Changeset 6cd6983 in git
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 Dec 1, 2008, 9:58:20 PM (15 years ago)
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 (u'spielwiese', '828514cf6e480e4bafc26df99217bf2a1ed1ef45')
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 a6a5d9ad89ba42bbc129bacba382bb6f463aae65
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 8c66fb0e0dc173c2eb5aff1e988308a8527bc045
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Singular/LIB/bfct.lib
r8c66fb r6cd6983 1 1 ////////////////////////////////////////////////////////////////////////////// 2 version="$Id: bfct.lib,v 1. 5 20081006 17:04:26 SingularExp $";2 version="$Id: bfct.lib,v 1.6 20081201 20:58:20 levandov Exp $"; 3 3 category="Noncommutative"; 4 4 info=" … … 18 18 MAIN PROCEDURES: 19 19 20 bfct(f[,s,t,v]); compute the global BernsteinSato polynomial of a given poly 21 bfctsyz(f[,r,s,t,u,v]); compute the global BernsteinSato polynomial of a given poly 22 bfctonestep(f[,s,t]); compute the global BernsteinSato polynomial of a given poly 23 bfctideal(I,w[,s,t]); compute the global bfunction of a given ideal w.r.t. a given weight 24 minpol(f,I); compute the minimal polynomial of the endormorphism in basering modulo ideal given by a poly 25 minpolsyz(f,I[,p,s,t]); compute the minimal polynomial of the endormorphism in basering modulo ideal given by a poly 26 linreduce(f,I[,s]); reduce a poly by linear reductions of its leading term 27 ncsolve(I[,s]); find and compute a linear dependency of the elements of an ideal 20 bfct(f[,s,t,v]); compute the global BernsteinSato polynomial of a given poly 21 bfctsyz(f[,r,s,t,u,v]); compute the global BernsteinSato polynomial of a given poly 22 bfctann(f[,s]); compute the global BernsteinSato polynomial of a given poly 23 bfctonestep(f[,s,t]); compute the global BernsteinSato polynomial of a given poly 24 bfctideal(I,w[,s,t]); compute the global bfunction of a given ideal w.r.t. a given weight 25 pintersect(f,I); compute the intersection of the ideal generated by a given poly with a given ideal 26 pintersectsyz(f,I[,p,s,t]); compute the intersection of the ideal generated by a given poly with a given ideal 27 linreduce(f,I[,s]); reduce a poly by linear reductions only 28 ncsolve(I[,s]); find and compute a linear dependency of the elements of an ideal 28 29 29 30 AUXILIARY PROCEDURES: … … 32 33 isin(l,i); check whether an element is a member of a list 33 34 scalarprod(v,w); compute the standard scalar product of two intvecs 35 vec2poly(v[,i]); convert a coefficient vector to a poly 34 36 35 37 SEE ALSO: dmod_lib, dmodapp_lib, gmssing_lib … … 38 40 39 41 LIB "qhmoduli.lib"; // for Max 40 LIB "dmodapp.lib"; // for initialideal etc42 LIB "dmodapp.lib"; // for initialideal etc 41 43 42 44 … … 48 50 example isin; 49 51 example scalarprod; 52 example vec2poly; 50 53 "MAIN PROCEDURES:"; 51 54 example bfct; 52 55 example bfctsyz; 56 example bfctann; 53 57 example bfctonestep; 54 58 example bfctideal; 55 example minpol;56 example minpolsyz;59 example pintersect; 60 example pintersectsyz; 57 61 example linreduce; 58 62 example ncsolve; … … 156 160 proc isin (list l, i) 157 161 "USAGE: isin(l,i); l a list, i an argument of any type 158 RETURN: 1 if i is a member of l, or 0 otherwise162 RETURN: an int, the position of the first appearance of i in l, or 0 if i is not a member of l 159 163 PURPOSE: check whether the second argument is a member of a list 160 164 EXAMPLE: example isin; shows an example … … 166 170 if (l[j]==i) 167 171 { 168 return( 