/p(a) */
951 number naMapPP(number a, const coeffs src, const coeffs dst)
952 {
953  if (n_IsZero(a, src)) return NULL;
954  assume(src == dst->extRing->cf);
955  poly result = p_One(dst->extRing);
956  p_SetCoeff(result, n_Copy(a, src), dst->extRing);
957  return (number)result;
958 }
959 
960 /* assumes that src = Z/u, dst = Z/p(a), where u != p */
961 number naMapUP(number a, const coeffs src, const coeffs dst)
962 {
963  if (n_IsZero(a, src)) return NULL;
964  /* mapping via intermediate int: */
965  int n = n_Int(a, src);
966  number q = n_Init(n, dst->extRing->cf);
967  poly result = p_One(dst->extRing);
968  p_SetCoeff(result, q, dst->extRing);
969  return (number)result;
970 }
971 
972 number naGenMap(number a, const coeffs cf, const coeffs dst)
973 {
974  if (a==NULL) return NULL;
975 
976  const ring rSrc = cf->extRing;
977  const ring rDst = dst->extRing;
978 
979  const nMapFunc nMap=n_SetMap(rSrc->cf,rDst->cf);
980  poly f = (poly)a;
981  poly g = prMapR(f, nMap, rSrc, rDst);
982 
983  n_Test((number)g, dst);
984  return (number)g;
985 }
986 
987 number naGenTrans2AlgExt(number a, const coeffs cf, const coeffs dst)
988 {
989  if (a==NULL) return NULL;
990 
991  const ring rSrc = cf->extRing;
992  const ring rDst = dst->extRing;
993 
994  const nMapFunc nMap=n_SetMap(rSrc->cf,rDst->cf);
995  fraction f = (fraction)a;
996  poly g = prMapR(NUM(f), nMap, rSrc, rDst);
997 
998  number result=NULL;
999  poly h = NULL;
1000 
1001  if (!DENIS1(f))
1002  h = prMapR(DEN(f), nMap, rSrc, rDst);
1003 
1004  if (h!=NULL)
1005  {
1006  result=naDiv((number)g,(number)h,dst);
1007  p_Delete(&g,dst->extRing);
1008  p_Delete(&h,dst->extRing);
1009  }
1010  else
1011  result=(number)g;
1012 
1013  n_Test((number)result, dst);
1014  return (number)result;
1015 }
1016 
1017 nMapFunc naSetMap(const coeffs src, const coeffs dst)
1018 {
1019  /* dst is expected to be an algebraic field extension */
1020  assume(getCoeffType(dst) == n_algExt);
1021 
1022  if( src == dst ) return ndCopyMap;
1023 
1024  int h = 0; /* the height of the extension tower given by dst */
1025  coeffs bDst = nCoeff_bottom(dst, h); /* the bottom field in the tower dst */
1026  coeffs bSrc = nCoeff_bottom(src, h); /* the bottom field in the tower src */
1027 
1028  /* for the time being, we only provide maps if h = 1 or 0 */
1029  if (h==0)
1030  {
1031  if ((src->rep==n_rep_gap_rat) && nCoeff_is_Q(bDst))
1032  return naMap00; /// Q or Z --> Q(a)
1033  if ((src->rep==n_rep_gap_gmp) && nCoeff_is_Q(bDst))
1034  return naMapZ0; /// Z --> Q(a)
1035  if (nCoeff_is_Zp(src) && nCoeff_is_Q(bDst))
1036  return naMapP0; /// Z/p --> Q(a)
1037  if (nCoeff_is_Q_or_BI(src) && nCoeff_is_Zp(bDst))
1038  return naMap0P; /// Q --> Z/p(a)
1039  if ((src->rep==n_rep_gap_gmp) && nCoeff_is_Zp(bDst))
1040  return naMapZ0; /// Z --> Z/p(a)
1041  if (nCoeff_is_Zp(src) && nCoeff_is_Zp(bDst))
1042  {
1043  if (src->ch == dst->ch) return naMapPP; /// Z/p --> Z/p(a)
1044  else return naMapUP; /// Z/u --> Z/p(a)
1045  }
1046  }
1047  if (h != 1) return NULL;
1048  if ((!nCoeff_is_Zp(bDst)) && (!nCoeff_is_Q(bDst))) return NULL;
1049  if ((!nCoeff_is_Zp(bSrc)) && (!nCoeff_is_Q_or_BI(bSrc))) return NULL;
1050 
1051  nMapFunc nMap=n_SetMap(src->extRing->cf,dst->extRing->cf);
1052  if (rSamePolyRep(src->extRing, dst->extRing) && (strcmp(rRingVar(0, src->extRing), rRingVar(0, dst->extRing)) == 0))
1053  {
1054  if (src->type==n_algExt)
1055  return ndCopyMap; // naCopyMap; /// K(a) --> K(a)
1056  else
1057  return naCopyTrans2AlgExt;
1058  }
1059  else if ((nMap!=NULL) && (strcmp(rRingVar(0,src->extRing),rRingVar(0,dst->extRing))==0) && (rVar (src->extRing) == rVar (dst->extRing)))
1060  {
1061  if (src->type==n_algExt)
1062  return naGenMap; // naCopyMap; /// K(a) --> K'(a)
1063  else
1064  return naGenTrans2AlgExt;
1065  }
1066 
1067  return NULL; /// default
1068 }
1069 
1070 int naParDeg(number a, const coeffs cf)
1071 {
1072  if (a == NULL) return -1;
1073  poly aa=(poly)a;
1074  return cf->extRing->pFDeg(aa,cf->extRing);
1075 }
1076 
1077 /// return the specified parameter as a number in the given alg. field
1078 number naParameter(const int iParameter, const coeffs cf)
1079 {
1081 
1082  const ring R = cf->extRing;
1083  assume( R != NULL );
1084  assume( 0 < iParameter && iParameter <= rVar(R) );
1085 
1086  poly p = p_One(R); p_SetExp(p, iParameter, 1, R); p_Setm(p, R);
1087 
1088  return (number) p;
1089 }
1090 
1091 
1092 /// if m == var(i)/1 => return i,
1093 int naIsParam(number m, const coeffs cf)
1094 {
1096 
1097  const ring R = cf->extRing;
1098  assume( R != NULL );
1099 
1100  return p_Var( (poly)m, R );
1101 }
1102 
1103 
1104 static void naClearContent(ICoeffsEnumerator& numberCollectionEnumerator, number& c, const coeffs cf)
1105 {
1106  assume(cf != NULL);
1108  assume(nCoeff_is_Q_algext(cf)); // only over (Q[a]/m(a)), while the default impl. is used over Zp[a]/m(a) !
1109 
1110  const ring R = cf->extRing;
1111  assume(R != NULL);
1112  const coeffs Q = R->cf;
1113  assume(Q != NULL);
1114  assume(nCoeff_is_Q(Q));
1115 
1116  numberCollectionEnumerator.Reset();
1117 
1118  if( !numberCollectionEnumerator.MoveNext() ) // empty zero polynomial?
