source: git/Singular/LIB/realrad.lib @ bee06d

jengelh-datetimespielwiese
Last change on this file since bee06d was bee06d, checked in by Hans Schoenemann <hannes@…>, 12 years ago
fix tr.280 git-svn-id: file:///usr/local/Singular/svn/trunk@13597 2c84dea3-7e68-4137-9b89-c4e89433aadc
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1///////////////////////////////////////////////////////////////////////////////
2version="$Id$";
3category="real algebra";
4info="
5LIBRARY:  realrad.lib   Computation of real radicals
6AUTHOR :  Silke Spang
7
8OVERVIEW:
9   Algorithms about the computation of the real
10   radical of an arbitary ideal over the rational numbers
11   and transcendetal extensions thereof
12
13PROCEDURES:
14 realpoly(f);     Computes the real part of the univariate polynomial f
15 realzero(j);     Computes the real radical of the zerodimensional ideal j
16 realrad(j);      Computes the real radical of an arbitary ideal over
17                  transcendental extension of the rational numbers
18";
19
20LIB "inout.lib";
21LIB "poly.lib";
22LIB "matrix.lib";
23LIB "general.lib";
24LIB "rootsur.lib";
25LIB "algebra.lib";
26LIB "standard.lib";
27LIB "primdec.lib";
28LIB "elim.lib";
29
30///////////////////////////////////////////////////////////////////////////////
31
32///////////////////////////////////////////////////////////////////////////////
33///////////////////////////////////////////////////////////////////////////////
34//// the main procedure //////////////////////////////////////////////////////
35//////////////////////////////////////////////////////////////////////////////
36proc realrad(ideal id)
37"USAGE: realrad(id), id an ideal of arbitary dimension
38RETURN: the real radical of id
39EXAMPE: example realrad; shows an example"
40{
41
42  def r=basering;
43  int n=nvars(basering);
44  // for faster Groebner basis and dimension compuations
45  string neuring ="ring schnell=("+charstr(r)+"),("+varstr(r)+"),dp;";
46  execute(neuring);
47  def ri=basering;
48
49  list reddim;//reduct dimension to 0
50  list lpar,lvar,sub;//for the ringchange
51  string pari,vari;
52  int i,siz,l,j;
53  string less="list lessvar="+varstr(r)+";";
54  execute(less);
55  ideal id=imap(r,id);
56  l=size(id);
57  for (i=1;i<=l;i++)
58  {
59    id[i]=simplify_gen(id[i]);
60  }
61  id=groebner(id);
62  if (dim(id)<=0)
63  {
64    id=realzero(id);
65    setring r;
66    id=imap(ri,id);
67    return(id);
68  }
69  //sub are the subsets of {x_1,...,x_n}
70  sub=subsets(n);
71  siz=size(sub)-1;//we dont want to localize on all variables
72
73  //for the empty set
74  reddim[1]=zeroreduct(id);
75  reddim[1]=realzero(reddim[1]);
76  for (i=1;i<=siz;i++)
77  {
78
79    lvar=lessvar;
80    lpar=list();
81    l=size(sub[i]);
82    for (j=1;j<=l;j++)
83    {
84      lpar=lpar+list(lvar[sub[i][j]-j+1]);
85      lvar=delete(lvar,sub[i][j]-j+1);
86    }
87    for(j=1;j<=l;j++)//there are l entries in lpar
88    {
89      pari=pari+","+string(lpar[j]);
90    }
91    l=n-l;//there are the remaining n-l entries in lvar
92    for(j=1;j<=l;j++)//there are l entries in lpar
93    {
94      vari=vari+","+string(lvar[j]);
95    }
96    vari=vari[2..size(vari)];
97    neuring="ring neu=("+charstr(r)+pari+"),("+vari+"),dp;";
98    execute(neuring);
99    ideal id=imap(r,id);
100    ideal buffer=zeroreduct(id);
101    buffer=realzero(buffer);
102    setring ri;
103    reddim[i+1]=imap(neu,buffer);
104    kill neu;
105    //compute the intersection of buffer with r
106    reddim[i+1]=contnonloc(reddim[i+1],pari,vari);
107    vari="";
108    pari="";
109  }
110  id=intersect(reddim[1..(siz+1)]);
111  id=timeStd(id,301);//simplify the output
112  setring r;
113  id=imap(ri,id);
114  return(id);
115
116}
117example
118{ "EXAMPLE:"; echo = 2;
119  ring r1=0,(x,y,z),lp;