1);172 return(j); 169 173 break; 170 174 } … … 216 220 217 221 proc linreduce(poly f, ideal I, list #) 218 "USAGE: linreduce(f, I [,s ]); f a poly, I an ideal, s an optional int219 RETURN: a poly obtained by linear reductions of the leading term of the given poly with an ideal220 PURPOSE: reduce a poly only by linear reductions of its leading term221 NOTE: If s<>0, a list consisting of the reduced poly and the vector of the used222 "USAGE: linreduce(f, I [,s,t]); f a poly, I an ideal, s,t optional ints 223 RETURN: a poly obtained by linear reductions of the given poly with the given ideal 224 PURPOSE: reduce a poly only by linear reductions 225 NOTE: If s<>0, a list consisting of the reduced poly and the coefficient vector of the used 222 226 @* reductions is returned. 227 @* If t<>0, only leading monomials are reduced, otherwise, and by default, all monomials 228 @* are reduced, if possible. 223 229 EXAMPLE: example linreduce; shows examples 224 230 " 225 231 { 232 int ppl = printlevel  voice + 2; 226 233 int remembercoeffs = 0; // default 234 int redlm = 0; // default 227 235 if (size(#)>0) 228 236 { … … 231 239 remembercoeffs = #[1]; 232 240 } 233 } 234 int i; 241 if (size(#)>1) 242 { 243 if (typeof(#[2])=="int"  typeof(#[2])=="number") 244 { 245 redlm = #[2]; 246 } 247 } 248 } 249 int i,j,k; 235 250 int sI = ncols(I); 236 251 ideal lmI,lcI; … … 241 256 } 242 257 vector v; 243 poly lm,c;244 int reduction ;245 lm = leadmonom(f);246 reduction = 1;247 while (reduction == 1) // while there was a reduction248 {249 reduction = 0;250 for (i=sI;i>=1;i)251 {252 if (lm <> 0 && lm == lmI[i])253 254 c = leadcoef(f)/lcI[i];255 f = f  c*I[i]; 256 lm = leadmonom(f); 257 reduction = 1;258 if (remembercoeffs <> 0)258 poly c; 259 int reduction = 1; 260 if (redlm == 0) 261 { 262 ideal monomf; 263 for (k=1; k<=size(f); k++) 264 { 265 monomf[k] = normalize(f[k]); 266 } 267 while (reduction == 1) // while there was a reduction 268 { 269 reduction = 0; 270 for (i=sI; i>=1; i) 271 { 272 dbprint(ppl,"testing ideal entry:",i); 273 for (j=1; j<=size(f); j++) 259 274 { 260 v = v  c * gen(i); 275 if (monomf[j] == lmI[i]) 276 { 277 c = leadcoef(f[j])/lcI[i]; 278 f = f  c*I[i]; 279 dbprint(ppl,"reducing poly to ",f); 280 monomf = 0; 281 for (k=1; k<=size(f); k++) 282 { 283 monomf[k] = normalize(f[k]); 284 } 285 reduction = 1; 286 if (remembercoeffs <> 0) 287 { 288 v = v  c * gen(i); 289 } 290 break; 291 } 292 } 293 if (reduction == 1) 294 { 295 break; 296 } 297 } 298 } 299 } 300 else // reduce only leading monomials 301 { 302 poly lm = leadmonom(f); 303 while (reduction == 1) // while there was a reduction 304 { 305 reduction = 0; 306 for (i=sI;i>=1;i) 307 { 308 if (lm <> 0 && lm == lmI[i]) 309 { 310 c = leadcoef(f)/lcI[i]; 311 f = f  c*I[i]; 312 lm = leadmonom(f); 313 reduction = 1; 314 if (remembercoeffs <> 0) 315 { 316 v = v  c * gen(i); 317 } 261 318 } 262 319 } … … 279 336 ideal I = 1,y,xy; 280 337 poly f = 5xy+7y+3; 281 poly g = 5y+7x+3; 282 linreduce(f,I); 338 poly g = 7x+5y+3; 283 339 linreduce(g,I); 340 linreduce(g,I,0,1); 284 341 linreduce(f,I,1); 285 342 } … … 386 443 } 387 444 388 proc minpol(poly s, ideal I)389 "USAGE: minpol(f, I); f a poly, I an ideal390 RETURN: coefficient vector of the m inimal polynomial of the endomorphism of basering modulo I defined by f391 PURPOSE: compute the minimal polynomial392 NOTE: If f does not define an endomorphism, this proc willnot terminate.445 proc pintersect (poly s, ideal I) 446 "USAGE: pintersect(f, I); f a poly, I an ideal 447 RETURN: coefficient vector of the monic generator of the intersection of the ideal generated by f with I 448 PURPOSE: compute the intersection of an ideal with a principal ideal defined by f 449 NOTE: If the intersection is zero, this proc might not terminate. 