1119  {
1120  c = n_Init(1, cf);
1121  return;
1122  }
1123 
1124  naTest(numberCollectionEnumerator.Current());
1125 
1126  // part 1, find a small candidate for gcd
1127  int s1; int s=2147483647; // max. int
1128 
1129  const BOOLEAN lc_is_pos=naGreaterZero(numberCollectionEnumerator.Current(),cf);
1130 
1131  int normalcount = 0;
1132 
1133  poly cand1, cand;
1134 
1135  do
1136  {
1137  number& n = numberCollectionEnumerator.Current();
1138  naNormalize(n, cf); ++normalcount;
1139 
1140  naTest(n);
1141 
1142  cand1 = (poly)n;
1143 
1144  s1 = p_Deg(cand1, R); // naSize?
1145  if (s>s1)
1146  {
1147  cand = cand1;
1148  s = s1;
1149  }
1150  } while (numberCollectionEnumerator.MoveNext() );
1151 
1152 // assume( nlGreaterZero(cand,cf) ); // cand may be a negative integer!
1153 
1154  cand = p_Copy(cand, R);
1155  // part 2: compute gcd(cand,all coeffs)
1156 
1157  numberCollectionEnumerator.Reset();
1158 
1159  int length = 0;
1160  while (numberCollectionEnumerator.MoveNext() )
1161  {
1162  number& n = numberCollectionEnumerator.Current();
1163  ++length;
1164 
1165  if( (--normalcount) <= 0)
1166  naNormalize(n, cf);
1167 
1168  naTest(n);
1169 
1170 // p_InpGcd(cand, (poly)n, R);
1171 
1172  { // R->cf is QQ
1173  poly tmp=gcd_over_Q(cand,(poly)n,R);
1174  p_Delete(&cand,R);
1175  cand=tmp;
1176  }
1177 
1178 // cand1 = p_Gcd(cand,(poly)n, R); p_Delete(&cand, R); cand = cand1;
1179 
1180  assume( naGreaterZero((number)cand, cf) ); // ???
1181 /*
1182  if(p_IsConstant(cand,R))
1183  {
1184  c = cand;
1185 
1186  if(!lc_is_pos)
1187  {
1188  // make the leading coeff positive
1189  c = nlNeg(c, cf);
1190  numberCollectionEnumerator.Reset();
1191 
1192  while (numberCollectionEnumerator.MoveNext() )
1193  {
1194  number& nn = numberCollectionEnumerator.Current();
1195  nn = nlNeg(nn, cf);
1196  }
1197  }
1198  return;
1199  }
1200 */
1201 
1202  }
1203 
1204 
1205  // part3: all coeffs = all coeffs / cand
1206  if (!lc_is_pos)
1207  cand = p_Neg(cand, R);
1208 
1209  c = (number)cand; naTest(c);
1210 
1211  poly cInverse = (poly)naInvers(c, cf);
1212  assume(cInverse != NULL); // c is non-zero divisor!?
1213 
1214 
1215  numberCollectionEnumerator.Reset();
1216 
1217 
1218  while (numberCollectionEnumerator.MoveNext() )
1219  {
1220  number& n = numberCollectionEnumerator.Current();
1221 
1222  assume( length > 0 );
1223 
1224  if( --length > 0 )
1225  {
1226  assume( cInverse != NULL );
1227  n = (number) p_Mult_q(p_Copy(cInverse, R), (poly)n, R);
1228  }
1229  else
1230  {
1231  n = (number) p_Mult_q(cInverse, (poly)n, R);
1232  cInverse = NULL;
1233  assume(length == 0);
1234  }
1235 
1236  definiteReduce((poly &)n, naMinpoly, cf);
1237  }
1238 
1239  assume(length == 0);
1240  assume(cInverse == NULL); // p_Delete(&cInverse, R);
1241 
1242  // Quick and dirty fix for constant content clearing... !?
1243  CRecursivePolyCoeffsEnumerator<NAConverter> itr(numberCollectionEnumerator); // recursively treat the numbers as polys!
1244 
1245  number cc;
1246 
1247  n_ClearContent(itr, cc, Q); // TODO: get rid of (-LC) normalization!?
1248 
1249  // over alg. ext. of Q // takes over the input number
1250  c = (number) __p_Mult_nn( (poly)c, cc, R);
1251 // p_Mult_q(p_NSet(cc, R), , R);
1252 
1253  n_Delete(&cc, Q);
1254 
1255  // TODO: the above is not enough! need GCD's of polynomial coeffs...!
1256 /*
1257  // old and wrong part of p_Content
1258  if (rField_is_Q_a(r) && !CLEARENUMERATORS) // should not be used anymore if CLEARENUMERATORS is 1
1259  {
1260  // we only need special handling for alg. ext.
1261  if (getCoeffType(r->cf)==n_algExt)
1262  {
1263  number hzz = n_Init(1, r->cf->extRing->cf);
1264  p=ph;
1265  while (p!=NULL)
1266  { // each monom: coeff in Q_a
1267  poly c_n_n=(poly)pGetCoeff(p);
1268  poly c_n=c_n_n;
1269  while (c_n!=NULL)
1270  { // each monom: coeff in Q
1271  d=n_NormalizeHelper(hzz,pGetCoeff(c_n),r->cf->extRing->cf);
1272  n_Delete(&hzz,r->cf->extRing->cf);
1273  hzz=d;
1274  pIter(c_n);
1275  }
1276  pIter(p);
1277  }
1278  // hzz contains the 1/lcm of all denominators in c_n_n
1279  h=n_Invers(hzz,r->cf->extRing->cf);
1280  n_Delete(&hzz,r->cf->extRing->cf);
1281  n_Normalize(h,r->cf->extRing->cf);
1282  if(!n_IsOne(h,r->cf->extRing->cf))
1283  {
1284  p=ph;
1285  while (p!=NULL)
1286  { // each monom: coeff in Q_a
1287  poly c_n=(poly)pGetCoeff(p);
1288  while (c_n!=NULL)
1289  { // each monom: coeff in Q
1290  d=n_Mult(h,pGetCoeff(c_n),r->cf->extRing->cf);
1291  n_Normalize(d,r->cf->extRing->cf);
1292  n_Delete(&pGetCoeff(c_n),r->cf->extRing->cf);
1293  pGetCoeff(c_n)=d;
1294  pIter(c_n);
1295  }
1296  pIter(p);
1297  }
1298  }
1299  n_Delete(&h,r->cf->extRing->cf);
1300  }
1301  }
1302 */
1303 
1304 
1305 // c = n_Init(1, cf); assume(FALSE); // TODO: NOT YET IMPLEMENTED!!!
1306 }
1307 
1308 
1309 void naClearDenominators(ICoeffsEnumerator& numberCollectionEnumerator, number& c, const coeffs cf)
1310 {
1311  assume(cf != NULL);
1313  assume(nCoeff_is_Q_algext(cf)); // only over (Q[a]/m(a)), while the default impl. is used over Zp[a]/m(a) !
