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Macros | Functions
maps_ip.cc File Reference
#include "kernel/mod2.h"
#include "coeffs/numbers.h"
#include "coeffs/coeffs.h"
#include "polys/monomials/ring.h"
#include "polys/monomials/maps.h"
#include "polys/matpol.h"
#include "polys/prCopy.h"
#include "polys/ext_fields/transext.h"
#include "misc/options.h"
#include "kernel/GBEngine/kstd1.h"
#include "kernel/maps/gen_maps.h"
#include "maps_ip.h"
#include "ipid.h"
#include "lists.h"
#include "tok.h"
#include "ipshell.h"

Go to the source code of this file.

Macros

#define TRANSEXT_PRIVATES
 

Functions

BOOLEAN maApplyFetch (int what, map theMap, leftv res, leftv w, ring preimage_r, int *perm, int *par_perm, int P, nMapFunc nMap)
 
poly pSubstPar (poly p, int par, poly image)
 
ideal idSubstPar (ideal id, int n, poly e)
 
poly pSubstPoly (poly p, int var, poly image)
 
ideal idSubstPoly (ideal id, int n, poly e)
 

Macro Definition Documentation

◆ TRANSEXT_PRIVATES

#define TRANSEXT_PRIVATES

Definition at line 7 of file maps_ip.cc.

Function Documentation

◆ idSubstPar()

ideal idSubstPar ( ideal  id,
int  n,
poly  e 
)

Definition at line 372 of file maps_ip.cc.

373 {
374  int k=MATROWS((matrix)id)*MATCOLS((matrix)id);
375  ideal res=(ideal)mpNew(MATROWS((matrix)id),MATCOLS((matrix)id));
376 
377  res->rank = id->rank;
378  for(k--;k>=0;k--)
379  {
380  res->m[k]=pSubstPar(id->m[k],n,e);
381  }
382  return res;
383 }
int k
Definition: cfEzgcd.cc:99
poly * m
Definition: matpol.h:18
CanonicalForm res
Definition: facAbsFact.cc:60
poly pSubstPar(poly p, int par, poly image)
Definition: maps_ip.cc:252
matrix mpNew(int r, int c)
create a r x c zero-matrix
Definition: matpol.cc:37
#define MATROWS(i)
Definition: matpol.h:26
#define MATCOLS(i)
Definition: matpol.h:27

◆ idSubstPoly()

ideal idSubstPoly ( ideal  id,
int  n,
poly  e 
)

Definition at line 411 of file maps_ip.cc.

412 {
413 
414 #ifdef HAVE_PLURAL
415  if (rIsPluralRing(currRing))
416  {
417  int k=MATROWS((matrix)id)*MATCOLS((matrix)id);
418  ideal res=(ideal)mpNew(MATROWS((matrix)id),MATCOLS((matrix)id));
419  res->rank = id->rank;
420  for(k--;k>=0;k--)
421  {
422  res->m[k]=pSubst(pCopy(id->m[k]),n,e);
423  }
424  return res;
425  }
426 #endif
427 #ifdef HAVE_SHIFTBBA
428  if (rIsLPRing(currRing))
429  {
430  int k=MATROWS((matrix)id)*MATCOLS((matrix)id);
431  ideal res=(ideal)mpNew(MATROWS((matrix)id),MATCOLS((matrix)id));
432  res->rank = id->rank;
433  for(k--;k>=0;k--)
434  {
435  res->m[k]=pSubst(pCopy(id->m[k]),n,e);
436  }
437  return res;
438  }
439 #endif
440  return id_SubstPoly(id,n,e,currRing,currRing,ndCopyMap);
441 }
number ndCopyMap(number a, const coeffs src, const coeffs dst)
Definition: numbers.cc:254
ideal id_SubstPoly(ideal id, int var, poly image, const ring preimage_r, const ring image_r, const nMapFunc nMap)
Definition: subst_maps.cc:68
VAR ring currRing
Widely used global variable which specifies the current polynomial ring for Singular interpreter and ...
Definition: polys.cc:13
#define pSubst(p, n, e)
Definition: polys.h:366
#define pCopy(p)
return a copy of the poly
Definition: polys.h:185
static BOOLEAN rIsPluralRing(const ring r)
we must always have this test!
Definition: ring.h:400
static BOOLEAN rIsLPRing(const ring r)
Definition: ring.h:411

◆ maApplyFetch()

BOOLEAN maApplyFetch ( int  what,
map  theMap,
leftv  res,
leftv  w,
ring  preimage_r,
int *  perm,
int *  par_perm,
int  P,
nMapFunc  nMap 
)

Definition at line 45 of file maps_ip.cc.