120  //dimension 0
121  ideal i0=(x2+1)*(x3-2),(y3-2)*(y2+y+1),z3+2;
122  //dimension 1
123  ideal i1=(y3+3y2+y+1)*(y2+4y+4)*(x2+1),(x2+y)*(x2-y2)*(x2+2xy+y2)*(y2+y+1);
124  ideal i=intersect(i0,i1);
125  realrad(i);
126}
127
128
129/*static*/ proc zeroreduct(ideal i)
130"USAGE:zeroreduct(i), i an arbitary ideal
131RETURN: an ideal j of dimension <=0 s.th. i is contained in
132        j and j is contained in i_{Iso} which is the zariski closure
133        of all real isolated points of i
134"
135{
136 list equi;
137 int d,n,di;
138 n=nvars(basering);
139 def r=basering;
140
141 //chance ring to get faster groebner bases computation for dimensions
142
143 string rneu="ring neu=("+charstr(r)+"),("+varstr(r)+"),dp;";
144 execute(rneu);
145 ideal i=imap(r,i);
146
147 i=groebner(i);
148 while (dim(i)> 0)
149 {
150   equi=equidim(i);
151   d=size(equi);
152   equi[d]=radical(equi[d]);
153   di=dim(std(equi[d]));
154   equi[d]=equi[d],minor(jacob(equi[d]),n-di);
155   equi[d]=radical(equi[d]);
156   i=intersect(equi[1..d]);
157   i=groebner(i);
158 }
159
160 setring r;
161 i=imap(neu,i);
162 i=timeStd(i,301);
163 return(i);
164}
165//////////////////////////////////////////////////////////////////////////////
166///////the zero-dimensional case /////////////////////////////////////////////
167//////////////////////////////////////////////////////////////////////////////
168proc realzero(ideal j)
169"USAGE:    realzero(j); a zero-dimensional ideal j
170RETURN:    j: a zero dimensional ideal, which is the real radical
171              of i, if dim(i)=0
172           0: otherwise
173           this acts via
174           primary decomposition (i=1)
175           listdecomp (i=2) or facstd (i=3)
176EXAMPLE:   example realzero; shows an example"
177
178
179{
180 list prim,prepared,nonshape,realu;
181 int r;//counter
182 int l;//number of first polynomial with degree >1 or even
183 l=size(j);
184 for (r=1;r<=l;r++)
185 {
186   j[r]=simplify_gen(j[r]);
187   if (j[r]==1)
188   {
189     return(ideal(1));
190   }
191 }
192 option(redSB);
193 //j=groebner(j);
194 //special case
195 //if (j==1)
196 //{
197 //  return(j);
198 //}
199 if (nvars(basering)==1)
200 {
201   j=groebner(j);
202   j=realpoly(j[1]);
203   return(j);
204 }
205
206
207 //if (dim(j)>0) {return(0);}
208
209 def r_alt=basering;
210 //store the ring
211 //for a ring chance to the ordering lp;
212 execute("ring r_neu =("+charstr(basering)+"),("+varstr(basering)+"),lp;");
213 setring r_neu;
214 ideal boeser,max;
215 prepared[1]=ideal(1);
216 ideal j=imap(r_alt,j);
217 //ideal j=fglm(r_alt,j);
218 prim=primdecGTZ(j);
219 for (r=1;r<=size(prim);r++)
220 {
221   max=prim[r][2];
222   max=groebner(max);
223   realu=prepare_max(max);
224   max=realu[1];
225   if (max!=1)
226   {
227     if (realu[2]==1)
228     {
229       prepared=insert(prepared,max);
230     }
231     else
232     {
233       nonshape=insert(nonshape,max);
234     }
235   }
236 }
237 j=intersect(prepared[1..size(prepared)]);
238
239 //use a variable change into general position to obtain
240 //the shape via radzero
241 if (size(nonshape)>0)
242 {
243   boeser=GeneralPos(nonshape);
244   j=intersect(j,boeser);
245 }
246 j=timeStd(j,301);
247 setring r_alt;
248 j=fetch(r_neu,j);
249 return(j);
250}
251example
252{ "EXAMPLE:"; echo = 2;
253  //in non parametric fields
254  ring r=0,(x,y),dp;
255  ideal i=(y3+3y2+y+1)*(y2+4y+4)*(x2+1),(x2+y)*(x2-y2)*(x2+2xy+y2)*(y2+y+1);
256  realzero(i);
257  ideal j=(y3+3y2+y+1)*(y2-2y+1),(x2+y)*(x2-y2);
258  realzero(j);
259
260  //to get every path
261  ring r1=(0,t),(x,y),lp;
262  ideal m1=x2+1-t,y3+t2;
263  ideal m2=x2+t2+1,y2+t;
264  ideal m3=x2+1-t,y2-t;
265  ideal m4=x^2+1+t,y2-t;
266  ideal i=intersect(m1,m2,m3,m4);
267  realzero(i);
268
269}
270
271static proc GeneralPos(list buffer)