1314 
1315  assume(cf->extRing != NULL);
1316  const coeffs Q = cf->extRing->cf;
1317  assume(Q != NULL);
1318  assume(nCoeff_is_Q(Q));
1319  number n;
1320  CRecursivePolyCoeffsEnumerator<NAConverter> itr(numberCollectionEnumerator); // recursively treat the numbers as polys!
1321  n_ClearDenominators(itr, n, Q); // this should probably be fine...
1322  c = (number)p_NSet(n, cf->extRing); // over alg. ext. of Q // takes over the input number
1323 }
1324 
1326 {
1327  if ((--cf->extRing->ref) == 0)
1328  rDelete(cf->extRing);
1329 }
1330 
1331 char* naCoeffString(const coeffs r) // currently also for tranext.
1332 {
1333  const char* const* p=n_ParameterNames(r);
1334  int l=0;
1335  int i;
1336  for(i=0; i<n_NumberOfParameters(r);i++)
1337  {
1338  l+=(strlen(p[i])+1);
1339  }
1340  char *s=(char *)omAlloc(l+10+1);
1341  s[0]='\0';
1342  snprintf(s,10+1,"%d",r->ch); /* Fp(a) or Q(a) */
1343  char tt[2];
1344  tt[0]=',';
1345  tt[1]='\0';
1346  for(i=0; i<n_NumberOfParameters(r);i++)
1347  {
1348  strcat(s,tt);
1349  strcat(s,p[i]);
1350  }
1351  return s;
1352 }
1353 
1354 char* naCoeffName(const coeffs r) // currently also for tranext.
1355 {
1356  const char* const* p=n_ParameterNames(r);
1357  int l=0;
1358  int i;
1359  for(i=0; i<n_NumberOfParameters(r);i++)
1360  {
1361  l+=(strlen(p[i])+1);
1362  }
1363  STATIC_VAR char s[200];
1364  s[0]='\0';
1365  snprintf(s,10+1,"%d",r->ch); /* Fp(a) or Q(a) */
1366  char tt[2];
1367  tt[0]=',';
1368  tt[1]='\0';
1369  for(i=0; i<n_NumberOfParameters(r);i++)
1370  {
1371  strcat(s,tt);
1372  strcat(s,p[i]);
1373  }
1374  return s;
1375 }
1376 
1377 number naChineseRemainder(number *x, number *q,int rl, BOOLEAN /*sym*/,CFArray &inv_cache,const coeffs cf)
1378 {
1379  poly *P=(poly*)omAlloc(rl*sizeof(poly*));
1380  number *X=(number *)omAlloc(rl*sizeof(number));
1381  int i;
1382  for(i=0;i<rl;i++) P[i]=p_Copy((poly)(x[i]),cf->extRing);
1383  poly result=p_ChineseRemainder(P,X,q,rl,inv_cache,cf->extRing);
1384  omFreeSize(X,rl*sizeof(number));
1385  omFreeSize(P,rl*sizeof(poly*));
1386  return ((number)result);
1387 }
1388 
1389 number naFarey(number p, number n, const coeffs cf)
1390 {
1391  // n is really a bigint
1392  poly result=p_Farey(p_Copy((poly)p,cf->extRing),n,cf->extRing);
1393  return ((number)result);
1394 }
1395 
1396 
1397 BOOLEAN naInitChar(coeffs cf, void * infoStruct)
1398 {
1399  assume( infoStruct != NULL );
1400 
1401  AlgExtInfo *e = (AlgExtInfo *)infoStruct;
1402  /// first check whether cf->extRing != NULL and delete old ring???
1403 
1404  assume(e->r != NULL); // extRing;
1405  assume(e->r->cf != NULL); // extRing->cf;
1406 
1407  assume((e->r->qideal != NULL) && // minideal has one
1408  (IDELEMS(e->r->qideal) == 1) && // non-zero generator
1409  (e->r->qideal->m[0] != NULL) ); // at m[0];
1410 
1411  assume( cf != NULL );
1412  assume(getCoeffType(cf) == n_algExt); // coeff type;
1413 
1414  e->r->ref ++; // increase the ref.counter for the ground poly. ring!
1415  const ring R = e->r; // no copy!
1416  cf->extRing = R;
1417 
1418  /* propagate characteristic up so that it becomes
1419  directly accessible in cf: */
1420  cf->ch = R->cf->ch;
1421 
1422  cf->is_field=TRUE;
1423  cf->is_domain=TRUE;
1424  cf->rep=n_rep_poly;
1425 
1426  #ifdef LDEBUG
1427  p_Test((poly)naMinpoly, naRing);
1428  #endif
1429 
1430  cf->cfCoeffString = naCoeffString;
1431  cf->cfCoeffName = naCoeffName;
1432 
1433  cf->cfGreaterZero = naGreaterZero;
1434  cf->cfGreater = naGreater;
1435  cf->cfEqual = naEqual;
1436  cf->cfIsZero = naIsZero;
1437  cf->cfIsOne = naIsOne;
1438  cf->cfIsMOne = naIsMOne;
1439  cf->cfInit = naInit;
1440  cf->cfFarey = naFarey;
1441  cf->cfChineseRemainder= naChineseRemainder;
1442  cf->cfInt = naInt;
1443  cf->cfInpNeg = naNeg;
1444  cf->cfAdd = naAdd;
1445  cf->cfSub = naSub;
1446  cf->cfMult = naMult;
1447  cf->cfDiv = naDiv;
1448  cf->cfExactDiv = naDiv;
1449  cf->cfPower = naPower;
1450  cf->cfCopy = naCopy;
1451 
1452  cf->cfWriteLong = naWriteLong;
1453 
1454  if( rCanShortOut(naRing) )
1455  cf->cfWriteShort = naWriteShort;
1456  else
1457  cf->cfWriteShort = naWriteLong;
1458 
1459  cf->cfRead = naRead;
1460  cf->cfDelete = naDelete;
1461  cf->cfSetMap = naSetMap;
1462  cf->cfGetDenom = naGetDenom;
1463  cf->cfGetNumerator = naGetNumerator;
1464  cf->cfRePart = naCopy;
1465  cf->cfCoeffWrite = naCoeffWrite;
1466  cf->cfNormalize = naNormalize;
1467  cf->cfKillChar = naKillChar;
1468 #ifdef LDEBUG
1469  cf->cfDBTest = naDBTest;
1470 #endif
1471  cf->cfGcd = naGcd;
1472  cf->cfNormalizeHelper = naLcmContent;
1473  cf->cfSize = naSize;
1474  cf->nCoeffIsEqual = naCoeffIsEqual;
1475  cf->cfInvers = naInvers;
1476  cf->convFactoryNSingN=naConvFactoryNSingN;
1477  cf->convSingNFactoryN=naConvSingNFactoryN;
1478  cf->cfParDeg = naParDeg;
1479 
1480  cf->iNumberOfParameters = rVar(R);
1481  cf->pParameterNames = (const char**)R->names;
1482  cf->cfParameter = naParameter;
1483  cf->has_simple_Inverse= R->cf->has_simple_Inverse;
1484  /* cf->has_simple_Alloc= FALSE; */
1485 
1486  if( nCoeff_is_Q(R->cf) )
1487  {
1488  cf->cfClearContent = naClearContent;
1489  cf->cfClearDenominators = naClearDenominators;
1490  }
1491 
1492  return FALSE;
1493 }
1494 
1496 
1497 template class IAccessor<snumber*>;
1498 
1499 /* --------------------------------------------------------------------*/
1500 /****************************************
1501 * Computer Algebra System SINGULAR *
1502 ****************************************/
1503 /**
1504  * ABSTRACT: numbers as polys in the ring K[a]
1505  * Assuming that we have a coeffs object cf, then these numbers
1506  * are polynomials in the polynomial ring K[a] represented by
1507  * cf->extRing.
