1 | /////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: resolve.lib,v 1.5 2005-06-07 11:28:19 Singular Exp $"; |
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3 | category="Commutative Algebra"; |
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4 | info=" |
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5 | LIBRARY: resolve.lib Resolution of singularities (Desingularization) |
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6 | Algorithm of Villamayor |
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7 | AUTHORS: A. Fruehbis-Krueger, anne@mathematik.uni-kl.de, |
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8 | @* G. Pfister, pfister@mathematik.uni-kl.de |
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9 | |
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10 | MAIN PROCEDURES: |
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11 | blowUp(J,C[,W,E]) computes the blowing up of the variety V(J) inside V(W) in |
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12 | the center V(C) |
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13 | Center(J[,W,E]) computes 'Villamayor'-center for blow up |
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14 | resolve(J) computes the desingularization of the variety V(J) |
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15 | |
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16 | AUXILLARY PROCEDURES: |
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17 | blowUpBO(BO,C) computes the blowing up of the variety V(BO[1]) in the |
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18 | center V(C). BO is a list (basic object), C is an ideal |
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19 | createBO(J,W,E) creates basic object from input data |
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20 | CenterBO(BO) computes the center for the next blow-up of the |
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21 | given basic object |
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22 | Delta(BO) apply the Delta-operator of [Bravo,Encinas,Villamayor] |
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23 | DeltaList(BO) list of results of Delta^0 to Delta^bmax |
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24 | showBO(BO) prints the content of a BO in more human readable form |
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25 | "; |
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26 | LIB "elim.lib"; |
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27 | LIB "primdec.lib"; |
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28 | LIB "presolve.lib"; |
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29 | LIB "linalg.lib"; |
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30 | LIB "sing.lib"; |
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31 | /////////////////////////////////////////////////////////////////////////////// |
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32 | // Tasks: |
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33 | // 1) optimization of the local case |
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34 | // 2) optimization in Coeff |
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35 | // 3) change invariant to represent coeff=1 case |
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36 | // 4) create procedures marked by * in above list |
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37 | /////////////////////////////////////////////////////////////////////////////// |
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38 | |
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39 | proc createBO(ideal J,list #) |
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40 | "USAGE: createBO(J[,W][,E]); |
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41 | @* J,W = ideals |
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42 | @* E = list |
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43 | ASSUME: J = ideal containing W ( W = 0 if not specified) |
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44 | @* E = list of smooth hypersurfaces (e.g. exceptional divisors) |
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45 | RETURN: list BO representing a basic object : |
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46 | BO[1] ideal W, if W has been specified; ideal(0) otherwise |
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47 | BO[2] ideal J |
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48 | BO[3] intvec |
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49 | BO[4] the list E of exceptional divisors if specified; |
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50 | empty list otherwise |
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51 | BO[5] an ideal defining the identity map |
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52 | BO[6] an intvec |
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53 | BO[7] intvec |
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54 | BO[8] a matrix |
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55 | entries 3,5,6,7,8 are initialized appropriately for use of CenterBO |
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56 | and blowUpBO |
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57 | EXAMPLE: example createBO; shows an example |
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58 | " |
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59 | { |
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60 | ideal W; |
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61 | list E; |
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62 | ideal abb=maxideal(1); |
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63 | intvec v; |
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64 | intvec bvec; |
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65 | intvec w=-1; |
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66 | matrix intE; |
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67 | if(size(#)>0) |
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68 | { |
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69 | if(typeof(#[1])=="ideal") |
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70 | { |
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71 | W=#[1]; |
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72 | } |
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73 | if(typeof(#[1])=="list") |
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74 | { |
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75 | E=#[1]; |
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76 | } |
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77 | if(size(#)>1) |
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78 | { |
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79 | if((typeof(#[2])=="list") && (size(E)==0)) |
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80 | { |
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81 | E=#[2]; |
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82 | } |
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83 | if((typeof(#[2])=="ideal") && (size(W)==0)) |
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84 | { |
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85 | W=#[2]; |
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86 | } |
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87 | } |
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88 | } |
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89 | list BO=W,J,bvec,E,abb,v,w,intE; |
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90 | return(BO); |
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91 | } |
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92 | example |
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93 | {"EXAMPLE:"; |
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94 | echo = 2; |
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95 | ring R=0,(x,y,z),dp; |
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96 | ideal J=x2-y3; |
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97 | createBO(J,ideal(z)); |
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98 | } |
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99 | /////////////////////////////////////////////////////////////////////////////// |
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100 | |
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101 | proc blowUp(ideal J,ideal C,list #) |
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102 | "USAGE: blowUp(J,C[,W][,E]); |
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103 | @* J,C,W = ideals |
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104 | @* E = list |
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105 | ASSUME: J = ideal containing W ( W = 0 if not specified) |
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106 | @* C = ideal containing J |
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107 | @* E = list of smooth hypersurfaces (e.g. exceptional divisors) |
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108 | COMPUTE: the blowing up of W in C, the exceptional locus, the strict transform |
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109 | @* of J |
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110 | RETURN: list of size at most size(C), |
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111 | l[i] is a ring containing a basic object BO: |
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112 | BO[1] an ideal, say Wi, defining the ambient space of the i-th chart |
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113 | of the blowing up |
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114 | BO[2] an ideal defining the strict transform |
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115 | BO[3] intvec, the first integer b such that in the original object |
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116 | (Delta^b(BO[2]))==1 |
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117 | the subsequent integers have the same property for Coeff-Objects |
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118 | of BO[2] used when determining the center |
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119 | BO[4] the list of exceptional divisors |
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120 | BO[5] an ideal defining the map K[W] ----> K[Wi] |
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121 | BO[6] an intvec BO[6][j]=1 indicates that <BO[4][j],BO[2]>=1, i.e. the |
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122 | strict transform does not meet the j-th exceptional divisor |
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123 | BO[7] intvec, |
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124 | the index of the first blown-up object in the resolution process |
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125 | leading to this object for which the value of b was BO[3] |
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126 | the subsequent ones are the indices for the Coeff-Objects |
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127 | of BO[2] used when determining the center |
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128 | BO[8] a matrix indicating that BO[4][i] meets BO[4][j] by BO[8][i,j]=1 |
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129 | for i < j |
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130 | EXAMPLE: example blowUp; shows an example |
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131 | " |
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132 | { |
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133 | ideal W; |
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134 | list E; |
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135 | ideal abb=maxideal(1); |
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136 | intvec v; |
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137 | intvec bvec; |
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138 | intvec w=-1; |
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139 | matrix intE; |
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140 | if(size(#)>0) |
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141 | { |
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142 | if(typeof(#[1])=="ideal") |
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143 | { |
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144 | W=#[1]; |
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145 | } |
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146 | if(typeof(#[1])=="list") |
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147 | { |
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148 | E=#[1]; |
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149 | } |
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150 | if(size(#)>1) |
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151 | { |
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152 | if((typeof(#[2])=="list") && (size(E)==0)) |
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153 | { |
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154 | E=#[2]; |
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155 | } |
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156 | if((typeof(#[2])=="ideal") && (size(W)==0)) |
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157 | { |
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158 | W=#[2]; |
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159 | } |
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160 | } |
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161 | } |
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162 | list BO=W,J,bvec,E,abb,v,w,intE; |
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163 | int locaT; |