272"USAGE:    GeneralPos(buffer);
273           buffer a list of maximal ideals which failed the prepare_max-test
274RETURN:    j: the intersection of their realradicals
275EXAMPLE:   example radzero; shows no example"
276{
277 def r=basering;
278 int n,ll;
279 //for the mapping in general position
280 map phi,psi;
281 ideal j;
282 ideal jmap=randomLast(20);
283 string ri;
284 intvec @hilb;
285 ideal trans,transprep;// the transformation ideals
286 int nva=nvars(r);
287 int zz,k,l;//counter
288 poly randp;
289 for (zz=1;zz<nva;zz++)
290 {
291   if (npars(basering)>0)
292   {
293     randp=randp+(random(0,5)*par(1)+random(0,5)*par(1)^2+random(0,5))*var(zz);
294   }
295   else
296   {
297     randp=randp+random(0,5)*var(zz);
298   }
299 }
300 randp=randp+var(nva);
301
302 //now they are all irreducible in the non univariate case and
303 //real in the univariate case
304
305 int m=size(buffer);
306 for (l=1;l<=m;l++)
307 {
308   //searching first non univariate polynomial with an even degree
309   //for odd degree we could use the fundamental theorem of algebra and
310   //get real zeros
311
312   //this will act via a coordinate chance into general position
313   //denote that this random chance doesn't work allways
314   //the ideas for the transformation into general position are
315   //used from the primdec.lib
316   transprep=buffer[l];
317   if (voice>=10)
318   {
319     jmap[size(jmap)]=randp;
320   }
321
322
323   for (k=2;k<=n;k++)
324   {
325     if (ord(buffer[l][k])==1)
326     {
327       for (zz=1;zz<=nva;zz++)
328       {
329         if (lead(buffer[l][k])/var(zz)!=0)
330         {
331           transprep[k]=var(zz);
332         }
333       }
334       jmap[nva]=subst(jmap[nva],lead(buffer[l][k]),0);
335     }
336   }
337   phi =r,jmap;
338   for (k=1;k<=nva;k++)
339   {
340     jmap[k]=-(jmap[k]-2*var(k));
341   }
342   psi =r,jmap;
343
344   //coordinate chance
345   trans=phi(transprep);
346
347   //acting with the chanced ideal
348
349   trans=groebner(trans);
350   trans[1]=realpoly(trans[1]);
351
352   //special case
353   if (trans==1)
354   {
355     buffer[l]=trans;
356   }
357   else
358   {
359     ri="ring rhelp=("+charstr(r)+ "),(" +varstr(r)+ ",@t),dp;";
360     execute(ri);
361     ideal trans=homog(imap(r,trans),@t);
362
363     ideal trans1=std(trans);
364     @hilb=hilb(trans1,1);
365     ri= "ring rhelp1=("
366                   +charstr(r)+ "),(" +varstr(rhelp)+ "),lp;";
367     execute(ri);
368     ideal trans=homog(imap(r,trans),@t);
369     kill rhelp;
370     trans=std(trans,@hilb);
371     trans=subst(trans,@t,1);//dehomogenising
372     setring r;
373     trans=imap(rhelp1,trans);
374     kill rhelp1;
375     trans=std(trans);
376     attrib(trans,"isSB",1);
377
378     trans=realzero(trans);
379
380     //going back
381     buffer[l]=psi(trans);
382     buffer[l]=timeStd(buffer[l],301);//timelimit for std computation
383   }
384 }
385 //option(returnSB);
386 j=intersect(buffer[1..m]);
387 return(j);
388
389}
390
391/*proc minAssReal(ideal i, int erg)
392{
393  int l,m,d,e,r,fac;
394  ideal buffer,factor;
395  list minreal;
396  l=size(i);
397  for (r=1;r<=l;r++)
398  {
399    i[r]=simplify_gen(i[r]);
400
401  }
402
403  list pr=primdecGTZ(i);
404  m=size(pr);
405  for (l=1;l<=m;l++)
406  {
407     d=dim(std(pr[l][2]));
408     buffer=realrad(pr[l][2]);
409     buffer=std(buffer);
410     e=dim(buffer);
411     if (d==e)
412     {
413         minreal=minreal+list(pr[l]);
414     }
415  }
416  if (erg==0)
417  {
418   return(minreal);
419  }
420  else
421  {
422     pr=list();
423     m=size(minreal);
424     for (l=1;l<=m;l++)
425     {
426       pr=insert(pr,minreal[l][2]);
427     }
428     i=intersect(pr[1..m]);
429     i=timeStd(i,301);
430     list realmin=minreal+list(i);