1508  * IMPORTANT ASSUMPTIONS:
1509  * 1.) So far we assume that cf->extRing is a valid polynomial
1510  * ring
1511  **/
1512 
1513 #ifdef LDEBUG
1514 #define n2pTest(a) n2pDBTest(a,__FILE__,__LINE__,cf)
1515 BOOLEAN n2pDBTest(number a, const char *f, const int l, const coeffs r);
1516 #else
1517 #define n2pTest(a) do {} while (0)
1518 #endif
1519 
1520 /* polynomial ring in which our numbers live */
1521 #define n2pRing cf->extRing
1522 
1523 /* coeffs object in which the coefficients of our numbers live;
1524  * methods attached to n2pCoeffs may be used to compute with the
1525  * coefficients of our numbers, e.g., use n2pCoeffs->nAdd to add
1526  * coefficients of our numbers */
1527 #define n2pCoeffs cf->extRing->cf
1528 
1529 #ifdef LDEBUG
1530 BOOLEAN n2pDBTest(number a, const char *f, const int l, const coeffs cf)
1531 {
1532  if (a == NULL) return TRUE;
1533  return p_Test((poly)a, n2pRing);
1534 }
1535 #endif
1536 
1537 void n2pNormalize(number &a, const coeffs cf)
1538 {
1539  poly aa=(poly)a;
1540  p_Normalize(aa,n2pRing);
1541 }
1542 
1543 /* TRUE iff (a != 0 and (b == 0 or deg(a) > deg(b) or (deg(a)==deg(b) && lc(a)>lc(b))) */
1544 number n2pMult(number a, number b, const coeffs cf)
1545 {
1546  n2pTest(a); n2pTest(b);
1547  if ((a == NULL)||(b == NULL)) return NULL;
1548  poly aTimesB = pp_Mult_qq((poly)a, (poly)b, n2pRing);
1549  return (number)aTimesB;
1550 }
1551 
1552 number n2pDiv(number a, number b, const coeffs cf)
1553 {
1554  n2pTest(a); n2pTest(b);
1555  if (b == NULL) WerrorS(nDivBy0);
1556  if (a == NULL) return NULL;
1557  poly p=singclap_pdivide((poly)a,(poly)b,n2pRing);
1558  return (number)p;
1559 }
1560 
1561 void n2pPower(number a, int exp, number *b, const coeffs cf)
1562 {
1563  n2pTest(a);
1564 
1565  *b= (number)p_Power((poly)a,exp,n2pRing);
1566 }
1567 
1568 const char * n2pRead(const char *s, number *a, const coeffs cf)
1569 {
1570  poly aAsPoly;
1571  const char * result = p_Read(s, aAsPoly, n2pRing);
1572  *a = (number)aAsPoly;
1573  return result;
1574 }
1575 
1576 /* expects *param to be castable to AlgExtInfo */
1577 static BOOLEAN n2pCoeffIsEqual(const coeffs cf, n_coeffType n, void * param)
1578 {
1579  if (n_polyExt != n) return FALSE;
1580  AlgExtInfo *e = (AlgExtInfo *)param;
1581  /* for extension coefficient fields we expect the underlying
1582  polynomial rings to be IDENTICAL, i.e. the SAME OBJECT;
1583  this expectation is based on the assumption that we have properly
1584  registered cf and perform reference counting rather than creating
1585  multiple copies of the same coefficient field/domain/ring */
1586  if (n2pRing == e->r)
1587  return TRUE;
1588  // NOTE: Q(a)[x] && Q(a)[y] should better share the _same_ Q(a)...
1589  if( rEqual(n2pRing, e->r, TRUE) ) // also checks the equality of qideals
1590  {
1591  rDelete(e->r);
1592  return TRUE;
1593  }
1594  return FALSE;
1595 }
1596 
1598 {
1599  const char* const* p=n_ParameterNames(cf);
1600  int l=0;
1601  int i;
1602  for(i=0; i<rVar(n2pRing);i++)
1603  {
1604  l+=(strlen(p[i])+1);
1605  }
1606  char *cf_s=nCoeffString(n2pRing->cf);
1607  char *s=(char *)omAlloc(l+5+strlen(cf_s));
1608  s[0]='\0';
1609  snprintf(s,strlen(cf_s)+2,"%s",cf_s);
1610  omFree(cf_s);
1611  char tt[2];
1612  tt[0]='[';
1613  tt[1]='\0';
1614  strcat(s,tt);
1615  tt[0]=',';
1616  for(i=0; i<rVar(n2pRing);i++)
1617  {
1618  strcat(s,p[i]);
1619  if (i+1!=rVar(n2pRing)) strcat(s,tt);
1620  else { tt[0]=']'; strcat(s,tt); }
1621  }
1622  return s;
1623 }
1624 
1625 char* n2pCoeffName(const coeffs cf)
1626 {
1627  const char* const* p=n_ParameterNames(cf);
1628  int l=0;
1629  int i;
1630  for(i=0; i<rVar(n2pRing);i++)
1631  {
1632  l+=(strlen(p[i])+1);
1633  }
1634  char *cf_s=nCoeffString(n2pRing->cf);
1635  STATIC_VAR char s[200];
1636  s[0]='\0';
1637  snprintf(s,strlen(cf_s)+2,"%s",cf_s);
1638  omFree(cf_s);
1639  char tt[2];
1640  tt[0]='[';
1641  tt[1]='\0';
1642  strcat(s,tt);
1643  tt[0]=',';
1644  for(i=0; i<rVar(n2pRing);i++)
1645  {
1646  strcat(s,p[i]);
1647  if (i+1!=rVar(n2pRing)) strcat(s,tt);
1648  else { tt[0]=']'; strcat(s,tt); }
1649  }
1650  return s;
1651 }
1652 
1653 void n2pCoeffWrite(const coeffs cf, BOOLEAN details)
1654 {
1655  assume( cf != NULL );
1656 
1657  const ring A = cf->extRing;
1658 
1659  assume( A != NULL );
1660  PrintS("// polynomial ring as coefficient ring :\n");
1661  rWrite(A);
1662  PrintLn();
1663 }
1664 
1665 number n2pInvers(number a, const coeffs cf)
1666 {
1667  poly aa=(poly)a;
1668  if(p_IsConstant(aa, n2pRing))
1669  {
1670  poly p=p_Init(n2pRing);
1672  return (number)p;
1673  }
1674  else
1675  {
1676  WerrorS("not invertible");
1677  return NULL;
1678  }
1679 }
1680 
1681 BOOLEAN n2pInitChar(coeffs cf, void * infoStruct)
1682 {
1683  assume( infoStruct != NULL );
1684 
1685  AlgExtInfo *e = (AlgExtInfo *)infoStruct;
1686  /// first check whether cf->extRing != NULL and delete old ring???