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164 | export locaT; |
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165 | list blow=blowUpBO(BO,C,0); |
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166 | kill locaT; |
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167 | return(blow); |
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168 | } |
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169 | example |
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170 | {"EXAMPLE:"; |
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171 | echo = 2; |
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172 | ring R=0,(x,y),dp; |
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173 | ideal J=x2-y3; |
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174 | ideal C=x,y; |
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175 | list blow=blowUp(J,C); |
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176 | def Q=blow[1]; |
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177 | setring Q; |
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178 | BO; |
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179 | } |
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180 | /////////////////////////////////////////////////////////////////////////////// |
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181 | |
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182 | proc Center(ideal J,list #) |
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183 | "USAGE: Center(J[,W][,E]) |
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184 | @* J,W = ideals |
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185 | @* E = list |
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186 | ASSUME: J = ideal containing W ( W = 0 if not specified) |
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187 | @* E = list of smooth hypersurfaces (e.g. exceptional divisors) |
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188 | COMPUTE: the center of the blow-up of J for the resolution algorithm |
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189 | of [Bravo,Encinas,Villamayor] |
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190 | EXAMPLE: example Center; shows an example |
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191 | " |
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192 | { |
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193 | ideal W; |
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194 | list E; |
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195 | ideal abb=maxideal(1); |
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196 | intvec v; |
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197 | intvec bvec; |
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198 | intvec w=-1; |
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199 | matrix intE; |
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200 | if(size(#)>0) |
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201 | { |
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202 | if(typeof(#[1])=="ideal") |
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203 | { |
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204 | W=#[1]; |
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205 | } |
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206 | if(typeof(#[1])=="list") |
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207 | { |
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208 | E=#[1]; |
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209 | } |
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210 | if(size(#)>1) |
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211 | { |
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212 | if((typeof(#[2])=="list") && (size(E)==0)) |
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213 | { |
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214 | E=#[2]; |
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215 | } |
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216 | if((typeof(#[2])=="ideal") && (size(W)==0)) |
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217 | { |
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218 | W=#[2]; |
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219 | } |
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220 | if(size(#)==3){bvec=#[3];} |
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221 | } |
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222 | } |
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223 | |
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224 | list BO=W,J,bvec,E,abb,v,w,intE,intvec(0); |
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225 | if(defined(invSat)){kill invSat;} |
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226 | list invSat=ideal(0),intvec(0); |
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227 | export(invSat); |
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228 | |
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229 | list re=CenterBO(BO); |
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230 | ideal cent=re[1]; |
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231 | return(cent); |
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232 | } |
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233 | example |
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234 | { "EXAMPLE:"; |
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235 | echo = 2; |
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236 | ring R=0,(x,y),dp; |
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237 | ideal J=x2-y3; |
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238 | Center(J); |
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239 | } |
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240 | /////////////////////////////////////////////////////////////////////////////// |
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241 | |
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242 | proc blowUpBO(list BO, ideal C,int e) |
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243 | "USAGE: blowUpBO (BO,C,e); |
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244 | @* BO = basic object, a list: ideal W, |
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245 | @* ideal J, |
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246 | @* intvec b, |
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247 | @* list Ex, |
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248 | @* ideal ab, |
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249 | @* intvec v, |
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250 | @* intvec w, |
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251 | @* matrix M |
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252 | @* C = ideal |
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253 | @* e = integer (0 usual blowing up, 1 deleting extra charts, 2 deleting |
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254 | @* no charts ) |
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255 | ASSUME: R = basering, a polynomial ring, W an ideal of R, |
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256 | @* J = ideal containing W, |
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257 | @* C = ideal containing J |
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258 | COMPUTE: the blowing up of BO[1] in C, the exeptional locus, the strict |
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259 | transform of BO[2] |
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260 | NOTE: blowUpBO may be applied to basic objects in the sense of |
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261 | @* [Bravo, Encinas, Villamayor] in the following referred to as BO and |
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262 | @* to presentations in the sense of [Bierstone, Milman] in the following |
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263 | @* referred to as BM. |
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264 | RETURN: a list l of length at most size(C), |
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265 | l[i] is a ring containing an object BO resp. BM: |
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266 | BO[1]=BM[1] an ideal, say Wi, defining the ambient space of the |
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267 | i-th chart of the blowing up |
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268 | BO[2]=BM[2] an ideal defining the strict transform |
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269 | BO[3] intvec, the first integer b such that in the original object |
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270 | (Delta^b(BO[2]))==1 |
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271 | the subsequent integers have the same property for Coeff-Objects |
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272 | of BO[2] used when determining the center |
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273 | BM[3] intvec, BM[3][i] is the assigned multiplicity of BM[2][i] |
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274 | BO[4]=BM[4] the list of exceptional divisors |
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275 | BO[5]=BM[5] an ideal defining the map K[W] ----> K[Wi] |
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276 | BO[6]=BM[6] an intvec BO[6][j]=1 indicates that <BO[4][j],BO[2]>=1, |
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277 | i.e. the strict transform does not meet the j-th exceptional |
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278 | divisor |
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279 | BO[7] intvec, |
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280 | the index of the first blown-up object in the resolution process |
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281 | leading to this object for which the value of b was BO[3] |
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282 | the subsequent ones are the indices for the Coeff-Objects |
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283 | of BO[2] used when determining the center |
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284 | BM[7] intvec, BM[7][i] is the index at which the (2i-1)st entry |
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285 | of the invariant first reached its current maximal value |
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286 | BO[8]=BM[8] a matrix indicating that BO[4][i] meets BO[4][j] by |
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287 | BO[8][i,j]=1 for i < j |
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288 | BO[9] empty |
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289 | BM[9] the invariant |
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290 | |
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291 | EXAMPLE: example blowUpBO; shows an example |
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292 | " |
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293 | { |
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294 | //--------------------------------------------------------------------------- |
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295 | // Initialization and sanity checks |
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296 | //--------------------------------------------------------------------------- |
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297 | def R0=basering; |
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298 | if(!defined(locaT)){int locaT;} |
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299 | if(locaT){poly pp=@p;} |
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300 | intvec v; |
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301 | int shortC=defined(shortcut); |
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302 | int invS=defined(invSat); |
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303 | int eq,hy; |
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304 | int extra,noDel,keepDiv; |
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305 | if(e==1){extra=1;} |
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306 | //---keeps all charts |
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307 | if(e==2){noDel=1;} |
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308 | //---this is only for curves and surfaces |
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309 | //---keeps all charts with relevant informations on the exceptional divisors |
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310 | if(e==3){keepDiv=1;} |
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311 | if( typeof(attrib(BO[2],"isEqui"))=="int" ) |
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312 | { |
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313 | eq=attrib(BO[2],"isEqui"); |
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314 | } |
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315 | if( typeof(attrib(BO[2],"isHy"))=="int" ) |
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316 | { |
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317 | hy=attrib(BO[2],"isHy"); |
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318 | } |
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319 | string newvar; |
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320 | int n=nvars(R0); |
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321 | int i,j,l,m,x,jj,ll,haveCenters,co; |
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322 | //---the center should neither be the whole space nor empty |
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323 | if((size(C)==0)||(deg(C[1])==0)) |
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324 | { |
