Changeset e372e5a in git
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 May 7, 2013, 6:00:08 PM (10 years ago)
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 (u'jengelhdatetime', 'ceac47cbc86fe4a15902392bdbb9bd2ae0ea02c6')(u'spielwiese', 'a800fe4b3e9d37a38c5a10cc0ae9dfa0c15a4ee6')
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 47fff556be2c2414e0228fdd969a7369db9d35fd
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 6387507201165a90fa7face76757ff5d55772233
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Singular/LIB/grobcov.lib
r638750 re372e5a 13 13 \"Groebner Bases for Polynomial Systems with parameters\". 14 14 Journal of Symbolic Computation 45 (2010) 13911425. 15 The locus algorithm and definitions will be 16 published in a forthcoming paper by Abanades, Botana, Montes, Recio: 17 \''Geometrical locus of points using the Groebner cover\''. 15 18 16 19 The central routine is grobcov. Given a parametric … … 117 120 Normal, Special, Accumulation point, Degenerate. 118 121 The output are the components given in Pcanonical form 119 of at most 4 constructible sets: Normal, Special, Accumulation, 120 Degenerate. 122 of two constructible sets: Normal, and NonNormal 123 The Normal Set has Normal and Special components 124 The NonNormal set has Accumulation and Degenerate components. 121 125 The description of the algorithm and definitions will be 122 given in a forthcoming paper by Abanades, Botana, Montes Recio. 126 given in a forthcoming paper by Abanades, Botana, Montes, Recio: 127 ''Geometrical locus of points using the Groebner cover'' 123 128 124 129 locusto: Transforms the output of locus to a string that … … 1894 1899 if(tes==0){break;} 1895 1900 m++; 1896 } //"T_BB="; BB;1901 } 1897 1902 crep=PtoCrep(prep); 1898 1903 if(tes==0) … … 3819 3824 // AddCons: given a set of locally closed components of a selection of 3820 3825 // segments of the Grobner cover, it builds the canonical Prepresentation 3821 // of the whole set.3826 // of the corresponding constructible set, including levels it they are 3822 3827 // input: a list L of a selection of segments of the Groebner cover 3823 3828 // given a a set of components of the form … … 3828 3833 // First step: Selecting the top varieties 3829 3834 list L1; list L2; list LL; int i; int j; int t; 3830 //"T_before first step L="; L;3831 3835 list Lend; 3832 3836 for(i=1;i<=size(L);i++) … … 3839 3843 if(containedideal(L[j][1],L[i][1])==1) 3840 3844 {t=0; 3841 // " l'ideal "; L[j][1]; "esta contingut a l'ideal "; L[i][1];3845 // "the ideal "; L[j][1]; "is contained in ideal "; L[i][1]; 3842 3846 j=size(L); 3843 3847 } … … 3862 3866 { 3863 3867 LL=addtolocalclosed(L1,L2); 3864 //"T_LL="; LL;3865 3868 for(i=1;i<=size(LL[1]);i++) 3866 3869 { … … 3959 3962 {alreadyadded=sort(elimrepeatedvl(alreadyadded))[1];} 3960 3963 if(Added==intvec(0)){Added=alreadyadded;} 3961 else{Added=sort(elimrepeatedvl(Added,alreadyadded))[1];} 3964 else{ 3965 Added=Added,alreadyadded; 3966 Added=sort(elimrepeatedvl(Added))[1]; 3967 } 3962 3968 h=1,1; 3963 3969 while((h[1]==1) and (Added!=intvec(0))) … … 4016 4022 4017 4023 // locus(G): Special routine for determining the locus of points 4018 // of objects. Given a parametric ideal J with 4019 // parameters (a_1,..a_m) and variables (x_1,..,xn), 4020 // representing the system determining 4021 // the locus of points (a_1,..,a_m)) who verify certain 4022 // properties, computing the grobcov G of 4023 // J and applying to it locus, determines the different 4024 // classes of locus components. They can be 4025 // Normal, Special, Accumulation point, Degenerate. 