1687 
1688  assume(e->r != NULL); // extRing;
1689  assume(e->r->cf != NULL); // extRing->cf;
1690 
1691  assume( cf != NULL );
1692 
1693  e->r->ref ++; // increase the ref.counter for the ground poly. ring!
1694  const ring R = e->r; // no copy!
1695  cf->extRing = R;
1696 
1697  /* propagate characteristic up so that it becomes
1698  directly accessible in cf: */
1699  cf->ch = R->cf->ch;
1700  cf->is_field=FALSE;
1701  cf->is_domain=TRUE;
1702 
1703  cf->cfCoeffString = n2pCoeffString;
1704  cf->cfCoeffName = n2pCoeffName;
1705 
1706  cf->cfGreaterZero = naGreaterZero;
1707  cf->cfGreater = naGreater;
1708  cf->cfEqual = naEqual;
1709  cf->cfIsZero = naIsZero;
1710  cf->cfIsOne = naIsOne;
1711  cf->cfIsMOne = naIsMOne;
1712  cf->cfInit = naInit;
1713  cf->cfFarey = naFarey;
1714  cf->cfChineseRemainder= naChineseRemainder;
1715  cf->cfInt = naInt;
1716  cf->cfInpNeg = naNeg;
1717  cf->cfAdd = naAdd;
1718  cf->cfSub = naSub;
1719  cf->cfMult = n2pMult;
1720  cf->cfDiv = n2pDiv;
1721  cf->cfPower = n2pPower;
1722  cf->cfCopy = naCopy;
1723 
1724  cf->cfWriteLong = naWriteLong;
1725 
1726  if( rCanShortOut(n2pRing) )
1727  cf->cfWriteShort = naWriteShort;
1728  else
1729  cf->cfWriteShort = naWriteLong;
1730 
1731  cf->cfRead = n2pRead;
1732  cf->cfDelete = naDelete;
1733  cf->cfSetMap = naSetMap;
1734  cf->cfGetDenom = naGetDenom;
1735  cf->cfGetNumerator = naGetNumerator;
1736  cf->cfRePart = naCopy;
1737  cf->cfCoeffWrite = n2pCoeffWrite;
1738  cf->cfNormalize = n2pNormalize;
1739  cf->cfKillChar = naKillChar;
1740 #ifdef LDEBUG
1741  cf->cfDBTest = naDBTest;
1742 #endif
1743  cf->cfGcd = naGcd;
1744  cf->cfNormalizeHelper = naLcmContent;
1745  cf->cfSize = naSize;
1746  cf->nCoeffIsEqual = n2pCoeffIsEqual;
1747  cf->cfInvers = n2pInvers;
1748  cf->convFactoryNSingN=naConvFactoryNSingN;
1749  cf->convSingNFactoryN=naConvSingNFactoryN;
1750  cf->cfParDeg = naParDeg;
1751 
1752  cf->iNumberOfParameters = rVar(R);
1753  cf->pParameterNames = (const char**)R->names;
1754  cf->cfParameter = naParameter;
1755  cf->has_simple_Inverse=FALSE;
1756  /* cf->has_simple_Alloc= FALSE; */
1757 
1758  if( nCoeff_is_Q(R->cf) )
1759  {
1760  cf->cfClearContent = naClearContent;
1761  cf->cfClearDenominators = naClearDenominators;
1762  }
1763 
1764  return FALSE;
1765 }
Rational pow(const Rational &a, int e)
Definition: GMPrat.cc:411
Concrete implementation of enumerators over polynomials.
number n2pDiv(number a, number b, const coeffs cf)
Definition: algext.cc:1552
BOOLEAN naGreater(number a, number b, const coeffs cf)
Definition: algext.cc:358
void heuristicReduce(poly &p, poly reducer, const coeffs cf)
Definition: algext.cc:560
static void p_Monic(poly p, const ring r)
returns NULL if p == NULL, otherwise makes p monic by dividing by its leading coefficient (only done ...
Definition: algext.cc:120
number naNeg(number a, const coeffs cf)
this is in-place, modifies a
Definition: algext.cc:332
number n2pMult(number a, number b, const coeffs cf)
Definition: algext.cc:1544
long naInt(number &a, const coeffs cf)
Definition: algext.cc:345
number naMap00(number a, const coeffs src, const coeffs dst)
Definition: algext.cc:848
number naCopy(number a, const coeffs cf)
Definition: algext.cc:296
#define naMinpoly
Definition: algext.cc:70
char * naCoeffString(const coeffs r)
Definition: algext.cc:1331
BOOLEAN naIsOne(number a, const coeffs cf)
Definition: algext.cc:315
CanonicalForm naConvSingNFactoryN(number n, BOOLEAN, const coeffs cf)
Definition: algext.cc:756
number naGcd(number a, number b, const coeffs cf)
Definition: algext.cc:770
void naClearDenominators(ICoeffsEnumerator &numberCollectionEnumerator, number &c, const coeffs cf)
Definition: algext.cc:1309
BOOLEAN naDBTest(number a, const char *f, const int l, const coeffs r)
Definition: algext.cc:233
number naInit(long i, const coeffs cf)
Definition: algext.cc:339
BOOLEAN naIsZero(number a, const coeffs cf)
Definition: algext.cc:272
const char * n2pRead(const char *s, number *a, const coeffs cf)
Definition: algext.cc:1568
number naGetNumerator(number &a, const coeffs cf)
Definition: algext.cc:304
static void naClearContent(ICoeffsEnumerator &numberCollectionEnumerator, number &c, const coeffs cf)
Definition: algext.cc:1104
number naGenMap(number a, const coeffs cf, const coeffs dst)
Definition: algext.cc:972
number naSub(number a, number b, const coeffs cf)
Definition: algext.cc:448
number naCopyTrans2AlgExt(number a, const coeffs src, const coeffs dst)
Definition: algext.cc:890
poly p_ExtGcd(poly p, poly &pFactor, poly q, poly &qFactor, ring r)
assumes that p and q are univariate polynomials in r, mentioning the same variable; assumes a global ...