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325 | list result=R0; |
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326 | return(result); |
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327 | } |
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328 | if(!defined(debugBlowUp)) |
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329 | { |
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330 | int debugBlowUp=0; |
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331 | } |
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332 | //--------------------------------------------------------------------------- |
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333 | // Drop unnecessary variables |
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334 | //--------------------------------------------------------------------------- |
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335 | //---step 1: substitution |
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336 | if(!((keepDiv)||(noDel))) |
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337 | { |
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338 | //!!! in case keepDiv and noDel: |
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339 | //!!! maybe simplify the situation by an appropriate coordinate change |
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340 | //!!! of this kind -- without dropping variables? |
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341 | list L=elimpart(BO[1]); |
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342 | if(size(L[2])!=0) |
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343 | { |
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344 | map psi=R0,L[5]; |
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345 | C=psi(C); |
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346 | BO=psi(BO); |
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347 | } |
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348 | |
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349 | if(size(BO[1])==0) |
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350 | { |
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351 | ideal LL; |
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352 | for(j=1;j<=size(BO[4]);j++) |
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353 | { |
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354 | LL=LL,BO[4][j]; |
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355 | } |
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356 | LL=findvars(LL); |
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357 | L=elimpart(BO[2]); |
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358 | if((size(L[2])!=0)&&(size(std(LL+L[2]))==size(L[2])+size(LL))) |
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359 | { |
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360 | map chi=R0,L[5]; |
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361 | C=chi(C); |
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362 | BO=chi(BO); |
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363 | } |
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364 | } |
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365 | //---step 2: dropping non-occurring variables |
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366 | int s=size(C); |
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367 | ideal K=BO[1],BO[2],C; |
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368 | for(j=1;j<=size(BO[4]);j++) |
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369 | { |
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370 | K=K,BO[4][j]; |
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371 | } |
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372 | list N=findvars(K,0); |
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373 | if(size(N[1])<n) |
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374 | { |
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375 | newvar=string(N[1]); |
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376 | v=N[4]; |
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377 | for(j=1;j<=size(v);j++){BO[5]=subst(BO[5],var(v[j]),0);} |
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378 | execute("ring R1=("+charstr(R0)+"),("+newvar+"),dp;"); |
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379 | list BO=imap(R0,BO); |
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380 | ideal C=imap(R0,C); |
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381 | n=nvars(R1); |
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382 | } |
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383 | else |
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384 | { |
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385 | def R1=basering; |
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386 | } |
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387 | } |
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388 | else |
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389 | { |
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390 | int s=size(C); |
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391 | def R1=basering; |
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392 | } |
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393 | if(debugBlowUp) |
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394 | { |
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395 | "---> In BlowUp: After dropping unnecessary variables"; |
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396 | "BO:"; |
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397 | BO; |
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398 | "C:"; |
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399 | C; |
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400 | } |
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401 | //--------------------------------------------------------------------------- |
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402 | // Do the actual blow-up |
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403 | //--------------------------------------------------------------------------- |
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404 | //--- control the names of the variables |
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405 | execute("ring R=("+charstr(R0)+"),(x(1..n)),dp;"); |
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406 | list BO=fetch(R1,BO); |
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407 | ideal C=fetch(R1,C); |
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408 | list Cmstd=mstd(C); |
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409 | C=Cmstd[2]; |
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410 | if(size(Cmstd[1])<=size(Cmstd[2])) |
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411 | { |
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412 | C=Cmstd[1]; |
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413 | } |
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414 | else |
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415 | { |
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416 | C=interred(C); |
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417 | } |
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418 | list result; |
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419 | //--- the blow-up process |
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420 | ideal W =BO[1]; |
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421 | ideal J =BO[2]; |
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422 | intvec bvec =BO[3]; |
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423 | list Ex =BO[4]; |
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424 | ideal abb=BO[5]; |
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425 | intvec wvec=BO[7]; |
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426 | ideal laM=maxideal(1); |
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427 | if((typeof(BO[9])=="intmat")||(typeof(BO[9])=="intvec")) |
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428 | { |
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429 | def @invmat=BO[9]; |
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430 | } |
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431 | if(size(BO)>9) |
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432 | { |
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433 | //--- check whether a previous center had been split into connected components |
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434 | if(size(BO[10])>0) |
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435 | { |
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436 | list knownCenters=BO[10]; |
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437 | haveCenters=1; |
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438 | } |
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439 | } |
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440 | |
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441 | matrix intE=BO[8]; |
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442 | Ex[size(Ex)+1]=var(1); |
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443 | //to have the list depending on R in case BO[4] is empty |
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444 | |
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445 | execute("ring S=("+charstr(R)+"),("+varstr(R)+",y(0..s-1)),dp;"); |
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446 | list resu; |
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447 | list B; |
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448 | execute("ring T=("+charstr(R)+"),("+varstr(R)+",y(0..s-1),t),dp;"); |
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449 | ideal C=imap(R,C); |
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450 | ideal W=imap(R,W); |
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451 | execute("map phi=S,"+varstr(R)+",t*C;") |
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452 | setring S; |
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453 | |
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454 | //--- the ideal describing the blow-up map |
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455 | ideal abb=imap(R,abb); |
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456 | ideal laM0=imap(R,laM); |
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457 | //--- the ideal of the blowing up of the ambient space |
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458 | ideal W=preimage(T,phi,W); |
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459 | //--- the ideal of the exceptional locus |
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460 | ideal E=imap(R,C); |
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461 | |
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462 | list E1=imap(R,Ex); |
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463 | E1[size(E1)]=E; |
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464 | ideal J=imap(R,J)+W; |
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465 | |
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466 | if(haveCenters){list kN=imap(R,knownCenters);} |
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467 | |
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468 | //--- the strict transform of the exceptional divisors |
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469 | for(j=1;j<size(E1);j++){E1[j]=sat(E1[j]+W,E)[1];} |
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470 | //--- the intersection matrix of the exceptional divisors |
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471 | matrix intEold=imap(R,intE); |
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472 | matrix intE[size(E1)][size(E1)]; |
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473 | ideal U,Jsub,sLstd; |
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474 | |
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475 | for(j=1;j<size(E1);j++) |
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476 | { |
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477 | for(l=j+1;l<=size(E1);l++) |
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478 | { |
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479 | if(deg(E1[j][1])==0) |
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480 | { |
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481 | if(l<size(E1)){intE[j,l]=intEold[j,l];} |
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482 | else {intE[j,l]=0;} |
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483 | } |
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484 | else |
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485 | { |
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486 | if(deg(E1[l][1])==0) |
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487 | { |
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488 | if(l<size(E1)){intE[j,l]=intEold[j,l];} |
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489 | } |
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490 | else |
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491 | { |
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492 | U= std(E1[l]+E1[j]); |
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493 | if(dim(U)>0){intE[j,l]=1;} |
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494 | else {intE[j,l]=0;} |
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495 | } |
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496 | } |
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497 | } |
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498 | } |
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499 | if(debugBlowUp) |