431     return(realmin);
432  }
433}*/
434//////////////////////////////////////////////////////////////////////////////
435///////the univariate case ///////////////////////////////////////////////////
436//////////////////////////////////////////////////////////////////////////////
437proc realpoly(poly f)
438"USAGE:    realpoly(f); a univariate polynomial f;
439RETURN:    poly f, where f is the real part of the input f
440EXAMPLE:   example realpoly; shows an example"
441{
442 def r=basering;
443 int tester;
444 if (size(parstr(r))!=0)
445 {
446   string changering="ring rneu=0,("+parstr(r)+","+varstr(r)+"),lp";
447   execute(changering);
448   poly f=imap(r,f);
449   tester=1;
450 }
451 f=simplify(f,1);//wlog f is monic
452 if (f==1)
453 {
454   setring r;
455   return(f);
456 }
457 ideal j=factorize(f,1);//for getting the squarefree factorization
458 poly erg=1;
459 for (int i=1;i<=size(j);i=i+1)
460 {
461  if (is_real(j[i])==1) {erg=erg*j[i];}
462  //we only need real primes
463 }
464 if (tester==1)
465 {
466   setring(r);
467   poly erg=imap(rneu,erg);
468 }
469 return(erg);
470}
471example
472{ "EXAMPLE:"; echo = 2;
473   ring r1 = 0,x,dp;
474   poly f=x5+16x2+x+1;
475   realpoly(f);
476   realpoly(f*(x4+2));
477   ring r2=0,(x,y),dp;
478   poly f=x6-3x4y2 + y6 + x2y2 -6y+5;
479   realpoly(f);
480   ring r3=0,(x,y,z),dp;
481   poly f=x4y4-2x5y3z2+x6y2z4+2x2y3z-4x3y2z3+2x4yz5+z2y2-2z4yx+z6x2;
482   realpoly(f);
483   realpoly(f*(x2+y2+1));
484}
485
486
487
488
489///////////////////////////////////////////////////////////////////////////////
490//// for semi-definiteness/////////////////////////////////////////////////////
491///////////////////////////////////////////////////////////////////////////////
492proc decision(poly f)
493" USAGE: decission(f); a multivariate polynomial f in Q[x_1,..,x_n] and lc f=0
494 RETURN: assume that the basering has a lexicographical ordering,
495         1 if f is positive semidefinite  0 if f is indefinite
496EXAMPLE: decision shows an example
497{
498   string ri,lessvar,parvar,perm;
499   ideal jac;
500   list varlist,buffer,isol,@s,lhelp,lhelp1,lfac,worklist;
501   poly p,g;
502   def rbuffer;
503   def r=basering;
504   //diverse zaehler
505   int @z,zz,count,tester;
506   int n=nvars(r);
507   //specialcases
508
509   if (leadcoef(f)<0)
510   {
511     return(0);
512   }
513   lfac=factorize(f,2);
514   ideal factor=lfac[1];
515   intvec @ex=lfac[2];
516   factor=factor[1];
517   zz=size(factor);
518   f=1;
519   for (@z=1;@z<=zz;@z++)
520   {
521     if ((@ex[@z] mod 2)==1)
522     {
523        f=f*factor[@z];
524     }
525   }
526   if (deg(f)<=0)
527   {
528     if (leadcoef(f)>=0)
529     {
530       return(1);
531     }
532     return(0);
533   }
534   //for recursion
535   if (n==1)
536   {
537    if (sturm(f,-length(f),length(f))==0)
538    {
539      return(1);
540    }
541    return(0);
542   }
543   //search for a p in Q[x_n] such that f is pos. sem. definite
544   //if and only if for every isolating setting S={a_1,...,a_r} holds that
545   //every f(x_1,..,x_n-1, a_i) is positiv semidefinite
546   //recursion of variables
547   ///////////////////////////////////////////////////////////////////////////
548   ///////////////////////////////////////////////////////////////////////////
549   ideal II = maxideal(1);
550   varlist = II[1..n-1];
551   lessvar=string(varlist);
552
553   parvar=string(var(n));
554   ri="ring r_neu="+charstr(r)+",(@t,"+parvar+","+lessvar+"),dp;";
555   execute(ri);
556   poly f=imap(r,f);
557   list varlist=imap(r,varlist);
558   ideal jac=jacob(@t+f);
559   jac=jac[3..(n+1)];
560   ideal eins=std(jac);
561   ideal i=@t+f,jac;
562   //use Wu method
563   if (eins==1)
564   {
565     zz=0;
566   }
567   else
568   {
569     matrix m=char_series(i);