393 450 @* I should be given as standard basis. 394 451 @* If printlevel=1, progress debug messages will be printed, 395 452 @* if printlevel>=2, all the debug messages will be printed. 396 EXAMPLE: example minpol; shows examples453 EXAMPLE: example pintersect; shows examples 397 454 " 398 455 { … … 400 457 attrib(I,"isSB",1); // set attribute for suppressing NF messages 401 458 int ppl = printlevelvoice+2; 459 // case 1: I = basering 460 if (size(I) == 1) 461 { 462 if (simplify(I,1) == ideal(1)) 463 { 464 return(gen(2)); // = s 465 } 466 } 402 467 def save = basering; 468 int n = nvars(save); 403 469 int i,j,k; 470 // case 2: intersection is zero 471 intvec degs = leadexp(s); 472 intvec possdegbounds; 473 list degI; 474 i = 1; 475 while (i <= ncols(I)) 476 { 477 if (i == ncols(I)+1) 478 { 479 break; 480 } 481 degI[i] = leadexp(I[i]); 482 for (j=1; j<=n; j++) 483 { 484 if (degs[j] == 0) 485 { 486 if (degI[i][j] <> 0) 487 { 488 break; 489 } 490 } 491 if (j == n) 492 { 493 k++; 494 possdegbounds[k] = Max(degI[i]); 495 } 496 } 497 i++; 498 } 499 int degbound = Min(possdegbounds); 500 dbprint(ppl,"a lower bound for the degree of the insection is:"); 501 dbprint(ppl,degbound); 502 if (degbound == 0) // lm(s) does not appear in lm(I) 503 { 504 return(vector(0)); 505 } 506 // case 3: intersection is nontrivial 507 ideal redNI = 1; 404 508 vector v; 405 509 list l,ll; 406 510 l[1] = vector(0); 407 poly toNF, tobracket, newNF, rednewNF; 408 ideal NI = 1; 511 poly toNF,tobracket,newNF,rednewNF,oldNF,secNF; 409 512 i = 1; 410 ideal redNI = 1; 411 newNF = NF(s,I); 412 dbprint(ppl+1,"minpol starts..."); 413 dbprint(ppl+1,"with ideal I=", I); 513 dbprint(ppl+1,"pintersect starts..."); 414 514 while (1) 415 515 { … … 417 517 if (i>1) 418 518 { 419 tobracket = s^(i1)NI[i]; 420 if (tobracket==0) // bracket doesn't like zeros 519 oldNF = newNF; 520 tobracket = s^(i1)  oldNF; 521 if (tobracket==0) // todo bug in bracket? 421 522 { 422 523 toNF = 0; … … 424 525 else 425 526 { 426 toNF = bracket(tobracket,NI[2]); 427 } 428 newNF = NF(toNF+NI[i]*NI[2],I); // = NF(s^i,I) 527 toNF = bracket(tobracket,secNF); 528 } 529 newNF = NF(toNF+oldNF*secNF,I); // = NF(s^i,I) 530 } 531 else 532 { 533 newNF = NF(s,I); 534 secNF = newNF; 429 535 } 430 536 ll = linreduce(newNF,redNI,1); 431 537 rednewNF = ll[1]; 432 538 l[i+1] = ll[2]; 433 dbprint(ppl +1,"newNF is:", newNF);434 dbprint(ppl +1,"rednewNF is:", rednewNF);539 dbprint(ppl,"newNF is:", newNF); 540 dbprint(ppl,"rednewNF is:", rednewNF); 435 541 if (rednewNF != 0) // no linear dependency 436 542 { 437 NI[i+1] = newNF;438 543 redNI[i+1] = rednewNF; 439 dbprint(ppl+1,"NI is:", NI);440 dbprint(ppl+1,"redNI is:", redNI);441 544 i++; 442 545 } 443 546 else // there is a linear dependency, hence we are done 444 547 { 445 dbprint(ppl+1,"the degree of the minimal polynomialis:", i);548 dbprint(ppl+1,"the degree of the generator of the intersection is:", i); 446 549 break; 447 550 } 448 551 } 449 552 dbprint(ppl,"used linear reductions:", l); 450 // we obtain the coefficients of the minimal polynomialby the used reductions:553 // we obtain the coefficients of the generator of the intersection by the used reductions: 451 554 ring @R = 0,(a(1..i+1)),dp; 452 555 setring @R; … … 461 564 } 462 565 } 463 for (j=i;j>=1;j)566 for (j=i;j>=1;j) 464 567 { 465 568 C[i+1]= subst(C[i+1],a(j),a(j)+C[j]); … … 474 577 v = imap(@R,v); 475 578 kill @R; 476 dbprint(ppl+1," minpolfinished");579 dbprint(ppl+1,"pintersect finished"); 477 580 return(v); 478 581 } … … 480 583 { 481 584 "EXAMPLE:"; echo = 2; 482 printlevel = 0;483 585 ring r = 0,(x,y),dp; 484 586 poly f = x^2+y^3+x*y^2; … … 486 588 setring D; 487 589 inF; 488 poly s = t*Dt; 489 minpol(s,inF); 490 } 491 492 proc minpolsyz (poly s, ideal II, list #) 493 "USAGE: minpolsyz(f, I [,p,s,t]); f a poly, I an ideal, p, t optial ints, p a prime number 494 RETURN: coefficient vector of the minimal polynomial of the endomorphism of basering modulo I defined by f 495 PURPOSE: compute the minimal polynomial 496 NOTE: If f does not define an endomorphism, this proc will not terminate. 590 pintersect(t*Dt,inF); 591 } 592 593 proc pintersectsyz (poly s, ideal II, list #) 594 "USAGE: pintersectsyz(f, I [,p,s,t]); f a poly, I an ideal, p, t optial ints, p a prime number 595 RETURN: coefficient vector of the monic generator of the intersection of the ideal generated by f with I 596 PURPOSE: compute the intersection of an ideal with a principal ideal defined by f 597 NOTE: If the intersection is zero, this proc might not terminate. 497 598 @* I should be given as standard basis. 498 @* If p>0 is given, th e proc computes the minimal polynomialin char p first and499 @* then only searches for a minimal polynomialof the obtained degree in the basering.599 @* If p>0 is given, this proc computes the generator of the intersection in char p first and 600 @* then only searches for a generator of the obtained degree in the basering. 500 601 @* Otherwise, it searched for all degrees. 501 602 @* This is done by computing syzygies. … … 506 607 @* If printlevel=1, progress debug messages will be printed, 507 608 @* if printlevel>=2, all the debug messages will be printed. 508 EXAMPLE: example minpolsyz; shows examples609 EXAMPLE: example pintersectsyz; shows examples 509 610 " 510 611 { … … 566 667 } 567 668 i = 1; 568 dbprint(ppl+1," minpolynomialstarts...");669 dbprint(ppl+1,"pintersectsyz starts..."); 569 670 dbprint(ppl+1,"with ideal I=", I); 570 671 while (1) … … 639 740 i++; 640 741 } 641 dbprint(ppl+1," minpolfinished");742 dbprint(ppl+1,"pintersectsyz finished"); 642 743 return(v); 643 744 } … … 645 746 { 646 747 "EXAMPLE:"; echo = 2; 647 printlevel = 0;648 748 ring r = 0,(x,y),dp; 649 749 poly f = x^2+y^3+x*y^2; … … 652 752 inF; 653 753 poly s = t*Dt; 654 minpolsyz(s,inF);754 pintersectsyz(s,inF); 655 755 int p = prime(20000); 656 minpolsyz(s,inF,p,0,0); 756 pintersectsyz(s,inF,p,0,0); 757 } 758 759 proc vec2poly (list #) 760 "USAGE: vec2poly(v [,i]); v a vector or an intvec, i an optional int 761 RETURN: a poly with coefficient vector v 762 PURPOSE: convert a coefficient vector to a poly 763 NOTE: If i>0 is given, the returned poly is an element of K[var(i)], 764 @* otherwise, and by default, @code{i=1} is used. 765 @* The first entry of v is the coefficient of 1. 