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500 | { |
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501 | "----> In BlowUp: After Blowing-up, before Clean-Up"; |
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502 | "W:"; |
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503 | W; |
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504 | "J:"; |
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505 | J; |
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506 | } |
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507 | //---------------------------------------------------------------------------- |
---|
508 | // generating and cleaning up the different charts |
---|
509 | //---------------------------------------------------------------------------- |
---|
510 | list M; |
---|
511 | map psi; |
---|
512 | list E2; |
---|
513 | ideal K,JJ,laM,LL,MM; |
---|
514 | n=nvars(S); |
---|
515 | list N; |
---|
516 | list Bstd; |
---|
517 | intvec delCharts,extraCharts; |
---|
518 | delCharts[s]=0; |
---|
519 | extraCharts[s]=0; |
---|
520 | ideal MA=y(0..s-1); |
---|
521 | list ZRes,ZlaM,ZsLstd; |
---|
522 | for(i=0;i<=s-1;i++) |
---|
523 | { |
---|
524 | if(haveCenters) |
---|
525 | { |
---|
526 | B[10]=kN; |
---|
527 | for(j=1;j<=size(kN);j++) |
---|
528 | { |
---|
529 | B[10][j][1]=subst(B[10][j][1],y(i),1); |
---|
530 | } |
---|
531 | } |
---|
532 | B[8]=intE; |
---|
533 | B[1]=std(subst(W,y(i),1)); |
---|
534 | if(deg(B[1][1])==0) |
---|
535 | { |
---|
536 | //--- subsets of the empty set are not really interesting! |
---|
537 | delCharts[i+1]=1; |
---|
538 | ZRes[i+1]=B; |
---|
539 | ZlaM[i+1]=laM; |
---|
540 | i++; |
---|
541 | continue; |
---|
542 | } |
---|
543 | Jsub=subst(J,y(i),1); |
---|
544 | attrib(Jsub,"isEqui",eq); |
---|
545 | attrib(Jsub,"isHy",hy); |
---|
546 | B[2]=Jsub; |
---|
547 | B[3]=bvec; |
---|
548 | for(j=1;j<size(E1);j++){E2[j]=subst(E1[j],y(i),1);} |
---|
549 | E2[size(E1)]=E+B[1]; |
---|
550 | B[4]=E2; |
---|
551 | M=elimpart(B[1]); |
---|
552 | B[5]=abb; |
---|
553 | laM=laM0; |
---|
554 | psi=S,maxideal(1); |
---|
555 | if(size(M[2])!=0) |
---|
556 | { |
---|
557 | psi=S,M[5]; |
---|
558 | B=psi(B); |
---|
559 | laM=psi(laM); |
---|
560 | } |
---|
561 | Jsub=B[2]; |
---|
562 | B[2]=sat(Jsub,std(psi(E)))[1]; |
---|
563 | if(!defined(MAtmp)){ideal MAtmp=MA;} |
---|
564 | MAtmp[i+1]=0; |
---|
565 | JJ=std(B[2]+MAtmp); |
---|
566 | if(deg(JJ[1])==0) |
---|
567 | { |
---|
568 | delCharts[i+1]=1; |
---|
569 | //--- the i-th chart will be marked for deleting because all informations |
---|
570 | //--- are already contained in the union of the remaining charts |
---|
571 | } |
---|
572 | else |
---|
573 | { |
---|
574 | if((eq)&&(dim(JJ)<dim(std(B[2])))) |
---|
575 | { |
---|
576 | extraCharts[i+1]=1; |
---|
577 | //--- compute the singular locus |
---|
578 | if((B[3][1]<=1)&&(hy)) |
---|
579 | //--- B[2] is a smooth hypersurface |
---|
580 | { |
---|
581 | sLstd=ideal(1); |
---|
582 | } |
---|
583 | else |
---|
584 | { |
---|
585 | Bstd=mstd(B[2]); |
---|
586 | if(n-dim(Bstd[1])>4) |
---|
587 | { |
---|
588 | //--- in this case the singular locus is too complicated |
---|
589 | sLstd=ideal(0); |
---|
590 | } |
---|
591 | JJ=Bstd[2]; |
---|
592 | attrib(JJ,"isEqui",eq); |
---|
593 | B[2]=JJ; |
---|
594 | sLstd=slocusE(B[2]); |
---|
595 | } |
---|
596 | m=0; |
---|
597 | if(deg(std(sLstd+MAtmp)[1])==0) |
---|
598 | { |
---|
599 | //--- the singular locus of B[2] is in the union of the remaining charts |
---|
600 | m=1; |
---|
601 | for(l=1;l<=size(B[4]);l++) |
---|
602 | { |
---|
603 | if(deg(std(B[2]+B[4][l]+MAtmp)[1])!=0) |
---|
604 | { |
---|
605 | //--- the exceptional divisor meets B[2] at the locus of MAtmp |
---|
606 | //--- we continue only if the option extra=1 and we have transversal |
---|
607 | //--- intersection |
---|
608 | m=0; |
---|
609 | break; |
---|
610 | } |
---|
611 | } |
---|
612 | } |
---|
613 | if(m) |
---|
614 | { |
---|
615 | //--- the i-th chart will be marked for deleting because all informations |
---|
616 | //--- are already contained in the union of the remaining charts |
---|
617 | delCharts[i+1]=1; |
---|
618 | } |
---|
619 | } |
---|
620 | } |
---|
621 | if(delCharts[i+1]==0) |
---|
622 | { |
---|
623 | MAtmp[i+1]=MA[i+1]; |
---|
624 | ZsLstd[i+1]=sLstd; |
---|
625 | } |
---|
626 | ZRes[i+1]=B; |
---|
627 | ZlaM[i+1]=laM; |
---|
628 | } |
---|
629 | //--------------------------------------------------------------------------- |
---|
630 | // extra = ignore uninteresting charts even if there is a normal |
---|
631 | // crosssing intersection in it |
---|
632 | //--------------------------------------------------------------------------- |
---|
633 | if(extra) |
---|
634 | { |
---|
635 | for(i=0;i<=s-1;i++) |
---|
636 | { |
---|
637 | if((delCharts[i+1]==0)&&(extraCharts[i+1])) |
---|
638 | { |
---|
639 | MAtmp[i+1]=0; |
---|
640 | B=ZRes[i+1]; |
---|
641 | sLstd=ZsLstd[i+1]; |
---|
642 | m=0; |
---|
643 | if(deg(std(sLstd+MAtmp)[1])==0) |
---|
644 | { |
---|
645 | //--- the singular locus of B[2] is in the union of the remaining charts |
---|
646 | m=1; |
---|
647 | for(l=1;l<=size(B[4]);l++) |
---|
648 | { |
---|
649 | if(deg(std(B[2]+B[4][l]+MAtmp)[1])!=0) |
---|
650 | { |
---|
651 | //--- the exceptional divisor meets B[2] at the locus of MAtmp |
---|
652 | //--- we continue only if the option extra=1 and we have transversal |
---|
653 | //--- intersection |
---|
654 | m=2; |
---|
655 | if(!transversalTB(B[2],list(B[4][l]),MAtmp)) |
---|
656 | { |
---|
657 | m=0;break; |
---|
658 | } |
---|
659 | } |
---|
660 | } |
---|
661 | } |
---|
662 | if(m) |
---|
663 | { |
---|
664 | if(m==1) |
---|
665 | { |
---|
666 | //--- the i-th chart will be marked for deleting because all informations |
---|
667 | //--- are already contained in the union of the remaining charts |
---|
668 | delCharts[i+1]=1; |
---|
669 | } |
---|
670 | else |
---|
671 | { |
---|
672 | //--- the option extra=1 and we have transversal intersection |
---|
673 | //--- we delete the chart in case of normal crossings |
---|
674 | if(normalCrossB(B[2],B[4],MAtmp)) |
---|
675 | { |
---|
676 | //--- in case of the option extra |
---|
677 | //--- the i-th chart will be marked for deleting because all informations |
---|
678 | //--- are already contained in the union of the remaining charts |
---|
679 | |
---|
680 | delCharts[i+1]=1; |
---|
681 | } |
---|
682 | } |
---|
683 | } |
---|
684 | if(delCharts[i+1]==0) |
---|
685 | { |
---|
686 | MAtmp[i+1]=MA[i+1]; |
---|
687 | } |
---|
688 | } |
---|
689 | } |
---|
690 | for(i=0;i<=s-1;i++) |
---|
691 | { |
---|
692 | if(!delCharts[i+1]){break;} |
---|
693 | } |
---|
694 | if(i==s){delCharts[s]=0;} |
---|
695 | } |
---|
696 | for(i=0;i<=s-1;i++) |
---|
697 | { |
---|
698 | B=ZRes[i+1]; |
---|
699 | laM=ZlaM[i+1]; |
---|
700 | if(noDel){delCharts[i+1]=0;} |
---|
701 | //---keeps chart if the exceptional divisor is not in any other chart |
---|
702 | if((delCharts[i+1])&&(keepDiv)) |
---|
703 | { |
---|
704 | for(j=1;j<=size(B[4]);j++) |
---|
705 | { |
---|
706 | if(deg(std(B[4][j])[1])>0) |
---|
707 | { |
---|
708 | x=0; |
---|
709 | for(l=0;l<=s-1;l++) |
---|
710 | { |
---|
711 | if((l!=i)&&(!delCharts[l+1])&&(deg(std(ZRes[l+1][4][j])[1])>0)) |
---|
712 | { |
---|
713 | x=1; |
---|
714 | break; |
---|
715 | } |
---|
716 | } |
---|
717 | if(!x) |
---|
718 | { |
---|
719 | delCharts[i+1]=0; |
---|
720 | //!!!evtl. diese Karten markieren und nicht weiter aufblasen??? |
---|
721 | break; |
---|
722 | } |
---|
723 | } |
---|
724 | } |
---|
725 | } |
---|
726 | //---keeps charts if the intersection of 2 divisors is not in any other chart |
---|
727 | if((delCharts[i+1])&&(keepDiv)) |
---|
728 | { |
---|
729 | for(j=1;j<=size(B[4])-1;j++) |
---|
730 | { |
---|
731 | for(l=j+1;l<=size(B[4]);l++) |
---|
732 | { |
---|
733 | if(deg(std(B[4][j]+B[4][l])[1])>0) |
---|
734 | { |
---|
735 | x=0; |
---|
736 | for(ll=0;ll<=s-1;ll++) |
---|
737 | { |
---|
738 | if((ll!=i)&&(!delCharts[ll+1])) |
---|
739 | { |
---|
740 | if(deg(std(ZRes[ll+1][4][j]+ZRes[ll+1][4][l])[1])>0) |
---|
741 | { |
---|
742 | x=1; |
---|
743 | break; |
---|
744 | } |
---|
745 | } |
---|
746 | } |
---|
747 | if(!x) |
---|
748 | { |
---|
749 | delCharts[i+1]=0; |
---|
750 | break; |
---|
751 | } |
---|
752 | } |
---|
753 | } |
---|
754 | if(!delCharts[i+1]){break;} |
---|
755 | } |
---|
756 | } |
---|
757 | //---keeps charts if the intersection of 3 divisors is not in any other chart |
---|
758 | if((delCharts[i+1])&&(keepDiv)) |
---|
759 | { |
---|
760 | for(j=1;j<=size(B[4])-2;j++) |
---|
761 | { |
---|
762 | for(l=j+1;l<=size(B[4])-1;l++) |
---|
763 | { |
---|
764 | for(ll=l+1;ll<=size(B[4]);ll++) |
---|
765 | { |
---|
766 | if(deg(std(B[4][j]+B[4][l]+B[4][ll])[1])>0) |
---|
767 | { |
---|
768 | x=0; |
---|
769 | for(jj=0;jj<=s-1;jj++) |
---|
770 | { |
---|
771 | if((jj!=i)&&(!delCharts[jj+1])) |
---|
772 | { |
---|
773 | if(deg(std(ZRes[jj+1][4][j] |
---|
774 | +ZRes[jj+1][4][l]+ZRes[jj+1][4][ll])[1])>0) |
---|
775 | { |
---|
776 | x=1; |
---|
777 | break; |
---|
778 | } |
---|
779 | } |
---|
780 | } |
---|
781 | if(!x) |
---|
782 | { |
---|
783 | delCharts[i+1]=0; |
---|
784 | break; |
---|
785 | } |
---|
786 | } |
---|
787 | } |
---|
788 | if(!delCharts[i+1]){break;} |
---|
789 | } |
---|
790 | if(!delCharts[i+1]){break;} |
---|
791 | } |
---|
792 | } |
---|
793 | if(delCharts[i+1]==0) |
---|
794 | { |
---|
795 | //--- try to decrease the number of variables by substitution |
---|
796 | if((!keepDiv)&&(!noDel)) |
---|
797 | { |
---|
798 | list WW=elimpart(B[1]); |
---|
799 | map phiW=basering,WW[5]; |
---|
800 | B=phiW(B); |
---|
801 | laM=phiW(laM); |
---|
802 | kill WW; |
---|
803 | kill phiW; |
---|
804 | if(size(B[1])==0) |
---|
805 | { |
---|
806 | LL=0; |
---|
807 | for(j=1;j<=size(B[4]);j++) |
---|
808 | { |
---|
809 | MM=std(B[4][j]); |
---|
810 | if(deg(MM[1])>0){LL=LL,MM;} |
---|
811 | } |
---|
812 | LL=findvars(LL); |
---|
813 | M=elimpart(B[2]); |
---|
814 | if((size(M[2])!=0)&&(size(std(LL+M[2]))==size(M[2])+size(LL))) |
---|
815 | { |
---|
816 | psi=S,M[5]; |
---|
817 | B=psi(B); |
---|
818 | laM=psi(laM); |
---|
819 | } |
---|
820 | } |
---|
821 | } |
---|
822 | //---- interreduce B[1],B[2] and all B[4][j] |
---|
823 | B[1]=interred(B[1]); |
---|
824 | B[2]=interred(B[2]); |
---|
825 | E2=B[4]; |
---|
826 | for(j=1;j<=size(E2);j++){E2[j]=interred(E2[j]);} |
---|
827 | B[4]=E2; |
---|
828 | v=0;v[size(E2)]=0; |
---|
829 | //--- mark those j for which B[4] does not meet B[2] |
---|
830 | for(j=1;j<=size(E2);j++) |
---|
831 | { |
---|
832 | K=E2[j],B[2]; |
---|
833 | K=std(K); |
---|
834 | if(deg(K[1])==0) |
---|
835 | { |
---|
836 | v[j]=1; |
---|
837 | } |
---|
838 | } |
---|
839 | B[6]=v; |
---|
840 | B[7]=wvec; |
---|
841 | //--- throw away variables which do not occur |
---|
842 | K=B[1],B[2],B[5]; //Aenderung!!! |
---|
843 | for(j=1;j<=size(B[4]);j++){K=K,B[4][j];} |
---|
844 | N=findvars(K,0); |
---|
845 | if(size(N[1])<n) |
---|
846 | { |
---|
847 | newvar=string(N[1]); |
---|
848 | v=N[4]; |
---|
849 | for(j=1;j<=size(v);j++) |
---|
850 | { |
---|
851 | B[5]=subst(B[5],var(v[j]),0); |
---|
852 | laM=subst(laM,var(v[j]),0); |
---|
853 | } |
---|
854 | execute("ring R2=("+charstr(S)+"),("+newvar+"),dp;"); |
---|
855 | list BO=imap(S,B); |
---|
856 | ideal laM=imap(S,laM); |
---|
857 | } |
---|
858 | else |
---|
859 | { |
---|
860 | def R2=basering; |
---|
861 | list BO=B; |
---|
862 | } |
---|
863 | ideal JJ=BO[2]; |
---|
864 | attrib(JJ,"isEqui",eq); |
---|
865 | attrib(JJ,"isHy",hy); |
---|
866 | BO[2]=JJ; |
---|
867 | //--- strict transforms of the known centers |
---|
868 | if(haveCenters) |
---|
869 | { |
---|
870 | ideal tt; |
---|
871 | list tList; |
---|
872 | for(j=1;j<=size(BO[10]);j++) |
---|
873 | { |
---|
874 | tt=std(BO[10][j][1]); |
---|
875 | if(deg(tt[1])>0) |
---|
876 | { |
---|
877 | tt=sat(tt,BO[4][size(BO[4])])[1]; |
---|
878 | } |
---|
879 | if((deg(tt[1])>0)&&(deg(std(tt+BO[2]+BO[1])[1])>0)) |
---|
880 | { |
---|
881 | tList[size(tList)+1]= |
---|
882 | list(tt,BO[10][j][2],BO[10][j][3],BO[10][j][4]); |
---|
883 | } |
---|
884 | } |
---|
885 | BO[10]=tList; |
---|
886 | kill tList; |
---|
887 | } |
---|
888 | //--- marking variables which do not occur in BO[1] and BO[2] |
---|
889 | //--- and occur in exactly one BO[4][j], which is the hyperplane given by |
---|
890 | //--- this variable |
---|
891 | //!!!! not necessarily in exactly one BO[4][j] |
---|
892 | list N=findvars(BO[1]+BO[2],0); |
---|
893 | if(size(N[1])<nvars(basering)) |
---|
894 | { |
---|
895 | v=N[4]; |
---|
896 | if(defined(H)){kill H;} |
---|
897 | if(defined(EE)){kill EE;} |
---|
898 | if(defined(vv)){kill vv;} |
---|
899 | list EE; |
---|
900 | intvec vv; |
---|
901 | ideal H=maxideal(1); |
---|
902 | for(j=1;j<=size(v);j++) |
---|
903 | { |
---|
904 | H[v[j]]=0; |
---|
905 | } |
---|
906 | H=std(H); |
---|
907 | for(l=1;l<=size(BO[4]);l++) |
---|
908 | { |
---|
909 | if(BO[6][l]==1){l++;continue;} |
---|
910 | if(size(ideal(reduce(BO[4][l],H)-BO[4][l]))==0) |
---|
911 | { |
---|
912 | //!!! need further cleanup: |
---|
913 | //!!! this part of the code is no longer used since it did not glue properly |
---|
914 | // BO[6][l]=2; |
---|
915 | EE[size(EE)+1]=BO[4][l]; |
---|
916 | vv[size(vv)+1]=l; |
---|
917 | } |
---|
918 | } |
---|
919 | if((size(vv)>dim(std(BO[2])))&&(deg(BO[2][1])>0)) |
---|
920 | { |
---|
921 | list BOtemp3=BO; |
---|
922 | BOtemp3[4]=EE; |
---|
923 | intvec ivtemp3; |
---|
924 | ivtemp3[size(BOtemp3[4])]=0; |
---|
925 | BOtemp3[6]=ivtemp3; |
---|
926 | BOtemp3[7][1]=-1; |
---|
927 | list iEtemp3=inters_E(BOtemp3); |
---|
928 | if(iEtemp3[2]>=dim(std(BOtemp3[2]))) |
---|
929 | { |
---|
930 | for(l=2;l<=size(vv);l++) |
---|
931 | { |
---|
932 | BO[6][vv[l]]=0; |
---|
933 | } |
---|
934 | } |
---|
935 | kill BOtemp3,ivtemp3,iEtemp3; |
---|
936 | } |
---|
937 | } |
---|
938 | list thisChart=ideal(0),i; |
---|
939 | export thisChart; |
---|
940 | |
---|