47 {
48  BOOLEAN use_mult=FALSE;
49 #ifdef HAVE_PLURAL
50  if ((what==IMAP_CMD)
52  && rIsPluralRing(preimage_r))
53  {
54  assume(perm!=NULL);
55  int i=1;
56  while((i<currRing->N)&&(perm[i]==0)) i++;
57  if (i<currRing->N)
58  {
59  int prev_nonnull=i;
60  i++;
61  for(;i<=currRing->N;i++)
62  {
63  if (perm[prev_nonnull] > perm[i])
64  {
65  if (TEST_V_ALLWARN)
66  {
67  Warn("imap not usable for permuting variables, use map (%s <-> %s)",currRing->names[prev_nonnull-1],currRing->names[i-1]);
68  }
69  use_mult=TRUE;
70  break;
71  }
72  else
73  prev_nonnull=i;
74  }
75  }
76  }
77 #endif
78  int i;
79  int N = preimage_r->N;
80 #if 0
81  Print("N=%d what=%s ",N,Tok2Cmdname(what));
82  if (perm!=NULL) for(i=1;i<=N;i++) Print("%d -> %d ",i,perm[i]);
83  PrintS("\n");
84  Print("P=%d ",P);
85  if (par_perm!=NULL) for(i=0;i<P;i++) Print("%d -> %d ",i,par_perm[i]);
86  PrintS("\n");
87 #endif
88 
89  void *data=w->Data();
90  res->rtyp = w->rtyp;
91  switch (w->rtyp)
92  {
93  case NUMBER_CMD:
94  if (P!=0)
95  {
96 // poly n_PermNumber(const number z, const int *par_perm, const int OldPar, const ring src, const ring dst);
97  res->data= (void *) n_PermNumber((number)data, par_perm, P, preimage_r, currRing);
98  res->rtyp=POLY_CMD;
99  if (nCoeff_is_algExt(currRing->cf))
100  res->data=(void *)p_MinPolyNormalize((poly)res->data, currRing);
101  pTest((poly) res->data);
102  }
103  else
104  {
105  assume( nMap != NULL );
106  number a = nMap((number)data, preimage_r->cf, currRing->cf);
107  if (nCoeff_is_Extension(currRing->cf))
108  {
109  n_Normalize(a, currRing->cf);
110 /*
111  number a = (number)res->data;
112  number one = nInit(1);
113  number product = nMult(a, one );
114  nDelete(&one);
115  nDelete(&a);
116  res->data=(void *)product;
117  */
118  }
119  #ifdef LDEBUG
120  n_Test(a, currRing->cf);
121  #endif
122  res->data=(void *)a;
123 
124  }
125  break;
126  case BUCKET_CMD:
127  if ((what==FETCH_CMD)&& (preimage_r->cf==currRing->cf))
128  res->data=(void *)prCopyR(sBucketPeek((sBucket_pt)data), preimage_r, currRing);
129  else
130  if ( (what==IMAP_CMD) || /*(*/ (what==FETCH_CMD) /*)*/) /* && (nMap!=nCopy)*/
131  res->data=(void *)p_PermPoly(sBucketPeek((sBucket_pt)data),perm,preimage_r,currRing, nMap,par_perm,P,use_mult);
132  else /*if (what==MAP_CMD)*/
133  {
134  res->data=(void*)maMapPoly(sBucketPeek((sBucket_pt)data),preimage_r,(ideal)theMap,currRing,nMap);
135  }
136  if (nCoeff_is_Extension(currRing->cf))
137  res->data=(void *)p_MinPolyNormalize(sBucketPeek((sBucket_pt)data), currRing);
138  break;
139  case POLY_CMD:
140  case VECTOR_CMD:
141  if ((what==FETCH_CMD)&& (preimage_r->cf==currRing->cf))
142  res->data=(void *)prCopyR( (poly)data, preimage_r, currRing);
143  else
144  if ( (what==IMAP_CMD) || /*(*/ (what==FETCH_CMD) /*)*/) /* && (nMap!=nCopy)*/
145  res->data=(void *)p_PermPoly((poly)data,perm,preimage_r,currRing, nMap,par_perm,P,use_mult);