570     zz=nrows(m);//number of rows
571   }
572   poly p=1;
573   for (@z=1;@z<=zz;@z++)
574   {
575     p=p*m[@z,1];
576   }
577   //trailing coefficient of p
578   p=subst(p,@t,0);
579   p=realpoly(p);
580   @s=subsets(n-1);
581   ideal jacs;
582   for (@z=1;@z<=size(@s);@z++)
583   {
584     perm="";
585     lhelp=list();
586
587     worklist=varlist;
588     buffer=jac[1..(n-1)];
589     //vorbereitungen fuer den Ringwechsel
590     //setze worklist=x_1,..,x_(n-1)
591
592     for (zz=1;zz<=size(@s[@z]);zz++)
593     {
594      buffer =delete(buffer ,@s[@z][zz]-zz+1);
595      worklist=delete(worklist,@s[@z][zz]-zz+1);
596      lhelp=lhelp+list(string(var(@s[@z][zz]+2)));
597      lhelp1=insert(lhelp,string(var(@s[@z][zz]+2)));
598     }
599     //worklist=(x_1,...,x_n-1)\(x_i1,...,x_ik)
600     //lhelp =(x_i1,...,x_ik)
601     //buffer=diff(f,x_i) i not in (i1,..,ik);
602
603     worklist=list("@t",string(var(2)))+lhelp+worklist;
604     for (zz=1;zz<=n+1;zz++)
605     {
606       perm=perm+","+string(worklist[zz]);
607     }
608     perm=perm[2..size(perm)];
609     if (size(buffer)!=0)
610     {
611      jacs=buffer[1..size(buffer)];
612      jacs=@t+f,jacs;
613     }
614     else
615     {
616      jacs=@t+f;
617     }
618     rbuffer=basering;
619     //perm=@t,x_n,x_1,..,x_ik,x\(x_i1,..,x_ik)
620     ri="ring rh=0,("+perm+"),dp;";
621     execute(ri);
622     ideal jacs=imap(rbuffer,jacs);
623     poly p=imap(rbuffer,p);
624     matrix m=char_series(jacs);
625     poly e=1;
626     for (count=1;count<=nrows(m);count++)
627     {
628       e=e*m[count,1];
629     }
630     //search for the leading coefficient of e in
631     //Q(@t,x_n)[x_@s[@z][1],..,x_@s[@z][size(@s[@z])]
632     intmat l[n-1][n-1];
633     for (zz=1;zz<n;zz++)
634     {
635       l[zz,n-zz]=1;
636     }
637     ri="ring rcoef="+"(0,@t,"+parvar+"),
638     ("+lessvar+"),M(l);";
639     execute(ri);
640     kill l;
641     poly e=imap(rh,e);
642     e=leadcoef(e);
643     setring rh;
644     e=imap(rcoef,e);
645     e=subst(e,@t,0);
646     e=realpoly(e);
647     p=p*e;
648     setring r_neu;
649     p=imap(rh,p);
650     kill rh,rcoef;
651   }
652   setring r;
653   p=imap(r_neu,p);
654   ///////////////////////////////////////////////////////////////////////////
655   ///////////found polynomial p /////////////////////////////////////////////
656   ///////////////////////////////////////////////////////////////////////////
657   //Compute an isolating set for p
658   ri="ring iso="+charstr(r)+","+parvar+",lp;";
659   execute(ri);
660   poly p=imap(r,p);
661   isol=isolset(p);
662   setring r;
663   list isol=imap(iso,isol);
664   tester=1;
665   for (@z=1;@z<=size(isol);@z++)
666   {
667     ri="ring rless="+charstr(r)+",("+lessvar+"),lp;";
668     g=subst(f,var(n),isol[@z]);
669     execute(ri);
670     poly g=imap(r,g);
671     tester=tester*decision(g);
672     setring r;
673     kill rless;
674   }
675   return(tester);
676}
677
678
679proc isolset(poly f)
680"USAGE:  isolset(f); f a univariate polynomial over the rational numbers
681RETURN: An isolating set of f
682NOTE: algorithm can be found in M-F. Roy,R: Pollack, S. Basu page 373
683EXAMPLE: example isolset; shows an example"
684{
685 int i,case;
686 number m;
687 list buffer;
688 //only real roots count
689 f=realpoly(f);
690 poly seppart=f;
691 seppart=simplify(seppart,1);
692 //int N=binlog(length(seppart));
693 //number zweihochN=exp(2,N+1);
694 number zweihochN=length(f);
695 //a special case
696 if (deg(seppart)==0)
697 {
698   return(list(number(0)));
699 }
700 if (sturm(seppart,-zweihochN,zweihochN)==1)
701 {
702  return(list(-zweihochN,zweihochN));
703 }
704 //getting bernstein coeffs
705 ideal id=isuni(f)-zweihochN;
706 map jmap=basering,id;
707 seppart=jmap(seppart);
708