941 | //---------------------------------------------------------------------------- |
---|
942 | // export the basic object and append the ring to the list of rings |
---|
943 | //---------------------------------------------------------------------------- |
---|
944 | if(debugBlowUp) |
---|
945 | { |
---|
946 | "----> In BlowUp: Adding a single chart"; |
---|
947 | "BO:"; |
---|
948 | BO; |
---|
949 | } |
---|
950 | if(locaT) |
---|
951 | { |
---|
952 | map locaPhi=R0,laM; |
---|
953 | poly @p=locaPhi(pp); |
---|
954 | export(@p); |
---|
955 | } |
---|
956 | ideal lastMap=laM; |
---|
957 | export lastMap; |
---|
958 | if(invS){list invSat=imap(R0,invSat);export invSat;} |
---|
959 | if(defined(@invmat)){BO[9]=@invmat;} |
---|
960 | if(shortC){list shortcut=imap(R0,shortcut);export(shortcut);} |
---|
961 | export BO; |
---|
962 | result[size(result)+1]=R2; |
---|
963 | setring S; |
---|
964 | kill R2; |
---|
965 | } |
---|
966 | } |
---|
967 | setring R0; |
---|
968 | return(result); |
---|
969 | } |
---|
970 | example |
---|
971 | {"EXAMPLE:"; |
---|
972 | echo = 2; |
---|
973 | ring R=0,(x,y),dp; |
---|
974 | |
---|
975 | ideal W; |
---|
976 | ideal J=x2-y3; |
---|
977 | intvec b=1; |
---|
978 | list E; |
---|
979 | ideal abb=maxideal(1); |
---|
980 | intvec v; |
---|
981 | intvec w=-1; |
---|
982 | matrix M; |
---|
983 | intvec ma; |
---|
984 | list BO=W,J,b,E,abb,v,w,M,ma; |
---|
985 | |
---|
986 | ideal C=CenterBO(BO)[1]; |
---|
987 | |
---|
988 | list blow=blowUpBO(BO,C,0); |
---|
989 | def Q=blow[1]; |
---|
990 | setring Q; |
---|
991 | BO; |
---|
992 | } |
---|
993 | ////////////////////////////////////////////////////////////////////////////// |
---|
994 | static |
---|
995 | proc slocusE(ideal i) |
---|
996 | "Internal procedure - no help and no example available |
---|
997 | " |
---|
998 | { |
---|
999 | //--- do slocus in equidimensional case directly -- speed up |
---|
1000 | if(size(i)==0){return(ideal(1));} |
---|
1001 | if( typeof(attrib(i,"isEqui"))=="int" ) |
---|
1002 | { |
---|
1003 | if(attrib(i,"isEqui")==1) |
---|
1004 | { |
---|
1005 | ideal j=std(i); |
---|
1006 | if(deg(j[1])==0){return(j);} |
---|
1007 | int cod = nvars(basering)-dim(j); |
---|
1008 | i = i+minor(jacob(i),cod); |
---|
1009 | return(i); |
---|
1010 | } |
---|
1011 | } |
---|
1012 | return(slocus(i)); |
---|
1013 | } |
---|
1014 | /////////////////////////////////////////////////////////////////////////////// |
---|
1015 | static |
---|
1016 | proc inters_E(list BO) |
---|
1017 | "USAGE: inters_E(BO); |
---|
1018 | @* BO = basic object, a list: ideal W, |
---|
1019 | @* ideal J, |
---|
1020 | @* intvec b, |
---|
1021 | @* list Ex, |
---|
1022 | @* ideal ab, |
---|
1023 | @* intvec v, |
---|
1024 | @* intvec w, |
---|
1025 | @* matrix M |
---|
1026 | ASSUME: R = basering, a polynomial ring, W an ideal of R, |
---|
1027 | @* J = ideal containing W, |
---|
1028 | @* BO in the setting of case 2 of [Bravo,Encinas,Villamayor] |
---|
1029 | @* BO[4]=E, BO[4][1..count]=E^- |
---|
1030 | @* BO[7][1]=count |
---|
1031 | COMPUTE: (W,(P,1),E^+) in the notation of [Bravo,Encinas,Villamayor] |
---|
1032 | RETURN: a list l , |
---|
1033 | l[1]: P = product of ideals I(H_i1)+..+I(H_in) over all |
---|
1034 | n-tuples of indices i1..in from 1..count |
---|
1035 | l[2]: n = maximal number of H_i from E^- meeting J simultaneously |
---|
1036 | l[3]: maximal number of H_i from E meeting J simultaneously |
---|
1037 | EXAMPLE: internal procedure - no example available |
---|
1038 | " |
---|
1039 | { |
---|
1040 | //--------------------------------------------------------------------- |
---|
1041 | // Initialization |
---|
1042 | //--------------------------------------------------------------------- |
---|
1043 | int kk,jj,ii,updated,n,count2,kkdiff; |
---|
1044 | def rb=basering; |
---|
1045 | def W=BO[1]; |
---|
1046 | ideal J=BO[1],BO[2]; |
---|
1047 | int nonnormal; |
---|
1048 | int maxkk=dim(std(J)); |
---|
1049 | int dimJ=maxkk; |
---|
1050 | ideal test2; |
---|
1051 | list merklist1,merklist2; |
---|
1052 | if(size(BO[4])==0) |
---|
1053 | { |
---|
1054 | list retlist=BO[2],n; |
---|
1055 | return(retlist); |
---|
1056 | } |
---|
1057 | def E=BO[4]; |
---|
1058 | intvec stoplist=BO[6]; |
---|
1059 | //--- fill in all known information about exceptional divisors not meeting |
---|
1060 | //--- current chart |
---|
1061 | for(ii=1;ii<=size(E);ii++) |
---|
1062 | { |
---|
1063 | if(deg(std(E[ii])[1])==0) |
---|
1064 | { |
---|
1065 | stoplist[ii]=1; |
---|
1066 | } |
---|
1067 | } |
---|
1068 | |
---|
1069 | int count=BO[7][1]; |
---|
1070 | if(!defined(debug_Inters_E)) |
---|
1071 | { |
---|
1072 | int debug_Inters_E=0; |
---|
1073 | } |
---|
1074 | //--------------------------------------------------------------------- |
---|
1075 | // we only want to look at E^-, not at all of E |
---|
1076 | //--------------------------------------------------------------------- |
---|
1077 | if (count>-1) |
---|
1078 | { |
---|
1079 | if (count>0) |
---|
1080 | { |
---|
1081 | list E_new=E[1..count]; |
---|
1082 | count2=size(E); |
---|
1083 | } |
---|
1084 | else |
---|
1085 | { |
---|
1086 | list E_new; |
---|
1087 | count2=size(E); |
---|
1088 | } |
---|
1089 | } |
---|
1090 | else |
---|
1091 | { |
---|
1092 | list E_new=E; |
---|
1093 | count=size(E_new); |
---|
1094 | count2=count; |
---|
1095 | } |
---|
1096 | //--------------------------------------------------------------------- |
---|
1097 | // combinatorics is expensive in an interpreted language, |
---|
1098 | // we leave it to the kernel by translating it into monomial |
---|
1099 | // ideals in a new ring with variables t(i) |
---|
1100 | //--------------------------------------------------------------------- |
---|
1101 | string rstr="ring rcomb=0,(t(1.." + string(size(E)) + ")),dp;"; |
---|
1102 | execute(rstr); |
---|
1103 | ideal potid,potid2; |
---|
1104 | list monlist,comblist,merkmon; |
---|
1105 | for(kk=1;kk<=count;kk++) |
---|
1106 | { |
---|
1107 | if(stoplist[kk]==0) |
---|
1108 | { |
---|
1109 | //**************************************************************************/ |
---|
1110 | // it does not make sense to intersect twice by the same E_i |
---|
1111 | // ===> reduce by t(i)^2 |
---|
1112 | //**************************************************************************/ |
---|
1113 | potid=potid,t(kk)^2; |
---|
1114 | } |
---|
1115 | else |
---|
1116 | { |
---|
1117 | //**************************************************************************/ |
---|
1118 | // it does not make sense to consider E_i with v[i]==1 ===> reduce by t(i) |
---|
1119 | //**************************************************************************/ |
---|
1120 | potid=potid,t(kk); |
---|
1121 | //**************************************************************************/ |
---|
1122 | // if stoplist[kk]==2 then J and all E_i automatically intersect E_kk |
---|
1123 | // hence we need not test it, but we have to lower maxkk by one |
---|
1124 | //**************************************************************************/ |
---|
1125 | if(stoplist[kk]==2) |
---|
1126 | { |
---|
1127 | maxkk=maxkk-1; |
---|
1128 | kkdiff++; // count these for dimension check later on |
---|
1129 | |
---|
1130 | } |
---|
1131 | } |
---|
1132 | } |
---|
1133 | |
---|
1134 | potid2=std(potid); |
---|
1135 | if(count2>count) |
---|
1136 | { |
---|
1137 | potid=potid,t((count+1)..count2); |
---|
1138 | for(kk=max(1,count);kk<=count2;kk++) |
---|
1139 | { |
---|
1140 | potid2=potid2,t(kk)^2; |
---|
1141 | } |
---|
1142 | } |
---|
1143 | potid=std(potid); |
---|
1144 | potid2=std(potid2); |
---|
1145 | for(kk=1;(((kk<=count)||(kk<=maxkk+1))&&(kk<=count2));kk++) |
---|
1146 | { |
---|
1147 | //------------------------------------------------------------------------- |
---|
1148 | // monlist[kk]=lists of kk entries of E_new, not containing an E_i twice, |
---|
1149 | // not containing an E_i where v[i]==1 |
---|
1150 | //------------------------------------------------------------------------- |
---|
1151 | monlist[kk]=redMax(kk,potid); |
---|
1152 | //*************************************************************************/ |
---|
1153 | // in the case of n<=maxkk we also need to know whether n would still be |
---|
1154 | // below this bound if we considered all of E instead of E_new |
---|
1155 | // ===> merkmon contains previously ignored tuples E_i1,..,E_im |
---|
1156 | //*************************************************************************/ |
---|
1157 | if(kk<=maxkk+1) |
---|
1158 | { |
---|
1159 | merkmon[kk]=redMax(kk,potid2); |
---|
1160 | merkmon[kk]=simplify(reduce(merkmon[kk],std(monlist[kk])),2); |
---|
1161 | } |
---|
1162 | } |
---|
1163 | if(debug_Inters_E) |
---|
1164 | { |
---|
1165 | "----> In Inters_E: the tuples"; |
---|
1166 | "tuples of E^-:"; |
---|
1167 | monlist; |
---|
1168 | "the remaining tuples:"; |
---|
1169 | merkmon; |
---|
1170 | } |
---|
1171 | //------------------------------------------------------------------------- |
---|
1172 | // check whether there is a kk-tuple of E_i intersecting J, |
---|
1173 | // kk running from 1 to count |
---|
1174 | //------------------------------------------------------------------------- |
---|
1175 | for(kk=1;kk<=count;kk++) |
---|
1176 | { |
---|
1177 | if(size(monlist[kk]==0)) break; |
---|
1178 | kill comblist; |
---|
1179 | list comblist; |
---|
1180 | //--- transscribe the tuples from monomial notation to intvec notation |
---|
1181 | for(jj=1;jj<=ncols(monlist[kk]);jj++) |
---|
1182 | { |
---|
1183 | comblist[jj]=leadexp(monlist[kk][jj]); |
---|
1184 | } |
---|
1185 | setring rb; |
---|
1186 | updated=0; |
---|
1187 | //------------------------------------------------------------------------ |
---|
1188 | // Do the intersections |
---|
1189 | //------------------------------------------------------------------------ |
---|
1190 | for(jj=1;jj<=size(comblist);jj++) |
---|
1191 | { |
---|
1192 | //--- jj-th tuple from list of tuples of kk E_i |
---|
1193 | test2=J; |
---|
1194 | for(ii=1;ii<=count;ii++) |
---|
1195 | { |
---|
1196 | if(comblist[jj][ii]==1) |
---|
1197 | { |
---|
1198 | test2=test2,E_new[ii]; |
---|
1199 | } |
---|
1200 | } |
---|
1201 | test2=std(test2); |
---|
1202 | //--- check whether this intersection is non-empty and store it accordingly |
---|
1203 | if(deg(test2[1])!=0) |
---|
1204 | { |
---|
1205 | //--- it is non-empty |
---|
1206 | if(updated!=0) |
---|
1207 | { |
---|
1208 | merklist1[size(merklist1)+1]=comblist[jj]; |
---|
1209 | } |
---|
1210 | else |
---|
1211 | { |
---|
1212 | kill merklist1; |
---|
1213 | list merklist1; |
---|
1214 | merklist1[1]=comblist[jj]; |
---|
1215 | updated=1; |
---|
1216 | n=kk; |
---|
1217 | } |
---|
1218 | if(dim(test2)!=maxkk-kk+kkdiff) |
---|
1219 | { |
---|
1220 | nonnormal=1; |
---|
1221 | } |
---|
1222 | } |
---|
1223 | else |
---|
1224 | { |
---|
1225 | //--- it is empty |
---|
1226 | merklist2[size(merklist2)+1]=jj; |
---|
1227 | } |
---|
1228 | } |
---|
1229 | setring rcomb; |
---|
1230 | ideal redid; |
---|
1231 | //--------------------------------------------------------------------- |
---|
1232 | // update monlist and merkmon by the knowledge what intersections are |
---|
1233 | // empty in the kk-th step |
---|
1234 | //--------------------------------------------------------------------- |
---|
1235 | for(jj=1;jj<=size(merklist2);jj++) |
---|
1236 | { |
---|
1237 | redid=redid,monlist[kk][merklist2[jj]]; |
---|
1238 | } |
---|
1239 | for(jj=kk+1;jj<=count;jj++) |
---|
1240 | { |
---|
1241 | monlist[jj]=simplify(reduce(monlist[jj],std(redid)),2); |
---|
1242 | if(jj<=maxkk+1) |
---|
1243 | { |
---|
1244 | merkmon[jj]=simplify(reduce(merkmon[jj],std(redid)),2); |
---|
1245 | } |
---|
1246 | } |
---|
1247 | kill redid; |
---|
1248 | kill merklist2; |
---|
1249 | list merklist2; |
---|
1250 | } |
---|
1251 | if(debug_Inters_E) |
---|
1252 | { |
---|
1253 | "----> In Inters_E: intersections found:"; |
---|
1254 | merklist1; |
---|
1255 | } |
---|
1256 | //--------------------------------------------------------------------- |
---|
1257 | // form the union of the intersections of the appropriate E_i |
---|
1258 | //--------------------------------------------------------------------- |
---|
1259 | setring rb; |
---|
1260 | ideal center,dummy; |
---|
1261 | list centlist; |
---|
1262 | for(kk=1;kk<=size(merklist1);kk++) |
---|
1263 | { |
---|
1264 | for(jj=1;jj<=size(merklist1[kk]);jj++) |
---|
1265 | { |
---|
1266 | if(merklist1[kk][jj]==1) |
---|
1267 | { |
---|
1268 | dummy=dummy,E_new[jj]; |
---|
1269 | } |
---|
1270 | } |
---|
1271 | if(size(center)==0) |
---|
1272 | { |
---|
1273 | center=dummy; |
---|
1274 | centlist[1]=dummy; |
---|
1275 | } |
---|
1276 | else |
---|
1277 | { |
---|
1278 | center=intersect(center,dummy); |
---|
1279 | centlist[size(centlist)+1]=dummy; |
---|
1280 | } |
---|
1281 | dummy=0; |
---|
1282 | } |
---|
1283 | if(debug_Inters_E) |
---|
1284 | { |
---|
1285 | "----> In Inters_E: intersection of E_i"; |
---|
1286 | "maximal number of E_i encountered in:"; |
---|
1287 | center; |
---|
1288 | "the components of this locus:"; |
---|
1289 | centlist; |
---|
1290 | "maximal number of E_i from E^- intersecting simultaneously:",n; |
---|
1291 | if(nonnormal) |
---|
1292 | { |
---|
1293 | "flag nonnormal is set"; |
---|
1294 | } |
---|
1295 | } |
---|
1296 | list retlist=center,n; |
---|
1297 | //------------------------------------------------------------------------- |
---|
1298 | // If n<=maxkk, then test if this is the case for all of E not just E_new |
---|
1299 | // using the pairs indicated by merkmon |
---|
1300 | //------------------------------------------------------------------------- |
---|
1301 | int ntotal=n; |
---|
1302 | if((n<=maxkk)&&(n<count2)&&(!nonnormal)) |
---|
1303 | { |
---|
1304 | //--- check kk-tuples |
---|
1305 | for(kk=n+1;kk<=maxkk+1;kk++) |
---|
1306 | { |
---|
1307 | setring rcomb; |
---|
1308 | //--- check if there are combinations to be checked |
---|
1309 | if(kk>size(merkmon)) |