146  else /*if (what==MAP_CMD)*/
147  {
148  p_Test((poly)data,preimage_r);
149  res->data=(void*)maMapPoly((poly)data,preimage_r,(ideal)theMap,currRing,nMap);
150  }
151  if (nCoeff_is_Extension(currRing->cf))
152  res->data=(void *)p_MinPolyNormalize((poly)res->data, currRing);
153  pTest((poly)res->data);
154  break;
155  case MODUL_CMD:
156  case MATRIX_CMD:
157  case IDEAL_CMD:
158  case MAP_CMD:
159  {
160  int C=((matrix)data)->cols();
161  int R;
162  if (w->rtyp==MAP_CMD) R=1;
163  else R=((matrix)data)->rows();
164  matrix m=mpNew(R,C);
165  char *tmpR=NULL;
166  if(w->rtyp==MAP_CMD)
167  {
168  tmpR=((map)data)->preimage;
169  ((matrix)data)->rank=((matrix)data)->rows();
170  }
171  if ((what==FETCH_CMD)&& (preimage_r->cf == currRing->cf))
172  {
173  for (i=R*C-1;i>=0;i--)
174  {
175  m->m[i]=prCopyR(((ideal)data)->m[i], preimage_r, currRing);
176  pTest(m->m[i]);
177  }
178  }
179  else if ((what==IMAP_CMD) || (what==FETCH_CMD))
180  {
181  for (i=R*C-1;i>=0;i--)
182  {
183  m->m[i]=p_PermPoly(((ideal)data)->m[i],perm,preimage_r,currRing,
184  nMap,par_perm,P,use_mult);
185  pTest(m->m[i]);
186  }
187  }
188  else /* (what==MAP_CMD) */
189  {
190  assume(what==MAP_CMD);
191  matrix s=mpNew(N,maMaxDeg_Ma((ideal)data,preimage_r));
192  for (i=R*C-1;i>=0;i--)
193  {
194  m->m[i]=maEval(theMap, ((ideal)data)->m[i], preimage_r, nMap, (ideal)s, currRing);
195  pTest(m->m[i]);
196  }
197  idDelete((ideal *)&s);
198  }
199  if(w->rtyp==MAP_CMD)
200  {
201  ((map)data)->preimage=tmpR;
202  ((map)m)->preimage=omStrDup(tmpR);
203  }
204  else
205  {
206  m->rank=((matrix)data)->rank;
207  }
208  res->data=(char *)m;
209  idTest((ideal) m);
210  break;
211  }
212 
213  case LIST_CMD:
214  {
215  lists l=(lists)data;
217  ml->Init(l->nr+1);
218  for(i=0;i<=l->nr;i++)
219  {
220  if (((l->m[i].rtyp>BEGIN_RING)&&(l->m[i].rtyp<END_RING))
221  ||(l->m[i].rtyp==LIST_CMD))
222  {
223  if (maApplyFetch(what,theMap,&ml->m[i],&l->m[i],
224  preimage_r,perm,par_perm,P,nMap))
225  {
226  ml->Clean();
228  res->rtyp=0;
229  return TRUE;
230  }
231  }
232  else
233  {
234  ml->m[i].Copy(&l->m[i]);
235  }
236  }
237  res->data=(char *)ml;
238  break;
239  }
240  default:
241  {
242  return TRUE;
243  }
244  }
245  return FALSE;
246 }
int BOOLEAN
Definition: auxiliary.h:87
#define TRUE
Definition: auxiliary.h:100
#define FALSE
Definition: auxiliary.h:96
void * ADDRESS
Definition: auxiliary.h:119
const CanonicalForm CFMap CFMap & N
Definition: cfEzgcd.cc:56
int l
Definition: cfEzgcd.cc:100
int m
Definition: cfEzgcd.cc:128
int i
Definition: cfEzgcd.cc:132
CanonicalForm map(const CanonicalForm &primElem, const Variable &alpha, const CanonicalForm &F, const Variable &beta)
map from to such that is mapped onto
Definition: cf_map_ext.cc:504
void Copy(leftv e)
Definition: subexpr.cc:685
Definition: lists.h:24
sleftv * m
Definition: lists.h:46
void Clean(ring r=currRing)
Definition: lists.h:26