709 id=2*zweihochN*var(1);
710 jmap=basering,id;
711 seppart=jmap(seppart);
712
713 matrix c=coeffs(seppart,var(1));
714 int s=size(c);
715 poly recproc;
716 //Reciprocal polynomial
717 for (i=1;i<=s;i++)
718 {
719  recproc=recproc+c[s+1-i,1]*(var(1)^(i-1));
720 }
721 jmap=basering,var(1)+1;
722 seppart=jmap(recproc);
723 list bernsteincoeffs,bern;
724 c=coeffs(seppart,var(1));
725 for (i=1;i<=s;i++)
726 {
727   bern[i]=number(c[s+1-i,1])/binomial(s-1,i-1);
728 }
729 bernsteincoeffs=bern,list(-zweihochN,zweihochN);
730 list POS;
731 POS[1]=bernsteincoeffs;
732 list L;
733 while (size(POS)!=0)
734 {
735  if (varsigns(POS[1][1])<2)
736  {
737    case=varsigns(POS[1][1]);
738  }
739  else
740  {
741    case=2;
742  }
743  //case Anweisung
744  buffer=POS[1];
745  POS=delete(POS,1);
746  while(1)
747  {
748   if (case==1)
749   {
750    L=L+buffer[2];
751    break;
752   }
753
754   if (case==2)
755   {
756    m=number(buffer[2][1]+buffer[2][2])/2;
757    bern=BernsteinCoefficients(buffer[1],buffer[2],m);
758    POS=bern+POS;
759    if (leadcoef(sign(leadcoef(subst(f,isuni(f),m))))==0)
760    {
761      number epsilon=1/10;
762      while (sturm(f,m-epsilon,m+epsilon)!=1)
763      {
764        epsilon=epsilon/10;
765      }
766      L=L+list(m-epsilon,m+epsilon);
767    }
768    break;
769   }
770  break;
771  }
772 }
773 i=1;
774 while (i<size(L))
775 {
776   if (L[i]==L[i+1])
777   {
778     L=delete(L,i);
779   }
780   else
781   {
782     i=i+1;
783   }
784 }
785 return(L);
786}
787
788static proc BernsteinCoefficients(list bern,list lr,number m)
789"USAGE :BernsteinCoefficients(bern,lr,m);
790        a list bern=b_0,...,b_p representing a polynomial P of degree <=p
791        in the Bernstein basis pf lr=(l,r) an a number m in Q
792 RETURN:a list erg=erg1,erg2 s.th. erg1=erg1[1],erg[2] and erg1[1] are
793        the bernstein coefficients of P w.r.t. to erg1[2]=(l,m) and erg2[1]
794        is one for erg2[2]=(m,r)
795 EXAMPLE: Bernsteincoefficients shows no example
796"
797{
798 //Zaehler
799 int i,j;
800 list erg,erg1,erg2;
801 number a=(lr[2]-m)/(lr[2]-lr[1]);
802 number b=(m-lr[1])/(lr[2]-lr[1]);
803 int p=size(bern);
804 list berns,buffer,buffer2;
805 berns[1]=bern;
806 for (i=2;i<=p;i++)
807 {
808  for (j=1;j<=p+1-i;j++)
809  {
810   buffer[j]=a*berns[i-1][j]+b*berns[i-1][j+1];
811  }
812  berns[i]=buffer;
813  buffer=list();
814 }
815
816 for (i=1;i<=p;i++)
817 {
818  buffer[i]=berns[i][1];
819  buffer2[i]=berns[p+1-i][i];
820 }
821 erg1=buffer,list(lr[1],m);
822 erg2=buffer2,list(m,lr[2]);
823 erg=erg1,erg2;
824 return(erg);
825}
826
827static proc binlog(number i)
828{
829 int erg;
830 if (i<2) {return(0);}
831 else
832 {
833  erg=1+binlog(i/2);
834  return(erg);
835 }
836}
837
838//////////////////////////////////////////////////////////////////////////////
839///////diverse Hilfsprozeduren ///////////////////////////////////////////////
840//////////////////////////////////////////////////////////////////////////////
841
842/////////////////////////////////////////////////////////////////////////////
843/////wichtig fuers Verstaendnis//////////////////////////////////////////////
844/////////////////////////////////////////////////////////////////////////////
845static proc is_real(poly f)
846"USAGE:     is_real(f);a univariate irreducible polynomial f;
847RETURN:    1: if f is real
848           0: is f is not real
849EXAMPLE:   example is_real; shows an example"
850
851{
852  int d,anz,i;
853  def r=basering;
854
855  if (f==1) {return(1);}
856  if (isuniv(f)==0)
857  {
858   for (i=1;i<=nvars(r);i++)
859   {
860     d=size(coeffs(f,var(i)))+1;
861     if ((d mod 2)==1)
862     {
863       return(1);
864     }
865   }
866   d=1-decision(f);
867   return(d);
868  }
869  d=deg(f) mod 2;
870  if (d==1)
871  {