---|
1310 | { |
---|
1311 | setring rb; |
---|
1312 | break; |
---|
1313 | } |
---|
1314 | if(size(merkmon[kk])!=0) |
---|
1315 | { |
---|
1316 | kill comblist; |
---|
1317 | list comblist; |
---|
1318 | //--- transscribe tuples from monomial notation to intvec notation |
---|
1319 | for(jj=1;jj<=size(merkmon[kk]);jj++) |
---|
1320 | { |
---|
1321 | comblist[jj]=leadexp(merkmon[kk][jj]); |
---|
1322 | } |
---|
1323 | setring rb; |
---|
1324 | //--- check jj-th tuple from the list of kk-tuples |
---|
1325 | for(jj=1;jj<=size(comblist);jj++) |
---|
1326 | { |
---|
1327 | test2=J; |
---|
1328 | for(ii=1;ii<=nvars(rcomb);ii++) |
---|
1329 | { |
---|
1330 | if(comblist[jj][ii]==1) |
---|
1331 | { |
---|
1332 | test2=test2,E[ii]; |
---|
1333 | } |
---|
1334 | } |
---|
1335 | test2=std(test2); |
---|
1336 | //--- as soon as we found one we can proceed to the subsequent kk |
---|
1337 | if(deg(test2[1])!=0) |
---|
1338 | { |
---|
1339 | ntotal=kk; |
---|
1340 | if(dim(test2)-kkdiff!=maxkk-kk) |
---|
1341 | { |
---|
1342 | nonnormal=2; |
---|
1343 | break; |
---|
1344 | } |
---|
1345 | } |
---|
1346 | } |
---|
1347 | //--- if we already know that too many E_i intersect simultaneously, |
---|
1348 | //--- we need not proceed any further |
---|
1349 | if(nonnormal) |
---|
1350 | { |
---|
1351 | break; |
---|
1352 | } |
---|
1353 | } |
---|
1354 | else |
---|
1355 | { |
---|
1356 | setring rb; |
---|
1357 | break; |
---|
1358 | } |
---|
1359 | } |
---|
1360 | } |
---|
1361 | //------------------------------------------------------------------------- |
---|
1362 | // update the result accordingly and return it |
---|
1363 | //------------------------------------------------------------------------- |
---|
1364 | if(maxkk<dimJ) |
---|
1365 | { |
---|
1366 | n=n+dimJ-maxkk; |
---|
1367 | ntotal=ntotal+dimJ-maxkk; |
---|
1368 | } |
---|
1369 | retlist[2]=n; |
---|
1370 | retlist[3]=ntotal; |
---|
1371 | if(n<=dimJ) |
---|
1372 | { |
---|
1373 | retlist[4]=centlist; |
---|
1374 | retlist[5]=merklist1; |
---|
1375 | if(nonnormal) |
---|
1376 | { |
---|
1377 | retlist[6]=nonnormal; |
---|
1378 | } |
---|
1379 | } |
---|
1380 | return(retlist); |
---|
1381 | } |
---|
1382 | /////////////////////////////////////////////////////////////////////////// |
---|
1383 | |
---|
1384 | proc Delta(list BO) |
---|
1385 | "USAGE: Delta (BO); |
---|
1386 | @* BO = basic object, a list: ideal W, |
---|
1387 | @* ideal J, |
---|
1388 | @* intvec b, |
---|
1389 | @* list Ex, |
---|
1390 | @* ideal ab, |
---|
1391 | @* intvec v, |
---|
1392 | @* intvec w, |
---|
1393 | @* matrix M |
---|
1394 | ASSUME: R = basering, a polynomial ring, W an ideal of R, |
---|
1395 | @* J = ideal containing W |
---|
1396 | COMPUTE: Delta-operator applied to J in the notation of |
---|
1397 | [Bravo,Encinas,Villamayor] |
---|
1398 | RETURN: ideal |
---|
1399 | EXAMPLE: example Delta; shows an example |
---|
1400 | " |
---|
1401 | { |
---|
1402 | //--------------------------------------------------------------------------- |
---|
1403 | // Initialization and sanity checks |
---|
1404 | //--------------------------------------------------------------------------- |
---|
1405 | ideal W=BO[1]; |
---|
1406 | ideal J=BO[2]; |
---|
1407 | ideal C=simplify(reduce(J,std(W)),2); |
---|
1408 | list LC; |
---|
1409 | int n=nvars(basering); |
---|
1410 | //--------------------------------------------------------------------------- |
---|
1411 | // Simple case: W is the empty set |
---|
1412 | //--------------------------------------------------------------------------- |
---|
1413 | if(size(W)==0) |
---|
1414 | { |
---|
1415 | C=C,jacob(J); |
---|
1416 | C=std(C); |
---|
1417 | return(C); |
---|
1418 | } |
---|
1419 | //--------------------------------------------------------------------------- |
---|
1420 | // General case: W is non-empty |
---|
1421 | // Step 1: Find a minor of the Jacobian of W which is not identically zero |
---|
1422 | // and look at the complement of the zero-set given by this minor; |
---|
1423 | // this leads to the system of local parameters |
---|
1424 | // Step 2: Form the derivatives w.r.t. this system of parameters |
---|
1425 | //--------------------------------------------------------------------------- |
---|
1426 | //--- Step 1 |
---|
1427 | list re=findMinor(W); |
---|
1428 | list L; |
---|
1429 | int ii,i,j,l,k; |
---|
1430 | J=C; |
---|
1431 | ideal D=ideal(1); |
---|
1432 | intvec v,w; |
---|
1433 | ideal V; |
---|
1434 | poly m; |
---|
1435 | |
---|
1436 | for(ii=1;ii<=size(re);ii++) |
---|
1437 | { |
---|
1438 | C=J; |
---|
1439 | L=re[ii]; |
---|
1440 | matrix A=L[1]; //(1/m)*A is the inverse matrix of the Jacobian of W |
---|
1441 | //corresponding to m |
---|
1442 | m=L[2]; //a k- minor of jacob(W), not identically zero |
---|
1443 | //k=n-dim(W) |
---|
1444 | V=L[3]; //the elements of W corresponding to m |
---|
1445 | v=L[4]; //the indices of variables corresponding to m |
---|
1446 | w=L[5]; //the indices of variables not corresponding to m |
---|
1447 | |
---|
1448 | //--- Step 2 |
---|
1449 | //--- first some initialization depending on results of step 1 |
---|
1450 | k=size(V); |
---|
1451 | matrix dg[1][k]; |
---|
1452 | matrix df[k][1]; |
---|
1453 | //--- derivatives of the generators of J w.r.t. system of parameters |
---|
1454 | for(i=1;i<=size(J);i++) |
---|
1455 | { |
---|
1456 | for(j=1;j<=n-k;j++) |
---|
1457 | { |
---|
1458 | for(l=1;l<=k;l++) |
---|
1459 | { |
---|
1460 | dg[1,l]=diff(V[l],var(w[j])); |
---|
1461 | df[l,1]=diff(J[i],var(v[l])); |
---|
1462 | } |
---|
1463 | C=C,m*diff(J[i],var(w[j]))-dg*A*df; |
---|
1464 | } |
---|
1465 | } |
---|
1466 | //--- everything should live in W, not just in the intersection of |
---|
1467 | //--- D(m) with W |
---|
1468 | C=C+W; |
---|
1469 | C=sat(C,m)[1]; |
---|
1470 | //--- intersect ideal with previously computed ones to make sure that no |
---|
1471 | //--- components are lost |
---|
1472 | D=intersect(D,C); |
---|
1473 | kill dg,df,A; |
---|
1474 | } |
---|
1475 | //--- return minimal set of generators of the result |
---|
1476 | list li=mstd(D); |
---|
1477 | D=li[2]; |
---|
1478 | if(size(li[1])<=size(D)){D=li[1];} |
---|
1479 | return(D); |
---|
1480 | } |
---|
1481 | example |
---|
1482 | { "EXAMPLE:"; |
---|
1483 | echo = 2; |
---|
1484 | ring R=0,(x,y,z),dp; |
---|
1485 | |
---|
1486 | ideal W=z^2-x; |
---|
1487 | ideal J=x*y^2+x^3; |
---|
1488 | intvec b=1; |
---|
1489 | list E; |
---|
1490 | ideal abb=maxideal(1); |
---|
1491 | intvec v; |
---|
1492 | intvec w=-1; |
---|
1493 | matrix M; |
---|
1494 | |
---|
1495 | list BO=W,J,b,E,abb,v,w,M; |
---|
1496 | |
---|
1497 | Delta(BO); |
---|
1498 | } |
---|
1499 | |
---|
1500 | ////////////////////////////////////////////////////////////////////////////// |
---|
1501 | static |
---|
1502 | proc redMax(int k,ideal J) |
---|
1503 | "Internal procedure - no help and no example available |
---|
1504 | " |
---|
1505 | { |
---|
1506 | //--- reduce maxideal(k) by J, more efficient approach |
---|
1507 | int i; |
---|
1508 | ideal K=simplify(reduce(maxideal(1),J),2); |
---|
1509 | for(i=2;i<=k;i++) |
---|
1510 | { |
---|
1511 | K=simplify(reduce(K*maxideal(1),J),2); |
---|
1512 | } |
---|
1513 | return(K); |
---|
1514 | } |
---|
1515 | |
---|
1516 | ////////////////////////////////////////////////////////////////////////////// |
---|
1517 | static |
---|
1518 | proc findMinor(ideal W) |
---|
1519 | "Internal procedure - no help and no example available |
---|
1520 | " |
---|
1521 | { |
---|
1522 | //--------------------------------------------------------------------------- |
---|
1523 | // Initialization and sanity checks |
---|
1524 | //--------------------------------------------------------------------------- |
---|
1525 | list L; |
---|
1526 | intvec v,w; |
---|
1527 | ideal Wstd=std(W); |
---|
1528 | int n=nvars(basering); // total number of columns of Jacobian |
---|
1529 | int k=n-dim(Wstd); // size of minors of Jacobian |
---|
1530 | int a=size(W); // total number of rows of Jacobian |
---|
1531 | matrix A[k][k]; |
---|
1532 | list LW=indexSet(a,k); // set of tuples of k rows |
---|
1533 | list LV=indexSet(n,k); // set of tuples of k columns |
---|
1534 | ideal IW,IV; |
---|
1535 | int i,j,l,e; |
---|
1536 | list re; |
---|
1537 | //--------------------------------------------------------------------------- |
---|
1538 | // We need to know which minor corresponds to which variable and to which |
---|
1539 | // generator of W - therefore we cannot use the function minor()! |
---|
1540 | //--------------------------------------------------------------------------- |
---|
1541 | //--- choose the generators which we want to differentiate |
---|
1542 | for(i=1;i<=size(LW);i++) |
---|
1543 | { |
---|
1544 | IW=0; |
---|
1545 | for(l=1;l<=a;l++) |
---|
1546 | { |
---|
1547 | if(LW[i][l]!=0){IW[size(IW)+1]=W[l];} |
---|
1548 | } |
---|
1549 | //--- choose the variables by which to differentiate and apply diff |
---|
1550 | for(j=1;j<=size(LV);j++) |
---|
1551 | { |
---|
1552 | IV=0;v=0;w=0; |
---|
1553 | for(l=1;l<=n;l++) |
---|
1554 | { |
---|
1555 | if(LV[j][l]!=0) |
---|
1556 | { |
---|
1557 | v[size(v)+1]=l; |
---|
1558 | IV[size(IV)+1]=var(l); |
---|
1559 | } |
---|
1560 | else |
---|
1561 | { |
---|
1562 | w[size(w)+1]=l; |
---|
1563 | |
---|
1564 | } |
---|
1565 | } |
---|
1566 | A=diff(IV,IW); // appropriate submatrix of Jacobian |
---|
1567 | //--- if the minor is non-zero, then it might be the one we need |
---|
1568 | //--- ==> put it in the list of candidates |
---|
1569 | if(det(A)!=0) |
---|
1570 | { |
---|
1571 | v=v[2..size(v)]; // first entry is zero for technical reasons |
---|
1572 | w=w[2..size(w)]; // first entry is zero for technical reasons |
---|
1573 | L=inverse_L(A); |
---|
1574 | L[3]=IW; |
---|
1575 | L[4]=v; |
---|
1576 | L[5]=w; |
---|
1577 | re[size(re)+1]=L; |
---|
1578 | } |
---|
1579 | } |
---|
1580 | } |
---|
1581 | //--------------------------------------------------------------------------- |
---|
1582 | // return the result |
---|
1583 | //--------------------------------------------------------------------------- |
---|
1584 | return(re); |
---|
1585 | } |
---|
1586 | |
---|
1587 | ///////////////////////////////////////////////////////////////////////////// |
---|
1588 | static |
---|
1589 | proc indexSet(int a, int b) |
---|
1590 | "Internal procedure - no help and no example available |
---|
1591 | " |
---|
1592 | { |
---|
1593 | //--------------------------------------------------------------------------- |
---|
1594 | // Find all tuples of size b containing pairwise distict elements from a |
---|
1595 | // list of a elements |
---|
1596 | //--------------------------------------------------------------------------- |
---|
1597 | //**************************************************************************/ |
---|
1598 | // Combinatorics is expensive in an interpreted language |
---|
1599 | // ==> shift it into the kernel |
---|
1600 | //**************************************************************************/ |
---|
1601 | def R=basering; |
---|
1602 | list L; |
---|
1603 | ring S=2,x(1..a),dp; |
---|
1604 | ideal I=maxideal(b); |
---|
1605 | int i; |
---|
1606 | ideal J=x(1)^2; |
---|
1607 | for(i=2;i<=a;i++){J=J,x(i)^2;} |
---|
1608 | attrib(J,"isSB",1); |
---|
1609 | I=reduce(I,J); |
---|
1610 | I=simplify(I,2); |
---|
1611 | for(i=1;i<=size(I);i++){L[i]=leadexp(I[i]);} |
---|
1612 | setring R; |
---|
1613 | return(L); |
---|
1614 | } |
---|
1615 | |
---|
1616 | ///////////////////////////////////////////////////////////////////////////// |
---|
1617 | |
---|
1618 | proc DeltaList(list BO) |
---|
1619 | "USAGE: DeltaList (BO); |
---|
1620 | @* BO = basic object, a list: ideal W, |
---|
1621 | @* ideal J, |
---|
1622 | @* intvec b, |
---|
1623 | @* list Ex, |
---|
1624 | @* ideal ab, |
---|
1625 | @* intvec v, |
---|
1626 | @* intvec w, |
---|
1627 | @* matrix M |
---|
1628 | ASSUME: R = basering, a polynomial ring, W an ideal of R, |
---|
1629 | @* J = ideal containing W |
---|
1630 | COMPUTE: Delta-operator iteratively applied to J in the notation of |
---|
1631 | [Bravo,Encinas,Villamayor] |
---|
1632 | RETURN: list l of length ((max w-ord) * b), |
---|
1633 | l[i+1]=Delta^i(J) |
---|
1634 | EXAMPLE: example DeltaList; shows an example |
---|
1635 | " |
---|
1636 | { |
---|
1637 | //---------------------------------------------------------------------------- |
---|
1638 | // Iteratively apply proc Delta |
---|
1639 | //---------------------------------------------------------------------------- |
---|
1640 | int i; |
---|
1641 | list L; |
---|
1642 | ideal C=BO[2]; |
---|
1643 | while(deg(C[1])!=0) |
---|
1644 | { |
---|
1645 | L[size(L)+1]=C; |
---|
1646 | C=Delta(BO); |
---|
1647 | BO[2]=C; |
---|
1648 | } |
---|
1649 | return(L); |
---|
1650 | } |
---|
1651 | example |
---|
1652 | { |
---|
1653 | "EXAMPLE:"; |
---|
1654 | echo = 2; |
---|
1655 | ring R=0,(x,y,z),dp; |
---|
1656 | |
---|
1657 | ideal W=z^2-x; |
---|
1658 | ideal J=x*y^2+x^3; |
---|
1659 | intvec b=1; |
---|
1660 | list E; |
---|
1661 | ideal abb=maxideal(1); |
---|
1662 | intvec v; |
---|
1663 | intvec w=-1; |
---|
1664 | matrix M; |
---|
1665 | |
---|
1666 | list BO=W,J,b,E,abb,v,w,M; |
---|
1667 | |
---|
1668 | DeltaList(BO); |
---|
1669 | } |
---|
1670 | ///////////////////////////////////////////////////////////////////////////// |
---|
1671 | proc CenterBM(list BM) |
---|
1672 | "USAGE: CenterBM(BM); |
---|
1673 | @* BM = object related to a presentation, |
---|
1674 | a list: ideal W, |
---|
1675 | @* ideal J, |
---|
1676 | @* intvec b, |
---|
1677 | @* list Ex, |
---|
1678 | @* ideal ab, |
---|
1679 | @* intvec v, |
---|
1680 | @* intvec w, |
---|
1681 | @* matrix M |
---|
1682 | ASSUME: R = basering, a polynomial ring, W an ideal of R, |
---|
1683 | @* J = ideal containing W |
---|
1684 | COMPUTE: the center of the next blow-up of BM in the resolution algorithm |
---|
1685 | of [Bierstone, Milman] |
---|
1686 | RETURN: list l, |
---|