4026 // The output are the components given in Pcanonical form 4027 // of at most 4 constructible sets: Normal, Special, Accumulation, 4028 // Degenerate. 4029 // The description of the algorithm and definitions will be 4030 // given in a forthcoming paper by Abanades, Botana, Montes Recio. 4031 4032 // input: 4024 // of objects. Given a parametric ideal J with 4025 // parameters (a_1,..a_m) and variables (x_1,..,xn), 4026 // representing the system determining 4027 // the locus of points (a_1,..,a_m)) who verify certain 4028 // properties, computing the grobcov G of 4029 // J and applying to it locus, determines the different 4030 // classes of locus components. They can be 4031 // Normal, Special, Accumulation point, Degenerate. 4032 // The output are the components given in Pcanonical form 4033 // of at most 4 constructible sets: Normal, Special, Accumulation, 4034 // Degenerate. 4035 // The description of the algorithm and definitions will be 4036 // given in a forthcoming paper by Abanades, Botana, Montes Recio. 4037 4038 // input: The output G of the grobcov (in generic representation, which is the default) 4039 // It allows an extra pair of arguments as option: locus(G,"minus",N) 4040 // where N is the ideal representing some excluded point of the mover 4041 // if there is some point that the mover should avoid. 4033 4042 // output: 4034 // list, the canonical Prepresentation of the 4 types of the locus: 4043 // list, the canonical Prepresentation of the Normal and NonNormal locus: 4044 // The Normal locus has two kind of components: Normal and Special. 4045 // The Nonnormal locus has two kind of components: Accumulation and Degenerate. 4035 4046 // Normal components: for each point in the component, 4036 4047 // the number of solutions in the variables is finite, and … … 4046 4057 // The output components are given as 4047 4058 // ((p1,(p11,..p1s_1),type_1,level_1),..,(pk,(pk1,..pks_k),type_k,level_k) 4048 // The components are given in four groups: normal, special, accumulation, degenerate 4049 // and each of these groups represent the canonical Prepresentation of the subset. 4059 // The components are given in canonical Prepresentation of the subset. 4050 4060 // If all levels of a class of locus are 1, then the set is locally closed. Otherwise the level 4051 4061 // gives the depth of the component. 4052 proc locus(list G )4062 proc locus(list GG,list #) 4053 4063 "USAGE: locus(G); 4054 4064 The input must be the grobcov of a parametrical ideal 4055 RETURN: The locus. 4056 The output components are given as a list of (pi,(pi1,..pis_i),type_i,level_i) varying i. 4057 NOTE: It can only be called after computing the grobcov of the 4058 parametrical ideal in generic representation ('ext',0), 4059 which is the default. 4060 The basering R, must be of the form Q[a_1,..,a_m][x_1,..,x_n]. 4065 RETURN: The locus. 4066 The output are the components of two constructible subsets of the locus 4067 of the parametrical system.: Normal and Nonnormal. 4068 These components are 4069 given as a list of (pi,(pi1,..pis_i),type_i,level_i) varying i, 4070 where the p's are prime ideals, the type can be: Normal, Special, 4071 Accumulation, Degenerate. 4072 NOTE: It can only be called after computing the grobcov of the 4073 parametrical ideal in generic representation ('ext',0), 4074 which is the default. 4075 The basering R, must be of the form Q[a_1,..,a_m][x_1,..,x_n]. 