INLINE_THIS void Init(int l=0)
static FORCE_INLINE BOOLEAN nCoeff_is_Extension(const coeffs r)
Definition: coeffs.h:870
#define n_Test(a, r)
BOOLEAN n_Test(number a, const coeffs r)
Definition: coeffs.h:736
static FORCE_INLINE BOOLEAN nCoeff_is_algExt(const coeffs r)
TRUE iff r represents an algebraic extension field.
Definition: coeffs.h:934
static FORCE_INLINE void n_Normalize(number &n, const coeffs r)
inplace-normalization of n; produces some canonical representation of n;
Definition: coeffs.h:579
#define Print
Definition: emacs.cc:80
#define Warn
Definition: emacs.cc:77
const CanonicalForm int s
Definition: facAbsFact.cc:51
const CanonicalForm & w
Definition: facAbsFact.cc:51
poly maMapPoly(const poly map_p, const ring map_r, const ideal image_id, const ring image_r, const nMapFunc nMap)
polynomial map for poly (vector) map_p: the poly (vector) to map map_r: the base ring for map_p image...
Definition: gen_maps.cc:159
const char * Tok2Cmdname(int tok)
Definition: gentable.cc:140
@ END_RING
Definition: grammar.cc:310
@ IDEAL_CMD
Definition: grammar.cc:284
@ MATRIX_CMD
Definition: grammar.cc:286
@ BUCKET_CMD
Definition: grammar.cc:283
@ IMAP_CMD
Definition: grammar.cc:298
@ MAP_CMD
Definition: grammar.cc:285
@ BEGIN_RING
Definition: grammar.cc:282
@ MODUL_CMD
Definition: grammar.cc:287
@ VECTOR_CMD
Definition: grammar.cc:292
@ NUMBER_CMD
Definition: grammar.cc:288
@ POLY_CMD
Definition: grammar.cc:289
@ FETCH_CMD
Definition: grammar.cc:295
#define idDelete(H)
delete an ideal
Definition: ideals.h:29
#define idTest(id)
Definition: ideals.h:47
VAR omBin slists_bin
Definition: lists.cc:23
poly maEval(map theMap, poly p, ring preimage_r, nMapFunc nMap, ideal s, const ring dst_r)
Definition: maps.cc:115
int maMaxDeg_Ma(ideal a, ring preimage_r)
Definition: maps.cc:329
poly p_MinPolyNormalize(poly p, const ring r)
Definition: maps.cc:399
BOOLEAN maApplyFetch(int what, map theMap, leftv res, leftv w, ring preimage_r, int *perm, int *par_perm, int P, nMapFunc nMap)
Definition: maps_ip.cc:45
ip_smatrix * matrix
Definition: matpol.h:43
#define assume(x)
Definition: mod2.h:387
slists * lists
Definition: mpr_numeric.h:146
#define omStrDup(s)
Definition: omAllocDecl.h:263
#define omAllocBin(bin)
Definition: omAllocDecl.h:205
#define omFreeBin(addr, bin)
Definition: omAllocDecl.h:259
#define NULL
Definition: omList.c:12
#define TEST_V_ALLWARN
Definition: options.h:141
poly n_PermNumber(const number z, const int *par_perm, const int, const ring src, const ring dst)
Definition: p_polys.cc:4053
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:4156
#define p_Test(p, r)
Definition: p_polys.h:162
#define pTest(p)
Definition: polys.h:415
poly prCopyR(poly p, ring src_r, ring dest_r)
Definition: prCopy.cc:34
void PrintS(const char *s)
Definition: reporter.cc:284
poly sBucketPeek(sBucket_pt b)
Definition: sbuckets.cc:455
#define R
Definition: sirandom.c:27
@ LIST_CMD
Definition: tok.h:118