1687 | l[1]: ideal describing the center |
---|
1688 | l[2]: intvec w obtained in the process of determining l[1] |
---|
1689 | l[3]: intvec b obtained in the process of determining l[1] |
---|
1690 | l[4]: intmat invmat obtained in the process of determining l[1] |
---|
1691 | EXAMPLE: example CenterBM; shows an example |
---|
1692 | " |
---|
1693 | { |
---|
1694 | //!!! NOCH NICHT IN BETRIEB |
---|
1695 | ERROR("Not implemented yet"); |
---|
1696 | int i,j; |
---|
1697 | intmat tmat[2][1]=0,-1; |
---|
1698 | //--- re=center,E^- indices, b vector, n vector |
---|
1699 | list re=ideal(1),BM[7],BM[3],tmat; |
---|
1700 | ideal J=BM[2]; |
---|
1701 | if(size(J)==0) |
---|
1702 | { |
---|
1703 | re[1]=ideal(0); |
---|
1704 | return(re); |
---|
1705 | } |
---|
1706 | //--- find Delta^(b-1)(J) |
---|
1707 | if(size(reduce(J,std(BM[1])))!=0) |
---|
1708 | { |
---|
1709 | list L=DeltaList(BM); |
---|
1710 | } |
---|
1711 | else |
---|
1712 | { |
---|
1713 | list L; |
---|
1714 | L[1]=J; |
---|
1715 | } |
---|
1716 | if(!defined(debugCenter)) |
---|
1717 | { |
---|
1718 | int debugCenter; |
---|
1719 | } |
---|
1720 | if(debugCenter) |
---|
1721 | { |
---|
1722 | "----> In Center: after DeltaList"; |
---|
1723 | "W"; |
---|
1724 | BM[1]; |
---|
1725 | "J"; |
---|
1726 | BM[2]; |
---|
1727 | "The Delta List:"; |
---|
1728 | L; |
---|
1729 | } |
---|
1730 | int b=size(L); |
---|
1731 | if(b==0) |
---|
1732 | { |
---|
1733 | //--- if J=W, we do not need to do anything |
---|
1734 | //--- returning center=1 marks this chart as completed |
---|
1735 | return(re); |
---|
1736 | } |
---|
1737 | //--------------------------------------------------------------------------- |
---|
1738 | // check whether max ord is constant |
---|
1739 | //--------------------------------------------------------------------------- |
---|
1740 | if((BM[9][2,1]<0)||(BM[9][1,1]>b)) |
---|
1741 | { |
---|
1742 | //--- we are either at the beginning or the invariant has dropped |
---|
1743 | intvec tempvec=size(BM[4]); |
---|
1744 | BM[7]=tempvec; |
---|
1745 | //!!!! nur fuer hyperflaechen!!!!!!!! |
---|
1746 | tempvec=b; |
---|
1747 | BM[3]=tempvec; |
---|
1748 | //!!!! Ende !!!!!!!!!!!! |
---|
1749 | kill tempvec; |
---|
1750 | BM[9][1,1]=b; |
---|
1751 | BM[9][2,1]=1; |
---|
1752 | } |
---|
1753 | //--------------------------------------------------------------------------- |
---|
1754 | // prepare for intersection with E_i |
---|
1755 | //--------------------------------------------------------------------------- |
---|
1756 | ideal C=L[b]; |
---|
1757 | re[2]=BM[7]; |
---|
1758 | re[3]=BM[3]; |
---|
1759 | BM[2]=C; |
---|
1760 | if(debugCenter) |
---|
1761 | { |
---|
1762 | "----> In Center: before intersection with E_i:"; |
---|
1763 | "bmax:",b; |
---|
1764 | "Sing(J,bmax):"; |
---|
1765 | C; |
---|
1766 | "E:"; |
---|
1767 | BO[4]; |
---|
1768 | "list marking a priori known intersection properties:",BO[6]; |
---|
1769 | "index of last element of E^- in E:",BO[7][1]; |
---|
1770 | } |
---|
1771 | list E=inters_E(BM); |
---|
1772 | // !!!!!!!!! Drop Redundant fehlt noch!!!!!!!! |
---|
1773 | //--------------------------------------------------------------------------- |
---|
1774 | // Check whether it is a single point |
---|
1775 | //--------------------------------------------------------------------------- |
---|
1776 | ideal C1=std(ideal(L[b])+E[1]); |
---|
1777 | if(dim(C1)==0) |
---|
1778 | { |
---|
1779 | if(size(E[4])==1) |
---|
1780 | { |
---|
1781 | tmat[1,1]=BM[9][1,1]; |
---|
1782 | tmat[2,1]=BM[9][2,1]; |
---|
1783 | re[4]=tmat; |
---|
1784 | re[1]=radical(C1); |
---|
1785 | return(re); |
---|
1786 | } |
---|
1787 | } |
---|
1788 | if(size(BM[9])>2) |
---|
1789 | { |
---|
1790 | BM[9][1,2]=E[2]; |
---|
1791 | BM[9][2,2]=1; |
---|
1792 | } |
---|
1793 | else |
---|
1794 | { |
---|
1795 | intmat tempInt[2][1]=E[2],1; |
---|
1796 | BM[9]=concatInt(BM[9],tempInt); |
---|
1797 | kill tempInt; |
---|
1798 | } |
---|
1799 | BM[2]=J; |
---|
1800 | list BM1=dropDim(BM); |
---|
1801 | list BMlist,hilfList; |
---|
1802 | ideal hilf; |
---|
1803 | intvec tempvec; |
---|
1804 | |
---|
1805 | if(!attrib(BM1[1],"isSB")){BM1[1]=std(BM1[1]);} |
---|
1806 | for(i=1;i<=size(BM1[4]);i++) |
---|
1807 | { |
---|
1808 | hilf=simplify(reduce(BM1[4][i],BM1[1]),2); |
---|
1809 | if(size(hilf)>1){"Problem with BM1[4]in CenterBM";~;} |
---|
1810 | hilfList[i]=hilf[1]; |
---|
1811 | } |
---|
1812 | for(i=1;i<=size(E[4]);i++) |
---|
1813 | { |
---|
1814 | BMlist[i]=BM1; |
---|
1815 | for(j=size(E[5][i]);j>=1;j--) |
---|
1816 | { |
---|
1817 | if(E[5][i][j]!=0) |
---|
1818 | { |
---|
1819 | BMlist[i][2][size(BMlist[i][2])+1]=hilfList[j]; |
---|
1820 | BMlist[i][3][size(BMlist[i][3])+1]=1; |
---|
1821 | } |
---|
1822 | BMlist[i][4]=delete(BMlist[i][4],j); |
---|
1823 | BMlist[i][6]=deleteInt(BMlist[i][6],j,0); |
---|
1824 | } |
---|
1825 | BMlist[i][7]=deleteInt(BMlist[i][7],1,-1); |
---|
1826 | if(size(BMlist[i][9])>4) |
---|
1827 | { |
---|
1828 | intmat tempInt[2][ncols(BMlist[i][9])]=BMlist[i][9]; |
---|
1829 | intmat tempInt2[2][ncols(BMlist[i][9])-2]= |
---|
1830 | tempInt[1..2,3..ncols(BMlist[i][9])]; |
---|
1831 | BMlist[i][9]=tempInt2; |
---|
1832 | kill tempInt,tempInt2; |
---|
1833 | } |
---|
1834 | else |
---|
1835 | { |
---|
1836 | BMlist[i][9]=tmat; |
---|
1837 | } |
---|
1838 | } |
---|
1839 | kill hilfList;list hilfList; |
---|
1840 | hilfList[1]=CenterTail(BMlist[1],C); |
---|
1841 | intmat maxmat=hilfList[1][9]; |
---|
1842 | intvec maxiv=E[4][1]; |
---|
1843 | int pos=1; |
---|
1844 | for(i=2;i<=size(E[4]);i++) |
---|
1845 | { |
---|
1846 | hilfList[i]=CenterTail(BMlist[i],C); |
---|
1847 | if(invGreater(hilfList[i][4]),maxmat,E[4][i],maxiv) |
---|
1848 | { |
---|
1849 | maxmat=hilfList[i][4]; |
---|
1850 | maxiv=E[4][i]; |
---|
1851 | pos=i; |
---|
1852 | } |
---|
1853 | } |
---|
1854 | re[1]=hilfList[pos][1]; |
---|
1855 | intmat tempint=BM[9]; |
---|
1856 | intmat tempint1[2][2]=tempint[1..2,1..2]; |
---|
1857 | re[4]=concatInt(tempint1,maxmat); |
---|
1858 | re[2]=re[2][1],hilfList[pos][2]; |
---|
1859 | re[3]=b; |
---|
1860 | ~; |
---|
1861 | return(re); |
---|
1862 | } |
---|
1863 | example |
---|
1864 | { "EXAMPLE:"; |
---|
1865 | echo = 2; |
---|
1866 | ring R=0,(x,y),dp; |
---|
1867 | |
---|
1868 | ideal W; |
---|
1869 | ideal J=x2-y3; |
---|
1870 | intvec b=1; |
---|
1871 | list E; |
---|
1872 | ideal abb=maxideal(1); |
---|
1873 | intvec v; |
---|
1874 | intvec w=-1; |
---|
1875 | matrix M; |
---|
1876 | intmat invmat[2][1]=0,-1; |
---|
1877 | |
---|
1878 | list BM=W,J,b,E,abb,v,w,M,invmat; |
---|
1879 | |
---|
1880 | CenterBM(BM); |
---|
1881 | } |
---|
1882 | |
---|
1883 | ///////////////////////////////////////////////////////////////////////////// |
---|
1884 | static |
---|
1885 | proc invGreater(intmat M1, intmat M2, intvec iv1, intvec iv2) |
---|
1886 | { |
---|
1887 | // Auxilliary procedure, BM-algorithm |
---|
1888 | int i; |
---|
1889 | for(i=1;i<=min(ncols(M1),ncols(M2));i++) |
---|
1890 | { |
---|
1891 | if(M1[2,i]==-1) |
---|
1892 | { |
---|
1893 | if(M1[1,i]==0){ERROR("Invariant not set");} |
---|
1894 | if(M2[2,i]!=-1){return(1);} |
---|
1895 | if(M2[1,i]==0){ERROR("Invariant not set");} |
---|
1896 | break; |
---|
1897 | } |
---|
1898 | else |
---|
1899 | { |
---|
1900 | if(M2[2,i]==-1) |
---|
1901 | { |
---|
1902 | if(M2[1,i]==0){ERROR("Invariant not set");} |
---|
1903 | return(0); |
---|
1904 | } |
---|
1905 | if(M1[1,i]*M2[2,i]!= M2[1,i]*M1[2,i]) |
---|
1906 | { |
---|
1907 | return(M1[1,i]*M2[2,i]> M2[1,i]*M1[2,i]); |
---|
1908 | } |
---|
1909 | } |
---|
1910 | } |
---|
1911 | return(iv1>iv2); |
---|
1912 | } |
---|
1913 | ///////////////////////////////////////////////////////////////////////////// |
---|
1914 | proc CenterTail(list BM, ideal C) |
---|
1915 | { |
---|
1916 | //!!! Auxilliary procedure, BM-algorithm |
---|
1917 | //!!!!!!!!Rueckgabe im Zentrumsformat |
---|
1918 | int i,j,bmin; |
---|
1919 | int alpha=lcm(BM[3]); |
---|
1920 | vector w; |
---|
1921 | list re; |
---|
1922 | if(size(BM[2])==0) |
---|
1923 | { |
---|
1924 | re[1]=C+BM[1]; |
---|
1925 | intvec tvec; |
---|
1926 | re[3]=tvec; |
---|
1927 | intmat tmat[2][1]=-1,-1; |
---|
1928 | re[4]=tmat; |
---|
1929 | tvec=size(BM[4]); |
---|
1930 | re[2]=tvec; |
---|
1931 | return(re); |
---|
1932 | } |
---|
1933 | for(i=1;i<=size(BM[3]);i++) |
---|
1934 | { |
---|
1935 | if(BM[2][i]!=0) |
---|
1936 | { |
---|
1937 | w[size(w)+1]=BM[2][i]^(alpha/BM[3][i]); |
---|
1938 | } |
---|
1939 | } |
---|
1940 | module M=w; |
---|
1941 | intvec satex; |
---|
1942 | list satList; |
---|
1943 | for(i=1;i<=size(BM[4]);i++) |
---|
1944 | { |
---|
1945 | satList=sat(M,BM[4][i]); |
---|
1946 | satex[i]=satList[2]; |
---|
1947 | M=satList[1]; |
---|
1948 | } |
---|
1949 | //!!!!Hilfsobjekt G bilden!!!!!!!!!!!!!!!! |
---|
1950 | //!!!!Hilfsobjekt H,codim -1 bilden!!!!!!!!!!!!!!!! |
---|
1951 | //!!!! ???? an welcher stelle????????? |
---|
1952 | list deltaL; |
---|
1953 | list BMtemp=BM; |
---|
1954 | for(i=1;i<=nrows(M[1]);i++) |
---|
1955 | { |
---|
1956 | BMtemp[2]=ideal(M[1][i])+BMtemp[1]; |
---|
1957 | deltaL[i]=DeltaList(BMtemp); |
---|
1958 | if(i==1) |
---|
1959 | { |
---|
1960 | bmin=size(deltaL[i]); |
---|
1961 | } |
---|
1962 | else |
---|
1963 | { |
---|
1964 | bmin=min(size(deltaL[i]),bmin); |
---|
1965 | } |
---|
1966 | } |
---|
1967 | if(bmin==0) |
---|
1968 | { |
---|
1969 | re[1]=C+BM[1]; |
---|
1970 | intvec tvec; |
---|
1971 | re[3]=tvec; |
---|
1972 | if((BM[9][2,1]==-1)||(BM[9][1,1]!=0)) |
---|
1973 | { |
---|
1974 | tvec=size(BM[4]); |
---|
1975 | re[2]=tvec; |
---|
1976 | intmat tmat[2][1]=0,1; |
---|
1977 | re[4]=tmat; |
---|
1978 | } |
---|
1979 | else |
---|
1980 | { |
---|
1981 | re[2]=BM[7]; |
---|
1982 | re[4]=BM[9]; |
---|
1983 | } |
---|
1984 | return(re); |
---|
1985 | } |
---|
1986 | ideal Ctemp=ideal(1); |
---|
1987 | while(deg(Ctemp[1])==0) |
---|
1988 | { |
---|
1989 | Ctemp=C; |
---|
1990 | for(i=1;i<=nrows(M[1]);i++) |
---|
1991 | { |
---|
1992 | Ctemp=Ctemp,deltaL[i][bmin]; |
---|
1993 | } |
---|
1994 | Ctemp=std(Ctemp); |
---|
1995 | bmin--; |
---|
1996 | if(bmin==0){ERROR("empty set");} |
---|
1997 | } |
---|
1998 | //!!!!!!!!!!!!!Invariante ist bmin/alpha |
---|
1999 | // naechster Eintrag s_i wie in CenterBM |
---|
2000 | // dann dropDim .... |
---|
2001 | } |
---|
2002 | |
---|
2003 | ///////////////////////////////////////////////////////////////////////////// |
---|
2004 | static |
---|
2005 | proc deleteInt(intvec v,int i,int ini) |
---|
2006 | { |
---|
2007 | //!!! Should be in kernel of Singular |
---|
2008 | //--- delete i-th entry in intvec v, |
---|
2009 | //--- if necessary reinitializing v with value ini |
---|
2010 | int s=size(v); |
---|
2011 | intvec w; |
---|
2012 | if((i<s)&&(i>1)){w=v[1..i-1],v[i+1..s];} |
---|
2013 | if(s==1){w=ini;return(w);} |
---|
2014 | if(i==1){w=v[2..s];} |
---|
2015 | if(i==s){w=v[1..s-1];} |
---|
2016 | return(w); |
---|
2017 | } |
---|
2018 | ///////////////////////////////////////////////////////////////////////////// |
---|
2019 | static |
---|
2020 | proc concatInt(intmat A, intmat B) |
---|
2021 | { |
---|
2022 | //!!! Should be in kernel of Singular |
---|
2023 | //--- concatenate two intmats |
---|
2024 | if(nrows(A)!=nrows(B)){ERROR("could not concat, wrong number of rows");} |
---|
2025 | intmat tempmat[nrows(A)][ncols(A)+ncols(B)]; |
---|
2026 | tempmat[1..nrows(A),1..ncols(A)]=A[1..nrows(A),1..ncols(A)]; |
---|
2027 | tempmat[1..nrows(A),ncols(A)+1..ncols(tempmat)]=B[1..nrows(A),1..ncols(B)]; |
---|
2028 | return(tempmat); |
---|
2029 | } |
---|
2030 | ///////////////////////////////////////////////////////////////////////////// |
---|
2031 | proc dropDim(list BM) |
---|
2032 | { |
---|
2033 | ERROR("Not implemented yet"); |
---|
2034 | } |
---|
2035 | ///////////////////////////////////////////////////////////////////////////// |
---|
2036 | proc CenterBO(list BO,list #) |
---|
2037 | "USAGE: CenterBO(BO); |
---|
2038 | @* BO = basic object, a list: ideal W, |
---|
2039 | @* ideal J, |
---|
2040 | @* intvec b, |
---|
2041 | @* list Ex, |
---|
2042 | @* ideal ab, |
---|
2043 | @* intvec v, |
---|
2044 | @* intvec w, |
---|
2045 | @* matrix M |
---|
2046 | ASSUME: R = basering, a polynomial ring, W an ideal of R, |
---|
2047 | @* J = ideal containing W |
---|
2048 | COMPUTE: the center of the next blow-up of BO in the resolution algorithm |
---|
2049 | of [Bravo,Encinas,Villamayor] |
---|
2050 | RETURN: list l, |
---|
2051 | l[1]: ideal describing the center |
---|
2052 | l[2]: intvec w obtained in the process of determining l[1] |
---|
2053 | l[3]: intvec b obtained in the process of determining l[1] |
---|
2054 | l[4]: intvec inv obtained in the process of determining l[1] |
---|
2055 | EXAMPLE: example CenterBO; shows an example |
---|
2056 | " |
---|
2057 | { |
---|
2058 | //--------------------------------------------------------------------------- |
---|
2059 | // Initialization and sanity checks |
---|
2060 | //--------------------------------------------------------------------------- |
---|
2061 | int i,bo7save; |
---|
2062 | intvec tvec; |
---|
2063 | //--- re=center,E^- indices, b vector, n vector |
---|
2064 | list re=ideal(1),BO[7],BO[3],tvec; |
---|
2065 | ideal J=BO[2]; |
---|
2066 | if(size(J)==0) |
---|
2067 | { |
---|
2068 | re[1]=ideal(0); |
---|
2069 | return(re); |
---|
2070 | } |
---|
2071 | //--- find Delta^(b-1)(J) |
---|
2072 | if(size(reduce(J,std(BO[1])))!=0) |
---|
2073 | { |
---|
2074 | list L=DeltaList(BO); |
---|
2075 | } |
---|
2076 | else |
---|
2077 | { |
---|
2078 | list L; |
---|
2079 | L[1]=J; |
---|
2080 | } |
---|
2081 | if(!defined(debugCenter)) |
---|
2082 | { |
---|
2083 | int debugCenter; |
---|
2084 | } |
---|
2085 | if(debugCenter) |
---|
2086 | { |
---|
2087 | "----> In Center: after DeltaList"; |
---|
2088 | "W"; |
---|
2089 | BO[1]; |
---|
2090 | "J"; |
---|
2091 | BO[2]; |
---|
2092 | "The Delta List:"; |
---|
2093 | L; |
---|
2094 | } |
---|
2095 | int b=size(L); |
---|