Definition: algext.cc:216
static poly p_Gcd(const poly p, const poly q, const ring r)
Definition: algext.cc:165
BOOLEAN naEqual(number a, number b, const coeffs cf)
Definition: algext.cc:287
void naNormalize(number &a, const coeffs cf)
Definition: algext.cc:742
void naWriteShort(number a, const coeffs cf)
Definition: algext.cc:588
char * n2pCoeffString(const coeffs cf)
Definition: algext.cc:1597
number napNormalizeHelper(number b, const coeffs cf)
Definition: algext.cc:629
void naPower(number a, int exp, number *b, const coeffs cf)
Definition: algext.cc:493
number naChineseRemainder(number *x, number *q, int rl, BOOLEAN, CFArray &inv_cache, const coeffs cf)
Definition: algext.cc:1377
BOOLEAN n2pDBTest(number a, const char *f, const int l, const coeffs r)
Definition: algext.cc:1530
#define n2pRing
Definition: algext.cc:1521
void naKillChar(coeffs cf)
Definition: algext.cc:1325
number naMap0P(number a, const coeffs src, const coeffs dst)
Definition: algext.cc:938
number naInvers(number a, const coeffs cf)
Definition: algext.cc:818
void naWriteLong(number a, const coeffs cf)
Definition: algext.cc:570
void naDelete(number *a, const coeffs cf)
Definition: algext.cc:278
void naCoeffWrite(const coeffs cf, BOOLEAN details)
Definition: algext.cc:387
#define n2pCoeffs
Definition: algext.cc:1527
number naDiv(number a, number b, const coeffs cf)
Definition: algext.cc:469
number naGenTrans2AlgExt(number a, const coeffs cf, const coeffs dst)
Definition: algext.cc:987
number naLcmContent(number a, number b, const coeffs cf)
Definition: algext.cc:643
char * naCoeffName(const coeffs r)
Definition: algext.cc:1354
number naMult(number a, number b, const coeffs cf)
Definition: algext.cc:459
const char * naRead(const char *s, number *a, const coeffs cf)
Definition: algext.cc:606
number naGetDenom(number &a, const coeffs cf)
Definition: algext.cc:309
static BOOLEAN n2pCoeffIsEqual(const coeffs cf, n_coeffType n, void *param)
Definition: algext.cc:1577
char * n2pCoeffName(const coeffs cf)
Definition: algext.cc:1625
void definiteReduce(poly &p, poly reducer, const coeffs cf)
Definition: algext.cc:730
number naConvFactoryNSingN(const CanonicalForm n, const coeffs cf)
Definition: algext.cc:750
#define naCoeffs
Definition: algext.cc:67
static BOOLEAN naCoeffIsEqual(const coeffs cf, n_coeffType n, void *param)
Definition: algext.cc:678
nMapFunc naSetMap(const coeffs src, const coeffs dst)
Get a mapping function from src into the domain of this type (n_algExt)
Definition: algext.cc:1017
static poly p_ExtGcdHelper(poly &p, poly &pFactor, poly &q, poly &qFactor, ring r)
Definition: algext.cc:183
#define naRing
Definition: algext.cc:61
BOOLEAN naInitChar(coeffs cf, void *infoStruct)
Initialize the coeffs object.
Definition: algext.cc:1397
#define naTest(a)
Definition: algext.cc:54
number naMapPP(number a, const coeffs src, const coeffs dst)
Definition: algext.cc:951
BOOLEAN n2pInitChar(coeffs cf, void *infoStruct)
Definition: algext.cc:1681
number n2pInvers(number a, const coeffs cf)
Definition: algext.cc:1665
int naParDeg(number a, const coeffs cf)
Definition: algext.cc:1070
int naIsParam(number m, const coeffs cf)
if m == var(i)/1 => return i,
Definition: algext.cc:1093
number naAdd(number a, number b, const coeffs cf)
Definition: algext.cc:437
int naSize(number a, const coeffs cf)
Definition: algext.cc:712
number naParameter(const int iParameter, const coeffs cf)
return the specified parameter as a number in the given alg. field
Definition: algext.cc:1078
BOOLEAN naGreaterZero(number a, const coeffs cf)
forward declarations
Definition: algext.cc:378
void n2pCoeffWrite(const coeffs cf, BOOLEAN details)
Definition: algext.cc:1653
#define n2pTest(a)
ABSTRACT: numbers as polys in the ring K[a] Assuming that we have a coeffs object cf,...
Definition: algext.cc:1514
void n2pNormalize(number &a, const coeffs cf)
Definition: algext.cc:1537
number naFarey(number p, number n, const coeffs cf)
Definition: algext.cc:1389
number naMapP0(number a, const coeffs src, const coeffs dst)
Definition: algext.cc:870
static coeffs nCoeff_bottom(const coeffs r, int &height)
Definition: algext.cc:258
number naMapZ0(number a, const coeffs src, const coeffs dst)
Definition: algext.cc:858
number naMapUP(number a, const coeffs src, const coeffs dst)
Definition: algext.cc:961
BOOLEAN naIsMOne(number a, const coeffs cf)
Definition: algext.cc:323
void n2pPower(number a, int exp, number *b, const coeffs cf)
Definition: algext.cc:1561
static poly p_GcdHelper(poly &p, poly &q, const ring r)
see p_Gcd; additional assumption: deg(p) >= deg(q); must destroy p and q (unless one of them is retur...
Definition: algext.cc:145
ring r
Definition: algext.h:37
struct for passing initialization parameters to naInitChar
Definition: algext.h:37
All the auxiliary stuff.
int BOOLEAN
Definition: auxiliary.h:87
#define TRUE
Definition: auxiliary.h:100
#define FALSE
Definition: auxiliary.h:96
CanonicalForm lc(const CanonicalForm &f)
CanonicalForm pp(const CanonicalForm &)
CanonicalForm pp ( const CanonicalForm & f )
Definition: cf_gcd.cc:248
int l
Definition: cfEzgcd.cc:93
int m
Definition: cfEzgcd.cc:121
int i
Definition: cfEzgcd.cc:125
const CanonicalForm const CanonicalForm const CanonicalForm const CanonicalForm & cand
Definition: cfModGcd.cc:69
Variable x
Definition: cfModGcd.cc:4023
int p
Definition: cfModGcd.cc:4019
g
Definition: cfModGcd.cc:4031
CanonicalForm cf
Definition: cfModGcd.cc:4024
CanonicalForm b
Definition: cfModGcd.cc:4044
STATIC_VAR int theDegree
Definition: cf_char.cc:21
FILE * f
Definition: checklibs.c:9
CanonicalForm convSingPFactoryP(poly p, const ring r)
Definition: clapconv.cc:85
poly convFactoryPSingP(const CanonicalForm &f, const ring r)
Definition: clapconv.cc:40
poly singclap_pdivide(poly f, poly g, const ring r)
Definition: clapsing.cc:558
BOOLEAN singclap_extgcd(poly f, poly g, poly &res, poly &pa, poly &pb, const ring r)
Definition: clapsing.cc:423
poly singclap_gcd_r(poly f, poly g, const ring r)
Definition: clapsing.cc:42
go into polynomials over an alg. extension recursively
factory's main class
Definition: canonicalform.h:83
CF_NO_INLINE bool isZero() const
Definition: cf_inline.cc:372
Templated accessor interface for accessing individual data (for instance, of an enumerator).