766 EXAMPLE: example vec2poly; shows examples 767 " 768 { 769 def save = basering; 770 int i,ringvar; 771 ringvar = 1; // default 772 if (size(#) > 0) 773 { 774 if (typeof(#[1])=="vector"  typeof(#[1])=="intvec") 775 { 776 def v = #[1]; 777 } 778 else 779 { 780 ERROR("wrong input: expected vector/intvec expression"); 781 } 782 if (size(#) > 1) 783 { 784 if (typeof(#[2])=="int"  typeof(#[2])=="number") 785 { 786 ringvar = int(#[2]); 787 } 788 } 789 } 790 if (ringvar > nvars(save)) 791 { 792 ERROR("var out of range"); 793 } 794 poly p; 795 for (i=1; i<=nrows(v); i++) 796 { 797 p = p + v[i]*(var(ringvar))^(i1); 798 } 799 return(p); 800 } 801 example 802 { 803 "EXAMPLE:"; echo = 2; 804 ring r = 0,(x,y),dp; 805 vector v = gen(1) + 3*gen(3) + 22/9*gen(4); 806 intvec iv = 3,2,1; 807 vec2poly(v,2); 808 vec2poly(iv); 657 809 } 658 810 … … 705 857 } 706 858 707 static proc bfctengine (poly f, int whichengine, int methodord, int methodminpol, int minpolchar, int modengine, intvec u0)859 static proc bfctengine (poly f, int inorann, int whichengine, int methodord, int methodpintersect, int pintersectchar, int modengine, intvec u0) 708 860 { 709 861 int ppl = printlevel  voice +2; … … 711 863 def save = basering; 712 864 int n = nvars(save); 713 def DD = initialmalgrange(f,whichengine,methodord,1,u0); 714 setring DD; 715 ideal inI = inF; 716 kill inF; 717 poly s = t*Dt; 865 if (inorann == 0) // bfct using initial ideal 866 { 867 def D = initialmalgrange(f,whichengine,methodord,1,u0); 868 setring D; 869 ideal J = inF; 870 kill inF; 871 poly s = t*Dt; 872 } 873 else // bfct using Ann(f^s) 874 { 875 def D = SannfsBFCT(f,whichengine); 876 setring D; 877 ideal J = LD; 878 kill LD; 879 } 718 880 vector b; 719 881 // try it modular 720 if (method minpol <> 0) // minpolsyz721 { 722 if ( minpolchar == 0) // minpolsyz::modular882 if (methodpintersect <> 0) // pintersectsyz 883 { 884 if (pintersectchar == 0) // pintersectsyz::modular 723 885 { 724 886 int lb = 30000; … … 734 896 usedprimes = usedprimes,q; 735 897 dbprint(ppl,"used prime is: "+string(q)); 736 b = minpolsyz(s,inI,q,whichengine,modengine);898 b = pintersectsyz(s,J,q,whichengine,modengine); 737 899 } 738 900 i++; 739 901 } 740 902 } 741 else // minpolsyz::nonmodular742 { 743 b = minpolsyz(s,inI,0,whichengine);744 } 745 } 746 else // minpol: linreduce747 { 748 b = minpol(s,inI);903 else // pintersectsyz::nonmodular 904 { 905 b = pintersectsyz(s,J,0,whichengine); 906 } 907 } 908 else // pintersect: linreduce 909 { 910 b = pintersect(s,J); 749 911 } 750 912 setring save; 751 vector b = imap(DD,b); 752 list l = listofroots(b,1); 913 vector b = imap(D,b); 914 if (inorann == 0) 915 { 916 list l = listofroots(b,1); 917 } 918 else 919 { 920 list l = listofroots(b,0); 921 } 753 922 return(l); 754 923 } … … 758 927 RETURN: list of roots of the BernsteinSato polynomial bs(f) and their multiplicies 759 928 PURPOSE: compute the global BernsteinSato polynomial for a hypersurface, defined by f, according to the algorithm by Masayuki Noro 760 NOTE: In this proc, a system of linear equations is solved by linear reductions. 929 NOTE: In this proc, the initial Malgrange ideal is computed. 930 @* Further, a system of linear equations is solved by linear reductions. 761 931 @* If s<>0, @code{std} is used for Groebner basis computations, 762 932 @* otherwise, and by default, @code{slimgb} is used. … … 801 971 } 802 972 } 803 list b = bfctengine(f, whichengine,methodord,0,0,0,u0);973 list b = bfctengine(f,0,whichengine,methodord,0,0,0,u0); 804 974 return(b); 805 975 } … … 818 988 RETURN: list of roots of the BernsteinSato polynomial bs(f) and its multiplicies 819 989 PURPOSE: compute the global BernsteinSato polynomial for a hypersurface, defined by f, according to the algorithm by Masayuki Noro 820 NOTE: In this proc, a system of linear equations is solved by computing syzygies. 990 NOTE: In this proc, the initial Malgrange ideal is computed. 991 @* Further, a system of linear equations is solved by computing syzygies. 