872     return(1);//because of fundamental theorem of algebra
873  }
874  else
875  {
876   f=simplify(f,1);//wlog we can assume that f is monic
877   number a=leadcoef(sign(leadcoef(subst(f,isuni(f),-length(f)))));
878   number b=leadcoef(sign(leadcoef(subst(f,isuni(f),length(f)))));
879   if
880   (a*b!=1)
881   //polynomials are contineous so the image is an interval
882   //referres to analysis
883   {
884      return(1);
885   }
886   else
887   {
888      anz=sturm(f,-length(f),length(f));
889      if (anz==0) {return(0);}
890      else {return(1);}
891   }
892  }
893}
894example
895{ "EXAMPLE:"; echo = 2;
896   ring r1 = 0,x,dp;
897   poly f=x2+1;
898   is_real(f);
899
900}
901
902
903static proc prepare_max(ideal m)
904"USAGE: prepare_max(m); m a maximal ideal in Q(y_1,...,y_m)[x_1,...,x_n]
905RETURN: a list erg=(id,j); where id is the real radical of m if j=1 (i.e. m
906        satisfies the shape lemma in one variable x_i) else id=m and j=0;
907EXAMPLE: is_in_shape shows an exmaple;
908"
909
910{
911  int j,k,i,l,fakul;
912  def r=basering;
913  int n=nvars(r);
914  list erg,varlist,perm;
915  string wechsler,vari;
916  //option(redSB);
917
918  for (i=1;i<=n;i++)
919  {
920   varlist=varlist+list(var(i));
921  }
922  perm=permutation(varlist);
923  fakul=size(perm);
924  for (i=1;i<=fakul;i++)
925  {
926    for (j=1;j<=n;j++)
927    {
928      vari=vari+","+string(perm[i][j]);
929    }
930    vari=vari[2..size(vari)];
931    wechsler="ring r_neu=("+charstr(r)+"),("+vari+"),lp;";
932    execute(wechsler);
933    ideal id=imap(r,m);
934    id=groebner(id);
935    k=search_first(id,2,2);
936    setring r;
937    m=imap(r_neu,id);
938    m[1]=realpoly(m[1]);
939    if (m[1]==1)
940    {
941      erg[1]=ideal(1);
942      erg[2]=1;
943      return(erg);
944    }
945    if (k>n)
946    {
947      erg[1]=m;
948      erg[2]=1;
949      return(erg);
950    }
951    else
952    {
953      for (l=k;l<=n;l++)
954      {
955        if (realpoly(m[l])==1)
956        {
957          erg[1]=ideal(1);
958          erg[2]=1;
959          return(erg);
960        }
961      }
962    }
963    vari="";
964    kill r_neu;
965  }
966  if (size(parstr(r))==0)
967  {
968    erg[1]=m;
969    j=1;
970    for (i=1;i<=n;i++)
971    {
972      j=j*isuniv(m[i]);
973    }
974    erg[2]=j;
975    return(erg);
976  }
977  erg[1]=m;
978  erg[2]=0;
979  return(erg);
980}
981
982static proc length(poly f)
983"USAGE:    length(f); poly f;
984RETURN:    sum of the absolute Value of all coeffients of an irreducible
985           poly nomial f
986EXAMPLE:   example length; shows an example"
987
988{
989 number erg,buffer;
990 f=simplify(f,1);//wlog f is monic
991 int n=size(f);
992 for (int i=1;i<=n;i=i+1)
993 {
994   buffer= leadcoef(f[i]);
995   erg=erg + absValue(buffer);
996 }
997
998 return(erg);
999}
1000example
1001{ "EXAMPLE:"; echo = 2;
1002   ring r1 = 0,x,dp;
1003   poly f=x4-6x3+x2+1;
1004   norm(f);
1005
1006   ring r2=0,(x,y),dp;
1007   poly g=x2-y3;
1008   length(g);
1009
1010}
1011//////////////////////////////////////////////////////////////////////////////
1012//////////////weniger wichtig fuers Verstaendnis//////////////////////////////
1013//////////////////////////////////////////////////////////////////////////////
1014static proc isuniv(poly f)
1015{
1016  int erg;
1017  if (f==0)
1018  {
1019    erg=1;
1020  }
1021  else
1022  {
1023  erg=(isuni(f)!=0);
1024  }
1025  return(erg);
1026}
1027static proc search_first(ideal j,int start, int i)
1028"USAGE:    searchfirst(j, start, i);
1029           id a reduced groebner basis w.r.t. lex
1030RETURN:    if i=1 then turns the number of the first non univariate entry
1031           with order >1 in its leading term after start
1032           else the first non univariate of even order
1033EXAMPLE:   example norm; shows no example"