◆ pSubstPar()

poly pSubstPar ( poly  p,
int  par,
poly  image 
)

Definition at line 252 of file maps_ip.cc.

253 {
254  const ring R = currRing->cf->extRing;
255  ideal theMapI = idInit(rPar(currRing),1);
256  nMapFunc nMap = n_SetMap(R->cf, currRing->cf);
257  int i;
258  for(i = rPar(currRing);i>0;i--)
259  {
260  if (i != par)
261  theMapI->m[i-1]= p_NSet(n_Param(i, currRing), currRing);
262  else
263  theMapI->m[i-1] = p_Copy(image, currRing);
264  p_Test(theMapI->m[i-1],currRing);
265  }
266  //iiWriteMatrix((matrix)theMapI,"map:",1,currRing,0);
267 
268  map theMap=(map)theMapI;
269  theMap->preimage=NULL;
270 
272  sleftv tmpW;
273  poly res=NULL;
274 
276  if (currRing->cf->rep==n_rep_rat_fct )
277  {
278  while (p!=NULL)
279  {
280  memset(v,0,sizeof(sleftv));
281 
282  number d = n_GetDenom(pGetCoeff(p), currRing->cf);
283  p_Test((poly)NUM((fraction)d), R);
284 
285  if ( n_IsOne (d, currRing->cf) )
286  {
287  n_Delete(&d, currRing->cf); d = NULL;
288  }
289  else if (!p_IsConstant((poly)NUM((fraction)d), R))
290  {
291  WarnS("ignoring denominators of coefficients...");
292  n_Delete(&d, currRing->cf); d = NULL;
293  }
294 
295  number num = n_GetNumerator(pGetCoeff(p), currRing->cf);
296  memset(&tmpW,0,sizeof(sleftv));
297  tmpW.rtyp = POLY_CMD;
298  p_Test((poly)NUM((fraction)num), R);
299 
300  tmpW.data = NUM ((fraction)num); // a copy of this poly will be used
301 
302  p_Normalize(NUM((fraction)num),R);
303  if (maApplyFetch(MAP_CMD,theMap,v,&tmpW,R,NULL,NULL,0,nMap))
304  {
305  WerrorS("map failed");
306  v->data=NULL;
307  }
308  n_Delete(&num, currRing->cf);
309  //TODO check for memory leaks
310  poly pp = pHead(p);
311  //PrintS("map:");pWrite(pp);
312  if( d != NULL )
313  {
314  pSetCoeff(pp, n_Invers(d, currRing->cf));
315  n_Delete(&d, currRing->cf); // d = NULL;
316  }
317  else
318  pSetCoeff(pp, nInit(1));
319 
320  //PrintS("->");pWrite((poly)(v->data));
321  poly ppp = pMult((poly)(v->data),pp);
322  //PrintS("->");pWrite(ppp);
323  res=pAdd(res,ppp);
324  pIter(p);
325  }
326  }
327  else if (currRing->cf->rep==n_rep_poly )
328  {
329  while (p!=NULL)
330  {
331  memset(v,0,sizeof(sleftv));
332 
333  number num = n_GetNumerator(pGetCoeff(p), currRing->cf);
334  memset(&tmpW,0,sizeof(sleftv));
335  tmpW.rtyp = POLY_CMD;
336  p_Test((poly)num, R);
337 
338 
339  p_Normalize((poly)num,R);
340  if (num==NULL) num=(number)R->qideal->m[0];
341  tmpW.data = num; // a copy of this poly will be used
342  if (maApplyFetch(MAP_CMD,theMap,v,&tmpW,R,NULL,NULL,0,nMap))
343  {
344  WerrorS("map failed");
345  v->data=NULL;
346  }
347  if (num!=(number)R->qideal->m[0]) n_Delete(&num, currRing->cf);
348  //TODO check for memory leaks
349  poly pp = pHead(p);
350  //PrintS("map:");pWrite(pp);
351  pSetCoeff(pp,n_Init(1,currRing->cf));
352  //PrintS("cf->");pWrite((poly)(v->data));
353  poly ppp = pMult((poly)(v->data),pp);
354  //PrintS("->");pWrite(ppp);
355  res=pAdd(res,ppp);
356  pIter(p);
357  }
358  }
359  else
360  {
361  WerrorS("cannot apply subst for these coeffcients");
362  }
363  idDelete((ideal *)(&theMap));
365  return res;
366 }
CanonicalForm pp(const CanonicalForm &)
CanonicalForm pp ( const CanonicalForm & f )
Definition: cf_gcd.cc:676
CanonicalForm num(const CanonicalForm &f)
