Changeset 269b1a in git
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 Dec 8, 2000, 10:30:34 AM (22 years ago)
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 (u'spielwiese', '0d6b7fcd9813a1ca1ed4220cfa2b104b97a0a003')
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 5cfef2c9b6a5a17ba7edfb21ee59a14da6abb6e6
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 f9602b8f7b0a32b385ad995c2280574abe06cb5b
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Singular/LIB/paramet.lib
rf9602b r269b1a 1 // $Id: paramet.lib,v 1. 5 19990706 15:32:55Singular Exp $1 // $Id: paramet.lib,v 1.6 20001208 09:30:34 Singular Exp $ 2 2 // author : Thomas Keilen email: keilen@mathematik.unikl.de 3 // last change: 0 7.08.983 // last change: 05.12.2000 4 4 /////////////////////////////////////////////////////////////////////////////// 5 6 version="$Id: paramet.lib,v 1.5 19990706 15:32:55 Singular Exp $"; 5 version="$Id: paramet.lib,v 1.6 20001208 09:30:34 Singular Exp $"; 7 6 info=" 8 LIBRARY: paramet.lib PROCEDURES FOR PARAMETRIZATION OF PRIMARY 9 DECOMPOSITION, ETC. 10 11 PROCEDURES: 12 parametrize(I); parametrizes a prime ideal if possible via the 13 normalization 14 parametrizepd(I); calculates the primary decomp. and parametrizes 15 the components 7 LIBRARY: paramet.lib PROCEDURES FOR PARAMETRIZATIONS 8 AUTHOR: Thomas Keilen, email: keilen@mathematik.unikl.de 9 10 OVERVIEW: 11 A library to compute parametrizations of algebraic varieties (if possible) 12 with the aid of a primary decomposition, respectively to compute a 13 parametrization of a plane curve singularity with the aid of a 14 HamburgerNoether expansion. 15 16 PROCEDURES: 17 parametrize(I); parametrizes a prime ideal via the normalization 18 parametrizepd(I); calculates the prim.dec. and parametrizes the components 16 19 parametrizesing(f); parametrize an isolated plane curve singularity 17 20 "; … … 30 33 1 resp. 0 depending on whether the parametrization was successful 31 34 or not 32 NOTE: if successful, the basering is changed to the parametrization ring; 33 the result will be incorrect, if the parametrization needs more than 34 two variables 35 CREATE: If the parametrization is successful, the basering will be changed to 36 the parametrization ring, that is to the ring PR=0,(s,t),dp; 37 respectively PR=0,t(1..d),dp;, depending on the dimension of the 38 parametrized variety. 35 39 EXAMPLE: example parametrize; shows an example 36 40 " 37 41 { 38 def BAS=basering; 39 ideal newI=radical(std(I)); 40 int d=dim(std(newI)); 41 if (size(primdecGTZ(newI))==1) 42 { 43 list nor=normal(newI); 44 def N=nor[1]; 45 ring PR=0,(s,t),dp; 46 setring N; 47 // If the ideal is zero dimensional, the procedure works as well in good cases. 48 if ((size(norid)==0) or (d==0)) 49 { 50 // Map the parametrization to the parametrization basering PR. 51 setring PR; 52 map p=N,(s,t); 53 ideal para=p(normap); 54 export para; 55 // The ith list component contains the parametrization, the 56 // number of necessary variables, and the information, if 57 // the parametrization was successful. 58 list param=para,d,1; 42 intvec ov=option(get); 43 option(noredefine); 44 def BAS=basering; 45 ideal newI=radical(std(I)); 46 int d=dim(std(newI)); 47 if (size(primdecGTZ(newI))==1) 48 { 49 list nor=normal(newI); 50 def N=nor[1]; 51 if (d<=2) 52 { 53 ring PR=0,(s,t),dp; 54 } 55 else 56 { 57 ring PR=0,t(1..d),dp; 58 } 59 setring N; 60 // If the ideal is zero dimensional, the procedure works as well in good 61 // cases. 62 if ((size(norid)==0) or (d==0)) 63 { 64 // Map the parametrization to the parametrization basering PR. 65 setring PR; 66 map p=N,maxideal(1); 67 ideal para=p(normap); 68 export para; 69 // The ith list component contains the parametrization, the 70 // number of necessary variables, and the information, if 71 // the parametrization was successful. 72 list param=para,d,1; 59 73 // if (d==0) 60 74 // { 61 // Include sometime a test, whether the maximal ideal I is of the form75 // Include sometimes a test, whether the maximal ideal I is of the form 62 76 // (xa,yb,zc), since only then normap=(a,b,c). 63 77 // } 64 setring BAS; 65 export(PR); 66 keepring(PR); 67 } 68 else 69 { 70 list param=I,0,0; 71 } 72 } 73 else 74 { 75 setring BAS; 76 list param=I,0,0; 77 } 78 return(param); 78 setring BAS; 79 export(PR); 80 keepring(PR); 81 } 82 else 83 { 84 setring BAS; 85 list param=I,0,0; 86 } 87 } 88 else 89 { 90 setring BAS; 91 list param=I,0,0; 92 } 93 option(set,ov); 94 return(param); 79 95 } 80 96 example … … 82 98 ring RING=0,(x,y,z),dp; 83 99 ideal I=z2y2x2+x3; 84 parametrize(I); 100 parametrize(I);parametrize(I); 85 101 } 86 102 /////////////////////////////////////////////////////////////////////////////// … … 94 110 resp. 0, and 1 resp. 0 depending on whether the parametrization 95 111 of the component was successful or not 96 NOTE: the basering will be changed to PR=0,(s,t),dp 97 the result will be incorrect, if the parametrization needs more than two 98 variables 112 CREATE: If the parametrization is successful, the basering will be changed to 113 the parametrization ring, that is to the ring PR=0,(s,t),dp; 114 respectively PR=0,t(1..d),dp;, depending on the dimension of the 115 parametrized variety. 