1034{
1035  int n=size(j);
1036  int k=start;//counter
1037  j=j,0;
1038  if (i==1)
1039  {
1040    while
1041    ((k<=n)&&(ord(j[k])==1))
1042    {
1043      k=k+1;
1044    }
1045  }
1046  else
1047  {
1048    while
1049    ((k<=n)&&(ord(j[k]) mod 2==1))
1050    {
1051      k=k+1;
1052    }
1053
1054  }
1055 return(k);
1056}
1057
1058static proc subsets(int n)
1059"USAGE :subsets(n); n>=0 in Z
1060RETURN :l a list of all non-empty subsets of {1,..,n}
1061EXAMPLE:subsets(n) shows an example;
1062"
1063{
1064 list l,buffer;
1065 int i,j,binzahl;
1066 if (n<=0)
1067 {
1068   return(l);
1069 }
1070 int grenze=2**n-1;
1071 for (i=1;i<=grenze;i++)
1072 {
1073  binzahl=i;
1074  for (j=1;j<=n;j++)
1075  {
1076   if ((binzahl mod 2)==1)
1077   {
1078     buffer=buffer+list(j);
1079   }
1080   binzahl=binzahl div 2;
1081  }
1082  l[i]=buffer;
1083  buffer=list();
1084 }
1085 return(l);
1086}
1087example
1088{ "EXAMPLE:"; echo = 2;
1089  subsets(3);
1090  subsets(4);
1091}
1092
1093proc permutation(list L)
1094" USAGE: permutation(L); L a list
1095 OUTPUT: a list of all permutation lists of L
1096EXAMPLE: permutation(L) gives an example"
1097{
1098  list erg,buffer,permi,einfueger;
1099  int i,j,l;
1100  int n=size(L);
1101  if (n==0)
1102  {
1103    return(erg);
1104  }
1105  if (n==1)
1106  {
1107    erg=list(L);
1108    return(erg);
1109  }
1110  for (i=1;i<=n;i++)
1111  {
1112    buffer=delete(L,i);
1113    einfueger=permutation(buffer);
1114    l=size(einfueger);
1115    for (j=1;j<=l;j++)
1116    {
1117      permi=list(L[i])+einfueger[j];
1118      erg=insert(erg,permi);
1119    }
1120  }
1121  return(erg);
1122}
1123example
1124{ "EXAMPLE:"; echo = 2;
1125  list L1="Just","an","example";
1126  permutation(L1);
1127  list L2=1,2,3,4;
1128  permutation(L2);
1129}
1130static proc simplify_gen(poly f)
1131"USAGE : simplify_gen(f); f a polymimial in Q(y_1,..,y_m)[x_1,..,x_n]
1132RETURN : a polynomial g such that g is the square-free part of f  and
1133        every real univariate factor of f is cancelled out
1134EXAMPLE:simplify_gen gives no example"
1135{
1136  int i,l;
1137  ideal factor;
1138  poly g=1;
1139  factor=factorize(f,2)[1];
1140  l=size(factor);
1141  for (i=1;i<=l;i++)
1142  {
1143    if (isuniv(factor[i]))
1144    {
1145        g=g*realpoly(factor[i]);
1146    }
1147    else
1148    {
1149      g=g*factor[i];
1150    }
1151  }
1152  return(g);
1153}
1154static proc contnonloc(ideal id,string pari, string vari)
1155"INPUT : a radical ideal id in in F[pari+vari] which is radical in
1156         F(pari)[vari), pari and vari strings of variables
1157OUTPUT : the contraction ideal of id, i.e. idF(pari)[vari]\cap F[pari+vari]
1158EXAMPLE: contnonloc shows an example
1159"
1160{
1161  list pr;
1162  list contractpr;
1163  int i,l,tester;
1164  ideal primcomp;
1165  def r=basering;
1166  string neu="ring r_neu=("+charstr(r)+pari+"),("+vari+"),dp;";
1167  execute(neu);
1168  def r1=basering;
1169  ideal buffer;
1170  setring r;
1171  pr=primdecGTZ(id);
1172  l=size(pr);
1173  contractpr[1]=ideal(1);
1174  for (i=1;i<=l;i++)
1175  {
1176    primcomp=pr[i][2];
1177    setring r1;
1178    buffer=imap(r,primcomp);
1179    buffer=groebner(buffer);
1180    if (buffer==1)
1181    {
1182     tester=0;
1183    }
1184    else
1185    {
1186     tester=1;
1187    }
1188    setring r;
1189
1190    //id only consits of non units in F(pari)
1191    if (tester==1)
1192    {
1193     contractpr=insert(contractpr,primcomp);
1194    }
1195  }
1196  l=size(contractpr);
1197  id=intersect(contractpr[1..l]);
1198  return(id);
1199}
1200example
1201{ "EXAMPLE:"; echo = 2;
1202   ring  r = 0,(a,b,c),lp;
1203   ideal i=b3+c5,ab2+c3;
1204   ideal j=contnonloc(i,",b","a,c");
1205   j;
1206}
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