821 992 @* If r<>0, @code{std} is used for Groebner basis computations in characteristic 0, 822 993 @* otherwise, and by default, @code{slimgb} is used. 823 994 @* If s<>0, a matrix ordering is used for Groebner basis computations, 824 995 @* otherwise, and by default, a block ordering is used. 825 @* If t<>0, the minimal polynomial computation is solely performed over charasteristic 0,996 @* If t<>0, the computation of the intersection is solely performed over charasteristic 0, 826 997 @* otherwise and by default, a modular method is used. 827 998 @* If u<>0 and by default, @code{std} is used for Groebner basis computations in characteristic >0, … … 839 1010 // one for the engine used for Groebner basis computations in char 0, 840 1011 // one for M() ordering or its realization 841 // one for a modular method when computing the minimal polynomial1012 // one for a modular method when computing the intersection 842 1013 // and one for the engine used for Groebner basis computations in char >0 843 1014 // in # can also be the optional weight vector … … 846 1017 int whichengine = 0; // default 847 1018 int methodord = 0; // default 848 int minpolchar = 0; // default1019 int pintersectchar = 0; // default 849 1020 int modengine = 1; // default 850 1021 intvec u0 = 0; // default … … 865 1036 if (typeof(#[3])=="int"  typeof(#[3])=="number") 866 1037 { 867 minpolchar = int(#[3]);1038 pintersectchar = int(#[3]); 868 1039 } 869 1040 if (size(#)>3) … … 884 1055 } 885 1056 } 886 list b = bfctengine(f, whichengine,methodord,1,minpolchar,modengine,u0);1057 list b = bfctengine(f,0,whichengine,methodord,1,pintersectchar,modengine,u0); 887 1058 return(b); 888 1059 } … … 938 1109 s = s + w[i]*var(i)*var(n+i); 939 1110 } 940 vector b = minpol(s,J);1111 vector b = pintersect(s,J); 941 1112 list l = listofroots(b,0); 942 1113 return(l); … … 994 1165 setring DDh; 995 1166 dbprint(ppl, "the initial ideal:", string(matrix(inF))); 996 inF = nselect(inF,3..2*n+4); 997 inF = nselect(inF,1); 1167 intvec tonselect = 1; 1168 for (i=3; i<=2*n+4; i++) 1169 { 1170 tonselect = tonselect,i; 1171 } 1172 inF = nselect(inF,tonselect); 998 1173 dbprint(ppl, "generators containing only s:", string(matrix(inF))); 999 1174 inF = engine(inF, whichengine); // is now a principal ideal; … … 1013 1188 bfctonestep(f); 1014 1189 bfctonestep(f,1,1); 1190 } 1191 1192 proc bfctann (poly f, list #) 1193 "USAGE: bfctann(f [,r]); f a poly, r an optional int 1194 RETURN: list of roots of the BernsteinSato polynomial bs(f) and their multiplicies 1195 PURPOSE: compute the global BernsteinSato polynomial for a hypersurface, defined by f 1196 NOTE: In this proc, ann(f^s) is computed. 1197 @* If r<>0, @code{std} is used for Groebner basis computations, 1198 @* otherwise, and by default, @code{slimgb} is used. 1199 @* If printlevel=1, progress debug messages will be printed, 1200 @* if printlevel>=2, all the debug messages will be printed. 1201 EXAMPLE: example bfctann; shows examples 1202 " 1203 { 1204 def save = basering; 1205 int ppl = printlevel  voice + 2; 1206 int whichengine = 0; // default 1207 if (size(#)>0) 1208 { 1209 if (typeof(#[1])=="int"  typeof(#[1])=="number") 1210 { 1211 whichengine = int(#[1]); 1212 } 1213 } 1214 list b = bfctengine(f,1,whichengine,0,1,0,0,0); 1215 return(b); 1216 } 1217 example 1218 { 1219 "EXAMPLE:"; echo = 2; 1220 ring r = 0,(x,y),dp; 1221 poly f = x^2+y^3+x*y^2; 1222 bfctann(f); 1015 1223 } 1016 1224 … … 1037 1245 bfct(xyzreiffen45); 1038 1246 } 1247
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