99 116 EXAMPLE: example parametrizepd; shows an example 100 117 " 101 118 { 102 list primary,no,nor,para,param; 103 def BAS=basering; 104 ring PR=0,(s,t),dp; 105 ideal max=s,t; 106 setring BAS; 107 primary=primdecGTZ(I); 108 for (int ii=1; ii<=size(primary); ii=ii+1) 109 { 110 no=normal(std(primary[ii][2])); 111 nor[ii]=no[1]; 112 def N=nor[ii]; 113 setring N; 114 int d=dim(std(norid)); 115 // Test if the normalization is K, K[s] or K[s,t]. Then give back the parametrization. 116 if (size(norid)==0) 117 { 118 setring PR; 119 map p=N,max; 120 para[ii]=p(normap); 121 // export para[ii]; 122 // list inter=para[ii],nvars(N),1; 123 list inter=para[ii],d,1; 119 intvec ov=option(get); 120 option(noredefine); 121 list primary,no,nor,para,param; 122 def BAS=basering; 123 int d=dim(std(I)); 124 if (d<=2) 125 { 126 ring PR=0,(s,t),dp; 127 } 128 else 129 { 130 ring PR=0,t(1..d),dp; 131 } 132 ideal max=maxideal(1); 133 setring BAS; 134 primary=primdecGTZ(I); 135 for (int ii=1; ii<=size(primary); ii=ii+1) 136 { 137 no=normal(std(primary[ii][2])); 138 nor[ii]=no[1]; 139 def N=nor[ii]; 140 setring N; 141 d=dim(std(norid)); 142 // Test if the normalization is K, K[s] or K[s,t]. 143 // Then give back the parametrization. 144 if (size(norid)==0) 145 { 146 setring PR; 147 map p=N,max; 148 para[ii]=p(normap); 149 // export para[ii]; 150 // list inter=para[ii],nvars(N),1; 151 list inter=para[ii],d,1; 124 152 // if (d==0) 125 153 // { … … 127 155 // (xa,yb,zc), since only then normap=(a,b,c). 128 156 // } 129 param[ii]=inter; 130 kill inter; 131 setring BAS; 132 } 133 else 134 { 135 setring PR; 136 list inter=0,0,0; 137 param[ii]=inter; 138 kill inter; 139 setring BAS; 140 } 141 } 142 export nor; 143 setring PR; 144 export PR; 145 keepring PR; 146 return(param); 157 param[ii]=inter; 158 kill inter; 159 setring BAS; 160 } 161 else 162 { 163 setring PR; 164 list inter=0,0,0; 165 param[ii]=inter; 166 kill inter; 167 setring BAS; 168 } 169 } 170 export nor; 171 setring PR; 172 export PR; 173 keepring PR; 174 option(set,ov); 175 return(param); 147 176 } 148 177 example … … 156 185 157 186 158 159 187 proc parametrizesing(poly f) 160 "USAGE: parametrizesing(); f a polynomial in two variables withordering ls or ds188 "USAGE: parametrizesing(); f a polynomial in two variables,ordering ls or ds 161 189 RETURN: a list containing the parametrizations of the different branches of the 162 190 singularity at the origin resp. 0, if f was not of the desired kind 163 NOTE: if successful, the basering is changed to ring 0,(x,y),ls; 191 CREATE: If the parametrization is successful, the basering will be changed to 192 the parametrization ring, that is to the ring 0,(x,y),ls; 164 193 EXAMPLE: example parametrizesing; shows an example 165 194 " 166 195 { 196 intvec ov=option(get); 197 option(noredefine); 167 198 list hn,para; 168 199 if (nvars(basering)==2 and … … 182 213 } 183 214 keepring basering; 215 option(set,ov); 184 216 return(para); 185 217 } … … 190 222 parametrizesing(f); 191 223 } 192 /////////////////////////////////////////////////////////////////////////////// 193 194 195 196 197 198 /////////////////////////////////////////////////////////////////////////////// 224 225 //////////////////////////////////////////////////////////////////////////// 226 227 228 229 230 231 /////////////////////////////////////////////////////////////////////////// 199 232 //////// Examples 200 /////////////////////////////////////////////////////////////////////////// ////233 /////////////////////////////////////////////////////////////////////////// 201 234 /* 202 235 … … 350 383 /// Example 7  wrong ring ordering 351 384 352 ring r=0,(x,y), lp;385 ring r=0,(x,y),dp; 353 386 poly f=x2y3; 354 387 parametrizesing(f); … … 356 389 357 390 /// To do: 358 /// 1) Make sure that the result of parametrize/parametrizepd is correct 359 /// for any number of variables needed. 360 /// 2) Let these two print more detailed failure reasons. 361 /// 3) Let these two check, if the input is inside a ring with global ordering. 362 /// 4) Include a better check, whether some variable in the normalization can 363 /// be dropped. 391 /// 2) Let parametrize/parametrizepd print more detailed failure reasons. 392 /// 3) Let these two check, if the input is inside a ring with global 393 /// ordering. 394 /// 4) Include a better check, whether some variable in the 395 /// normalization can be dropped. 396 /// 5) Drop ordering lp in parametrizesing, as soon as the link to 397 /// MuPAD allows to give back data in ordering ls. 398 364 399 365 400
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