Definition: Enumerator.h:82
virtual reference Current()=0
Gets the current element in the collection (read and write).
virtual void Reset()=0
Sets the enumerator to its initial position: -1, which is before the first element in the collection.
virtual bool MoveNext()=0
Advances the enumerator to the next element of the collection. returns true if the enumerator was suc...
Templated enumerator interface for simple iteration over a generic collection of T's.
Definition: Enumerator.h:125
Coefficient rings, fields and other domains suitable for Singular polynomials.
static FORCE_INLINE number n_Mult(number a, number b, const coeffs r)
return the product of 'a' and 'b', i.e., a*b
Definition: coeffs.h:636
static FORCE_INLINE long n_Int(number &n, const coeffs r)
conversion of n to an int; 0 if not possible in Z/pZ: the representing int lying in (-p/2 ....
Definition: coeffs.h:547
static FORCE_INLINE number n_Copy(number n, const coeffs r)
return a copy of 'n'
Definition: coeffs.h:451
static FORCE_INLINE number n_NormalizeHelper(number a, number b, const coeffs r)
assume that r is a quotient field (otherwise, return 1) for arguments (a1/a2,b1/b2) return (lcm(a1,...
Definition: coeffs.h:717
static FORCE_INLINE char * nCoeffString(const coeffs cf)
TODO: make it a virtual method of coeffs, together with: Decompose & Compose, rParameter & rPar.
Definition: coeffs.h:981
static FORCE_INLINE void n_CoeffWrite(const coeffs r, BOOLEAN details=TRUE)
output the coeff description
Definition: coeffs.h:741
static FORCE_INLINE BOOLEAN nCoeff_is_Extension(const coeffs r)
Definition: coeffs.h:868
#define n_Test(a, r)
BOOLEAN n_Test(number a, const coeffs r)
Definition: coeffs.h:738
n_coeffType
Definition: coeffs.h:28
@ n_polyExt
used to represent polys as coeffcients
Definition: coeffs.h:35
@ n_algExt
used for all algebraic extensions, i.e., the top-most extension in an extension tower is algebraic
Definition: coeffs.h:36
static FORCE_INLINE number n_Invers(number a, const coeffs r)
return the multiplicative inverse of 'a'; raise an error if 'a' is not invertible
Definition: coeffs.h:564
static FORCE_INLINE BOOLEAN n_GreaterZero(number n, const coeffs r)
ordered fields: TRUE iff 'n' is positive; in Z/pZ: TRUE iff 0 < m <= roundedBelow(p/2),...
Definition: coeffs.h:494
static FORCE_INLINE BOOLEAN n_IsMOne(number n, const coeffs r)
TRUE iff 'n' represents the additive inverse of the one element, i.e. -1.
Definition: coeffs.h:472
static FORCE_INLINE nMapFunc n_SetMap(const coeffs src, const coeffs dst)
set the mapping function pointers for translating numbers from src to dst
Definition: coeffs.h:721
static FORCE_INLINE BOOLEAN nCoeff_is_Q_algext(const coeffs r)
is it an alg. ext. of Q?
Definition: coeffs.h:936
static FORCE_INLINE number n_Div(number a, number b, const coeffs r)
return the quotient of 'a' and 'b', i.e., a/b; raises an error if 'b' is not invertible in r exceptio...
Definition: coeffs.h:615
static FORCE_INLINE BOOLEAN nCoeff_is_Q(const coeffs r)
Definition: coeffs.h:828
static FORCE_INLINE BOOLEAN n_Greater(number a, number b, const coeffs r)
ordered fields: TRUE iff 'a' is larger than 'b'; in Z/pZ: TRUE iff la > lb, where la and lb are the l...
Definition: coeffs.h:511
static FORCE_INLINE char const ** n_ParameterNames(const coeffs r)
Returns a (const!) pointer to (const char*) names of parameters.
Definition: coeffs.h:800
static FORCE_INLINE BOOLEAN n_IsZero(number n, const coeffs r)
TRUE iff 'n' represents the zero element.
Definition: coeffs.h:464
static FORCE_INLINE void n_ClearDenominators(ICoeffsEnumerator &numberCollectionEnumerator, number &d, const coeffs r)
(inplace) Clears denominators on a collection of numbers number d is the LCM of all the coefficient d...
Definition: coeffs.h:957
static FORCE_INLINE BOOLEAN nCoeff_is_Q_or_BI(const coeffs r)
Definition: coeffs.h:851
static FORCE_INLINE n_coeffType getCoeffType(const coeffs r)
Returns the type of coeffs domain.
Definition: coeffs.h:421
static FORCE_INLINE void n_Delete(number *p, const coeffs r)
delete 'p'
Definition: coeffs.h:455
static FORCE_INLINE int n_NumberOfParameters(const coeffs r)
Returns the number of parameters.
Definition: coeffs.h:796
static FORCE_INLINE BOOLEAN nCoeff_is_Zp(const coeffs r)
Definition: coeffs.h:822
static FORCE_INLINE number n_Init(long i, const coeffs r)
a number representing i in the given coeff field/ring r
Definition: coeffs.h:538
static FORCE_INLINE void n_ClearContent(ICoeffsEnumerator &numberCollectionEnumerator, number &c, const coeffs r)
Computes the content and (inplace) divides it out on a collection of numbers number c is the content ...
Definition: coeffs.h:950
static FORCE_INLINE BOOLEAN nCoeff_is_algExt(const coeffs r)
TRUE iff r represents an algebraic extension field.
Definition: coeffs.h:932
@ n_rep_gap_rat
(number), see longrat.h
Definition: coeffs.h:111
@ n_rep_gap_gmp
(), see rinteger.h, new impl.
Definition: coeffs.h:112
@ n_rep_poly
(poly), see algext.h
Definition: coeffs.h:113
static FORCE_INLINE number n_SubringGcd(number a, number b, const coeffs r)
Definition: coeffs.h:688
number(* nMapFunc)(number a, const coeffs src, const coeffs dst)
maps "a", which lives in src, into dst
Definition: coeffs.h:73
static FORCE_INLINE void n_Normalize(number &n, const coeffs r)
inplace-normalization of n; produces some canonical representation of n;
Definition: coeffs.h:578
static FORCE_INLINE BOOLEAN n_IsOne(number n, const coeffs r)
TRUE iff 'n' represents the one element.
Definition: coeffs.h:468
static FORCE_INLINE BOOLEAN nCoeff_is_transExt(const coeffs r)
TRUE iff r represents a transcendental extension field.