int p
Definition: cfModGcd.cc:4080
Class used for (list of) interpreter objects.
Definition: subexpr.h:83
int rtyp
Definition: subexpr.h:91
void * data
Definition: subexpr.h:88
static FORCE_INLINE number n_Param(const int iParameter, const coeffs r)
return the (iParameter^th) parameter as a NEW number NOTE: parameter numbering: 1....
Definition: coeffs.h:807
static FORCE_INLINE number n_GetDenom(number &n, const coeffs r)
return the denominator of n (if elements of r are by nature not fractional, result is 1)
Definition: coeffs.h:604
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:565
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:723
static FORCE_INLINE void n_Delete(number *p, const coeffs r)
delete 'p'
Definition: coeffs.h:456
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:539
static FORCE_INLINE number n_GetNumerator(number &n, const coeffs r)
return the numerator of n (if elements of r are by nature not fractional, result is n)
Definition: coeffs.h:609
@ n_rep_poly
(poly), see algext.h
Definition: coeffs.h:114
@ n_rep_rat_fct
(fraction), see transext.h
Definition: coeffs.h:115
number(* nMapFunc)(number a, const coeffs src, const coeffs dst)
maps "a", which lives in src, into dst
Definition: coeffs.h:74
static FORCE_INLINE BOOLEAN n_IsOne(number n, const coeffs r)
TRUE iff 'n' represents the one element.
Definition: coeffs.h:469
#define WarnS
Definition: emacs.cc:78
const Variable & v
< [in] a sqrfree bivariate poly
Definition: facBivar.h:39
void WerrorS(const char *s)
Definition: feFopen.cc:24
EXTERN_VAR omBin sleftv_bin
Definition: ipid.h:145
if(yy_init)
Definition: libparse.cc:1420
#define pIter(p)
Definition: monomials.h:37
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 nInit(i)
Definition: numbers.h:24
void p_Normalize(poly p, const ring r)
Definition: p_polys.cc:3842
poly p_NSet(number n, const ring r)
returns the poly representing the number n, destroys n
Definition: p_polys.cc:1460
static BOOLEAN p_IsConstant(const poly p, const ring r)
Definition: p_polys.h:1971
static poly p_Copy(poly p, const ring r)
returns a copy of p
Definition: p_polys.h:812
#define pAdd(p, q)
Definition: polys.h:203
#define pHead(p)
returns newly allocated copy of Lm(p), coef is copied, next=NULL, p might be NULL
Definition: polys.h:67
#define pSetCoeff(p, n)
deletes old coeff before setting the new one
Definition: polys.h:31
#define pMult(p, q)
Definition: polys.h:207
@ NUM
Definition: readcf.cc:170
static int rPar(const ring r)
(r->cf->P)
Definition: ring.h:604
ideal idInit(int idsize, int rank)
initialise an ideal / module
Definition: simpleideals.cc:35
sleftv * leftv
Definition: structs.h:62

◆ pSubstPoly()

poly pSubstPoly ( poly  p,
int  var,
poly  image 
)

Definition at line 389 of file maps_ip.cc.

390 {
391  if (p==NULL) return NULL;
392 #ifdef HAVE_PLURAL
393  if (rIsPluralRing(currRing))
394  {
395  return pSubst(pCopy(p),var,image);
396  }
397 #endif
398 #ifdef HAVE_SHIFTBBA
399  if (rIsLPRing(currRing))
400  {
401  return pSubst(pCopy(p),var,image);
402  }
403 #endif
404  return p_SubstPoly(p,var,image,currRing,currRing,ndCopyMap);
405 }
poly p_SubstPoly(poly p, int var, poly image, const ring preimage_r, const ring image_r, const nMapFunc nMap, matrix cache=NULL)
Definition: subst_maps.cc:39