4061 4076 KEYWORDS: geometrical locus, locus, loci. 4062 4077 EXAMPLE: locus; shows an example" 4063 4078 { 4064 4079 int t1=1; int t2=1; 4080 list Opc=#; 4065 4081 def R=basering; 4066 4082 setglobalrings(); 4067 4083 string ty; 4068 4084 list G1; list G2; 4069 int i; int d; int j; int k; 4085 int i; int d; int j; int k; int i0=1; ideal minu; int te; 4086 for(i=1;i<=size(Opc) div 2;i++) 4087 { 4088 if(Opc[2*i1]=="minus") 4089 { 4090 minu=Opc[2*i]; 4091 setring(@RP); 4092 list nGP; 4093 def GP=imap(R,GG); 4094 ideal BP; 4095 def minup=imap(R,minu); 4096 for(j=1;j<=size(GP);j++) 4097 { 4098 te=1; k=1; 4099 BP=std(GP[j][2]); 4100 while((te==1) and (k<=size(minup))) 4101 { 4102 if((reduce(minup[k],BP))!=0){te=0;} 4103 k++; 4104 } 4105 if(te==0){nGP[size(nGP)+1]=GP[j];} 4106 } 4107 setring(R); 4108 GG=imap(@RP,nGP); 4109 } 4110 } 4111 list G; 4112 for(i=i0;i<=size(GG);i++) 4113 { 4114 G[i+1i0]=GG[i]; 4115 } 4070 4116 for(i=1;i<=size(G);i++) 4071 4117 { … … 4095 4141 { 4096 4142 attrib(G1RP[i][3][k][1],"IsSB",1); 4097 G1RP[i][3][k][1]=std(G1RP[i][3][k][1]); //"T_G1RP[i][2]="; G1RP[i][2];4143 G1RP[i][3][k][1]=std(G1RP[i][3][k][1]); 4098 4144 for(j=1;j<=size(G1RP[i][2]);j++) 4099 4145 { … … 4123 4169 } 4124 4170 setring @P; 4171 // CANVIAR 4125 4172 if(t1==1) 4126 4173 { 4127 def P1P=imap(R,P1); 4128 list CN; list CS; 4129 for(i=1;i<=size(P1P);i++) 4130 { 4131 d=dim(std(P1P[i][1])); 4132 if(d==0){P1P[i][3]="Normal";} 4133 if(P1P[i][3]=="Normal"){CN[size(CN)+1]=P1P[i];} 4134 else {CS[size(CS)+1]=P1P[i];} 4135 } 4136 def LN=AddCons(CN); 4137 def LS=AddCons(CS); 4138 } 4139 else{list P1P; list CN; list CS; list C2; list LN; list LS;} 4174 def C1=imap(R,P1); 4175 def L1=AddCons(C1); 4176 } 4177 else{list C1; list L1; kill P1; list P1;} 4140 4178 if(t2==1) 4141 4179 { … … 4143 4181 def L2=AddCons(C2); 4144 4182 } 4145 else{list L2; list C2; list P2;} 4146 list LA; list LD; 4147 4183 else{list L2; list C2; kill P2; list P2;} 4148 4184 for(i=1;i<=size(L2);i++) 4149 4185 { … … 4152 4188 { 4153 4189 L2[i][3]=string("Accumulation,",L2[i][3]); 4154 LA[size(LA)+1]=L2[i]; 4155 } 4156 else{L2[i][3]=string("Degenerate,",L2[i][3]); LD[size(LD)+1]=L2[i];} 4157 } 4158 4159 if(t1==0){list LN;} 4160 else{for(i=1;i<=size(LS);i++){LN[size(LN)+1]=LS[i];}} 4190 } 4191 else{L2[i][3]=string("Degenerate,",L2[i][3]);} 4192 } 4193 list LN; 4194 if(t1==1) 4195 { 4196 for(i=1;i<=size(L1);i++){LN[size(LN)+1]=L1[i];} 4197 } 4161 4198 if(t2==1) 4162 4199 { 4163 for(i=1;i<=size(LA);i++){LN[size(LN)+1]=LA[i];} 4164 for(i=1;i<=size(LD);i++){LN[size(LN)+1]=LD[i];} 4200 for(i=1;i<=size(L2);i++){LN[size(LN)+1]=L2[i];} 4165 4201 } 4166 4202 setring(R); 4167 4203 def L=imap(@P,LN); 4204 for(i=1;i<=size(L);i++){if(size(L[i][2])==0){L[i][2]=ideal(1);}} 4168 4205 kill @R; kill @RP; kill @P; 4169 4206 return(L); … … 4174 4211 short=0; 4175 4212 ideal H=x^2+y^24,(b2)*xa*y+2*a,(ax)^2+(by)^21; 4213 "System H = "; H; " "; 4176 4214 def G=grobcov(H); 4177 "grobcov(H)="; G; " "; 4178 def Gp=locus(G); 4179 "locus(G)="; Gp; 4215 //"grobcov(H)="; G; " "; 4216 def L=locus(G); 4217 "locus(G)="; L; " "; 4218 kill R; 4219 ring R=(0,x5,x6),(x1,x2,x3,x4),dp; 4220 ideal S=(x13)^2+(x21)^29, 4221 (4x2)*(x33)+(x13)*(x41), 4222 (3x1)*(x3x1)+(4x2)*(x4x2), 4223 (4x4)*x5+(x33)*x6+3*x44*x3, 4224 (x5x1)^2+(x6x2)^2(x1x3)^2(x2x4)^2; 4225 "System S="; S;" "; 4226 def G=grobcov(S); 4227 // "grobcov(S)="; G; 4228 ideal N=x13,x24; 4229 "N="; N;" "; 4230 def L=locus(G,"minus",N); 4231 "locus(G,minus,N)="; L;" "; 4180 4232 } 4181 4233
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