Definition: coeffs.h:940
BOOLEAN fa(leftv res, leftv args)
Definition: cohomo.cc:4392
#define Print
Definition: emacs.cc:80
return result
Definition: facAbsBiFact.cc:76
const CanonicalForm int s
Definition: facAbsFact.cc:55
const CanonicalForm int const CFList const Variable & y
Definition: facAbsFact.cc:57
CanonicalForm factor
Definition: facAbsFact.cc:101
void WerrorS(const char *s)
Definition: feFopen.cc:24
#define STATIC_VAR
Definition: globaldefs.h:7
static BOOLEAN length(leftv result, leftv arg)
Definition: interval.cc:263
STATIC_VAR Poly * h
Definition: janet.cc:971
STATIC_VAR jList * Q
Definition: janet.cc:30
number nlModP(number q, const coeffs, const coeffs Zp)
Definition: longrat.cc:1435
#define assume(x)
Definition: mod2.h:390
int dReportError(const char *fmt,...)
Definition: dError.cc:43
#define p_SetCoeff0(p, n, r)
Definition: monomials.h:60
#define pIter(p)
Definition: monomials.h:37
#define pNext(p)
Definition: monomials.h:36
static number & pGetCoeff(poly p)
return an alias to the leading coefficient of p assumes that p != NULL NOTE: not copy
Definition: monomials.h:44
#define p_GetCoeff(p, r)
Definition: monomials.h:50
gmp_float exp(const gmp_float &a)
Definition: mpr_complex.cc:357
The main handler for Singular numbers which are suitable for Singular polynomials.
number ndGcd(number, number, const coeffs r)
Definition: numbers.cc:161
number ndCopyMap(number a, const coeffs aRing, const coeffs r)
Definition: numbers.cc:251
const char *const nDivBy0
Definition: numbers.h:88
#define omFreeSize(addr, size)
Definition: omAllocDecl.h:260
#define omAlloc(size)
Definition: omAllocDecl.h:210
#define omFree(addr)
Definition: omAllocDecl.h:261
#define NULL
Definition: omList.c:12
poly p_Farey(poly p, number N, const ring r)
Definition: p_polys.cc:50
poly p_PolyDiv(poly &p, const poly divisor, const BOOLEAN needResult, const ring r)
assumes that p and divisor are univariate polynomials in r, mentioning the same variable; assumes div...
Definition: p_polys.cc:1845
poly p_ISet(long i, const ring r)
returns the poly representing the integer i
Definition: p_polys.cc:1287
poly p_PermPoly(poly p, const int *perm, const ring oldRing, const ring dst, nMapFunc nMap, const int *par_perm, int OldPar, BOOLEAN use_mult)
Definition: p_polys.cc:4036
poly p_Power(poly p, int i, const ring r)
Definition: p_polys.cc:2172
void p_Normalize(poly p, const ring r)
Definition: p_polys.cc:3732
const char * p_Read(const char *st, poly &rc, const ring r)
Definition: p_polys.cc:1360
int p_Var(poly m, const ring r)
Definition: p_polys.cc:4562
poly p_One(const ring r)
Definition: p_polys.cc:1303
poly p_NSet(number n, const ring r)
returns the poly representing the number n, destroys n
Definition: p_polys.cc:1455
long p_Deg(poly a, const ring r)
Definition: p_polys.cc:577
poly p_ChineseRemainder(poly *xx, number *x, number *q, int rl, CFArray &inv_cache, const ring R)
Definition: p_polys.cc:83
BOOLEAN p_EqualPolys(poly p1, poly p2, const ring r)
Definition: p_polys.cc:4418
static poly p_Neg(poly p, const ring r)
Definition: p_polys.h:1046
static poly p_Add_q(poly p, poly q, const ring r)
Definition: p_polys.h:895
static poly p_Mult_q(poly p, poly q, const ring r)
Definition: p_polys.h:1053
static unsigned long p_SetExp(poly p, const unsigned long e, const unsigned long iBitmask, const int VarOffset)
set a single variable exponent @Note: VarOffset encodes the position in p->exp
Definition: p_polys.h:487
void p_String0Long(const poly p, ring lmRing, ring tailRing)
print p in a long way
Definition: polys0.cc:203
void p_String0Short(const poly p, ring lmRing, ring tailRing)
print p in a short way, if possible
Definition: polys0.cc:184
static void p_Setm(poly p, const ring r)
Definition: p_polys.h:232
static number p_SetCoeff(poly p, number n, ring r)
Definition: p_polys.h:411
static long p_GetExp(const poly p, const unsigned long iBitmask, const int VarOffset)
get a single variable exponent @Note: the integer VarOffset encodes:
Definition: p_polys.h:468
static BOOLEAN p_IsConstant(const poly p, const ring r)
Definition: p_polys.h:1931
static void p_Delete(poly *p, const ring r)
Definition: p_polys.h:860
void p_Write0(poly p, ring lmRing, ring tailRing)
Definition: polys0.cc:332
static poly pp_Mult_qq(poly p, poly q, const ring r)
Definition: p_polys.h:1090
static poly p_Init(const ring r, omBin bin)
Definition: p_polys.h:1259
static poly p_Copy(poly p, const ring r)
returns a copy of p
Definition: p_polys.h:811
static long p_Totaldegree(poly p, const ring r)
Definition: p_polys.h:1446
#define p_Test(p, r)
Definition: p_polys.h:162
#define __p_Mult_nn(p, n, r)
Definition: p_polys.h:930
poly prMapR(poly src, nMapFunc nMap, ring src_r, ring dest_r)
Definition: prCopy.cc:45
void StringAppendS(const char *st)
Definition: reporter.cc:107
void PrintS(const char *s)
Definition: reporter.cc:284
void PrintLn()
Definition: reporter.cc:310
void rWrite(ring r, BOOLEAN details)
Definition: ring.cc:226
BOOLEAN rSamePolyRep(ring r1, ring r2)
returns TRUE, if r1 and r2 represents the monomials in the same way FALSE, otherwise this is an analo...
Definition: ring.cc:1712
void rDelete(ring r)
unconditionally deletes fields in r
Definition: ring.cc:448
BOOLEAN rEqual(ring r1, ring r2, BOOLEAN qr)
returns TRUE, if r1 equals r2 FALSE, otherwise Equality is determined componentwise,...
Definition: ring.cc:1660
static char * rRingVar(short i, const ring r)
Definition: ring.h:572
static BOOLEAN rCanShortOut(const ring r)
Definition: ring.h:581
static short rVar(const ring r)
#define rVar(r) (r->N)
Definition: ring.h:587
#define IDELEMS(i)
Definition: simpleideals.h:23
#define R
Definition: sirandom.c:27
#define A
Definition: sirandom.c:24
poly gcd_over_Q(poly f, poly g, const ring r)
helper routine for calling singclap_gcd_r
Definition: transext.cc:275