2096 | if(b==0) |
---|
2097 | { |
---|
2098 | //--- if J=W, we do not need to do anything |
---|
2099 | //--- returning center=1 marks this chart as completed |
---|
2100 | return(re); |
---|
2101 | } |
---|
2102 | //--------------------------------------------------------------------------- |
---|
2103 | // check whether max w-ord is constant |
---|
2104 | //--------------------------------------------------------------------------- |
---|
2105 | if(b==BO[3][1]) |
---|
2106 | { |
---|
2107 | //--- max w-ord is constant |
---|
2108 | if(BO[7][1]==-1) |
---|
2109 | { |
---|
2110 | //--- first got its value in the previous step ==> initialize BO[7] |
---|
2111 | tvec[1]=size(BO[4])-1; |
---|
2112 | for(i=2;i<=size(BO[7]);i++) |
---|
2113 | { |
---|
2114 | tvec[i]=BO[7][i]; |
---|
2115 | } |
---|
2116 | re[2]=tvec; |
---|
2117 | BO[7]=tvec; |
---|
2118 | } |
---|
2119 | } |
---|
2120 | else |
---|
2121 | { |
---|
2122 | //--- max w-ord changed ==> reset BO[7], correct BO[3] |
---|
2123 | tvec[1]=-1; |
---|
2124 | re[2]=tvec; |
---|
2125 | BO[7]=tvec; |
---|
2126 | tvec[1]=b; |
---|
2127 | BO[3]=tvec; |
---|
2128 | if(defined(invSat)) |
---|
2129 | { |
---|
2130 | invSat[2]=intvec(0); |
---|
2131 | } |
---|
2132 | } |
---|
2133 | re[3]=BO[3]; |
---|
2134 | //--------------------------------------------------------------------------- |
---|
2135 | // reduce from case 2 to case 1 of [Bravo, Encinas, Villamayor] |
---|
2136 | //--------------------------------------------------------------------------- |
---|
2137 | ideal C=L[b]; |
---|
2138 | BO[2]=C; |
---|
2139 | if(debugCenter) |
---|
2140 | { |
---|
2141 | "----> In Center: before intersection with E_i:"; |
---|
2142 | "bmax:",b; |
---|
2143 | "Sing(J,bmax):"; |
---|
2144 | C; |
---|
2145 | "E:"; |
---|
2146 | BO[4]; |
---|
2147 | "list marking a priori known intersection properties:",BO[6]; |
---|
2148 | "index of last element of E^- in E:",BO[7][1]; |
---|
2149 | } |
---|
2150 | //--- is intermediate result in iteration already good? |
---|
2151 | //--- return it to calling proc CenterBO |
---|
2152 | if(size(#)>0) |
---|
2153 | { |
---|
2154 | if(#[1]==2) |
---|
2155 | { |
---|
2156 | re[1]=C; |
---|
2157 | kill tvec; |
---|
2158 | intvec tvec=re[2][1]; |
---|
2159 | re[2]=tvec; |
---|
2160 | kill tvec; |
---|
2161 | intvec tvec=re[3][1]; |
---|
2162 | re[3]=tvec; |
---|
2163 | kill tvec; |
---|
2164 | intvec tvec; |
---|
2165 | re[4]=tvec; |
---|
2166 | return(re); |
---|
2167 | } |
---|
2168 | } |
---|
2169 | //--- do the reduction to case 1 |
---|
2170 | list E=inters_E(BO); |
---|
2171 | //--- if J is smooth and not too many E_i intersect simultaneously, let us |
---|
2172 | //--- try to drop redundant components of the candidate for the center |
---|
2173 | if((b==1)&&(size(E)>3)) |
---|
2174 | { |
---|
2175 | //--- if J is not smooth we do not want to drop any information |
---|
2176 | if((size(E[4])>0) && (dim(std(slocusE(BO[2])))<0)) |
---|
2177 | { |
---|
2178 | //--- BO[2]==J because b==1 |
---|
2179 | //--- DropRedundant is the counterpart to DropCoeff |
---|
2180 | //--- do not leave out one of them separately!!! |
---|
2181 | E=DropRedundant(BO,E); |
---|
2182 | if(size(E)==1) |
---|
2183 | { |
---|
2184 | kill tvec; |
---|
2185 | intvec tvec=re[2][1]; |
---|
2186 | re[2]=tvec; |
---|
2187 | tvec[1]=re[3][1]; |
---|
2188 | re[3]=tvec; |
---|
2189 | tvec[1]=re[4][1]; |
---|
2190 | re[4]=tvec; |
---|
2191 | re[1]=E[1]; |
---|
2192 | return(re); |
---|
2193 | } |
---|
2194 | } |
---|
2195 | } |
---|
2196 | //--- set n correctly |
---|
2197 | if(E[2]<BO[9][1]) |
---|
2198 | { |
---|
2199 | //--- if n dropped, the subsequent Coeff-object will not be the same |
---|
2200 | //--- ===> set BO[3][2] to zero to make sure that no previous data is used |
---|
2201 | if(defined(tvec)) {kill tvec;} |
---|
2202 | intvec tvec=BO[7][1],-1; |
---|
2203 | BO[7]=tvec; |
---|
2204 | tvec=BO[3][1],0; |
---|
2205 | BO[3]=tvec; |
---|
2206 | if(defined(invSat)) |
---|
2207 | { |
---|
2208 | invSat[2]=intvec(0); |
---|
2209 | } |
---|
2210 | } |
---|
2211 | re[4][1]=E[2]; |
---|
2212 | C=E[1]^b+J; |
---|
2213 | C=mstd(C)[2]; |
---|
2214 | ideal C1=std(ideal(L[b])+E[1]); |
---|
2215 | if(debugCenter) |
---|
2216 | { |
---|
2217 | "----> In Center: reduction of case 2 to case 1"; |
---|
2218 | "Output of inters_E, after dropping redundant components:"; |
---|
2219 | E; |
---|
2220 | "result of intersection with E^-, i.e.(E^-)^b+J:"; |
---|
2221 | C; |
---|
2222 | "candidate for center:"; |
---|
2223 | C1; |
---|
2224 | } |
---|
2225 | //--------------------------------------------------------------------------- |
---|
2226 | // Check whether we have a hypersurface component |
---|
2227 | //--------------------------------------------------------------------------- |
---|
2228 | if(dim(C1)==dim(std(BO[1]))-1) |
---|
2229 | { |
---|
2230 | if((size(reduce(J,C1))==0)&&(size(reduce(C1,std(J)))==0)) |
---|
2231 | { |
---|
2232 | //--- C1 equals J and is of codimension 1 in W |
---|
2233 | re[1]=C1; |
---|
2234 | } |
---|
2235 | else |
---|
2236 | { |
---|
2237 | //--- C1 has a codimension 1 (in W) component |
---|
2238 | re[1]=std(equiRadical(C1)); |
---|
2239 | } |
---|
2240 | kill tvec; |
---|
2241 | intvec tvec=re[2][1]; |
---|
2242 | re[2]=tvec; |
---|
2243 | tvec[1]=re[3][1]; |
---|
2244 | re[3]=tvec; |
---|
2245 | tvec[1]=re[4][1]; |
---|
2246 | re[4]=tvec; |
---|
2247 | //--- is the codimension 1 component a good choice or do we need to reset |
---|
2248 | //--- the information from the previous steps |
---|
2249 | if(transversalT(re[1],BO[4])) |
---|
2250 | { |
---|
2251 | if(size(E)>2) |
---|
2252 | { |
---|
2253 | if(E[3]>E[2]) |
---|
2254 | { |
---|
2255 | if(defined(shortcut)){kill shortcut;} |
---|
2256 | list shortcut=ideal(0),size(BO[4]),BO[7]; |
---|
2257 | export(shortcut); |
---|
2258 | } |
---|
2259 | } |
---|
2260 | return(re); |
---|
2261 | } |
---|
2262 | |
---|
2263 | ERROR("reset in Center, please send the example to the authors."); |
---|
2264 | } |
---|
2265 | //--------------------------------------------------------------------------- |
---|
2266 | // Check whether it is a single point |
---|
2267 | //--------------------------------------------------------------------------- |
---|
2268 | if(dim(C1)==0) |
---|
2269 | { |
---|
2270 | C1=std(radical(C1)); |
---|
2271 | if(vdim(C1)==1) |
---|
2272 | { |
---|
2273 | //--- C1 is one point |
---|
2274 | re[1]=C1; |
---|
2275 | kill tvec; |
---|
2276 | intvec tvec=re[2][1]; |
---|
2277 | re[2]=tvec; |
---|
2278 | kill tvec; |
---|
2279 | intvec tvec=re[3][1]; |
---|
2280 | re[3]=tvec; |
---|
2281 | return(re); |
---|
2282 | } |
---|
2283 | } |
---|
2284 | //--------------------------------------------------------------------------- |
---|
2285 | // Prepare input for forming the Coeff-Ideal |
---|
2286 | //--------------------------------------------------------------------------- |
---|
2287 | BO[2]=C; |
---|
2288 | if(size(BO[2])>5) |
---|
2289 | { |
---|
2290 | BO[2]=mstd(BO[2])[2]; |
---|
2291 | } |
---|
2292 | //--- drop leading entry of BO[3] |
---|
2293 | tvec=BO[3]; |
---|
2294 | if(size(tvec)>1) |
---|
2295 | { |
---|
2296 | tvec=tvec[2..size(tvec)]; |
---|
2297 | BO[3]=tvec; |
---|
2298 | } |
---|
2299 | else |
---|
2300 | { |
---|
2301 | BO[3][1]=0; |
---|
2302 | } |
---|
2303 | tvec=BO[9]; |
---|
2304 | if(size(tvec)>1) |
---|
2305 | { |
---|
2306 | tvec=tvec[2..size(tvec)]; |
---|
2307 | BO[9]=tvec; |
---|
2308 | } |
---|
2309 | else |
---|
2310 | { |
---|
2311 | BO[9][1]=0; |
---|
2312 | } |
---|
2313 | bo7save=BO[7][1]; // original value needed for result |
---|
2314 | if(defined(shortcut)) |
---|
2315 | { |
---|
2316 | if((bo7save!=shortcut[3][1])&&(size(shortcut[3])!=1)) |
---|
2317 | { |
---|
2318 | kill shortcut; |
---|
2319 | } |
---|
2320 | else |
---|
2321 | { |
---|
2322 | shortcut[2]=shortcut[2]-bo7save; |
---|
2323 | tvec=shortcut[3]; |
---|
2324 | if(size(tvec)>1) |
---|
2325 | { |
---|
2326 | tvec=tvec[2..size(tvec)]; |
---|
2327 | shortcut[3]=tvec; |
---|
2328 | } |
---|
2329 | else |
---|
2330 | { |
---|
2331 | shortcut[3]=intvec(shortcut[2]); |
---|
2332 | } |
---|
2333 | } |
---|
2334 | } |
---|
2335 | if(BO[7][1]>-1) |
---|
2336 | { |
---|
2337 | //--- drop E^- and the corresponding information from BO[6] |
---|
2338 | for(i=1;i<=BO[7][1];i++) |
---|
2339 | { |
---|
2340 | BO[4]=delete(BO[4],1); |
---|
2341 | intvec bla1=BO[6]; |
---|
2342 | BO[6]=intvec(bla1[2..size(bla1)]); |
---|
2343 | kill bla1; |
---|
2344 | } |
---|
2345 | //--- drop leading entry of BO[7] |
---|
2346 | tvec=BO[7]; |
---|
2347 | if(size(tvec)>1) |
---|
2348 | { |
---|
2349 | tvec=tvec[2..size(tvec)]; |
---|
2350 | BO[7]=tvec; |
---|
2351 | } |
---|
2352 | else |
---|
2353 | { |
---|
2354 | BO[7][1]=-1; |
---|
2355 | } |
---|
2356 | } |
---|
2357 | else |
---|
2358 | { |
---|
2359 | if(BO[7][1]==-1) |
---|
2360 | { |
---|
2361 | list tplist; |
---|
2362 | BO[4]=tplist; |
---|
2363 | kill tplist; |
---|
2364 | } |
---|
2365 | } |
---|
2366 | if(debugCenter) |
---|
2367 | { |
---|
2368 | "----> In Center: Input to Coeff"; |
---|
2369 | "b:",b; |
---|
2370 | "BO:"; |
---|
2371 | BO; |
---|
2372 | } |
---|
2373 | //--- prepare the third entry of the invariant tuple |
---|
2374 | int invSatSave=invSat[2][1]; |
---|
2375 | tvec=invSat[2]; |
---|
2376 | if(size(tvec)>1) |
---|
2377 | { |
---|
2378 | tvec=tvec[2..size(tvec)]; |
---|
2379 | invSat[2]=tvec; |
---|
2380 | } |
---|
2381 | else |
---|
2382 | { |
---|
2383 | invSat[2][1]=0; |
---|
2384 | } |
---|
2385 | //--------------------------------------------------------------------------- |
---|
2386 | // Form the Coeff-ideal, if possible and useful; otherwise use the previous |
---|
2387 | // candidate for the center |
---|
2388 | //--------------------------------------------------------------------------- |
---|
2389 | list BO1=Coeff(BO,b); |
---|
2390 | if(debugCenter) |
---|
2391 | { |
---|
2392 | "----> In Center: Output of Coeff"; |
---|
2393 | BO1; |
---|
2394 | } |
---|
2395 | //--- Coeff returns int if something went wrong |
---|
2396 | if(typeof(BO1[1])=="int") |
---|
2397 | { |
---|
2398 | if(BO1[1]==0) |
---|
2399 | { |
---|
2400 | //--- Coeff ideal was already resolved |
---|
2401 | re[1]=C1; |
---|
2402 | return(re); |
---|
2403 | } |
---|
2404 | else |
---|
2405 | { |
---|
2406 | //--- no global hypersurface found |
---|
2407 | re=CoverCenter(BO,b,BO1[2]); |
---|
2408 | kill tvec; |
---|
2409 | intvec tvec=invSatSave; |
---|
2410 | for(i=1;i<=size(invSat[2]);i++) |
---|
2411 | { |
---|
2412 | tvec[i+1]=invSat[2][i]; |
---|
2413 | } |
---|
2414 | invSat[2]=tvec; |
---|
2415 | return(re); |
---|
2416 | } |
---|
2417 | } |
---|
2418 | int coeff_invar; |
---|
2419 | ideal Idropped=1; |
---|
2420 | //--- if b=1 drop redundant components of the Coeff-ideal |
---|
2421 | if(b==1) |
---|
2422 | { |
---|
2423 | //--- Counterpart to DropRedundant -- do not leave out one of them separately |
---|
2424 | Idropped=DropCoeff(BO1); // blow-up in these components |
---|
2425 | // is unnecessary |
---|
2426 | } |
---|
2427 | //--- to switch off DropCoeff, set Idropped=1; |
---|
2428 | BO1[2]=sat(BO1[2],Idropped)[1]; |
---|
2429 | if(deg(BO1[2][1])==0) |
---|
2430 | { |
---|
2431 | //--- Coeff ideal is trivial |
---|
2432 | C1=radical(C1); |
---|
2433 | ideal C2=sat(C1,Idropped)[1]; |
---|
2434 | if(deg(std(C2)[1])!=0) |
---|
2435 | { |
---|
2436 | C1=C2; |
---|
2437 | } |
---|
2438 | //Aenderung: Strategie: nur im Notfall ganze except. Divisoren |
---|
2439 | if(deg(std(BO1[2])[1])==0) |
---|
2440 | { |
---|
2441 | list BOtemp=BO; |
---|
2442 | int bo17save=BO1[7][1]; |
---|
2443 | BOtemp[7]=0; |
---|
2444 | BO1=Coeff(BOtemp,b,int(0)); |
---|
2445 | BO1[2]=sat(BO1[2],Idropped)[1]; |
---|
2446 | if(deg(std(BO1[2])[1])==0) |
---|
2447 | { |
---|
2448 | //--- there is really nothing left to do for the Coeff ideal |
---|
2449 | //--- the whole original BO1[2], i.e. Idropped, is the upcoming center |
---|
2450 | re[1]=Idropped; |
---|
2451 | re[2]=intvec(bo7save); |
---|
2452 | re[3]=intvec(b); |
---|
2453 | re[4]=intvec(E[2]); |
---|
2454 | return(re); |
---|
2455 | } |
---|
2456 | if(deg(std(slocus(radical(BO1[2])))[1])==0) |
---|
2457 | { |
---|
2458 | re[1]=BO1[2]; |
---|
2459 | // re[2]=intvec(bo7save,BO1[7][1]); |
---|
2460 | re[2]=intvec(bo7save,bo17save); |
---|
2461 | re[3]=intvec(b,1); |
---|
2462 | re[4]=intvec(E[2],1); |
---|
2463 | invSat[2]=intvec(1,0); |
---|
2464 | return(re); |
---|
2465 | } |
---|
2466 | //!!! effizienter machen??? |
---|
2467 | list pr=primdecGTZ(BO1[2]); |
---|
2468 | ideal Itemp1=1; |
---|
2469 | int aa,bb; |
---|
2470 | for(aa=1;aa<=size(pr);aa++) |
---|
2471 | { |
---|
2472 | if(dim(std(pr[aa][2])) < (dim(std(BO1[1]))-1)) |
---|
2473 | { |
---|
2474 | //--- drop components which are themselves exceptional diviosrs |
---|
2475 | Itemp1=intersect(Itemp1,pr[aa][1]); |
---|
2476 | } |
---|
2477 | } |
---|
2478 | if(deg(std(Itemp1)[1])!=0) |
---|
2479 | { |
---|
2480 | //--- treat the remaining components of the weak Coeff ideal |
---|
2481 | BO1[2]=Itemp1; |
---|
2482 | } |
---|
2483 | BO1[7]=BO[7]; |
---|
2484 | for(aa=1;aa<=size(BO1[4]);aa++) |
---|
2485 | { |
---|
2486 | if(deg(std(BO1[4][aa])[1])==0){aa++;continue;} |
---|
2487 | if(defined(satlist)){kill satlist;} |
---|
2488 | list satlist=sat(BO1[2],BO1[4][aa]+BO1[1]); |
---|
2489 | if(deg(std(satlist[1])[1])==0) |
---|
2490 | { |
---|
2491 | coeff_invar++; |
---|
2492 | if(satlist[2]!=0) |
---|
2493 | { |
---|
2494 | for(bb=1;bb<=satlist[2]-1;bb++) |
---|
2495 | { |
---|
2496 | BO1[2]=quotient(BO1[2],BO1[4][aa]+BO1[1]); |
---|
2497 | } |
---|
2498 | } |
---|
2499 | else |
---|
2500 | { |
---|
2501 | ERROR("J of temporary object had unexpected value; |
---|
2502 | please send this example to the authors."); |
---|
2503 | } |
---|
2504 | } |
---|
2505 | else |
---|
2506 | { |
---|
2507 | BO1[2]=satlist[1]; |
---|
2508 | } |
---|
2509 | } |
---|
2510 | if(deg(std(Itemp1)[1])==0) |
---|
2511 | { |
---|
2512 | re[1]=BO1[2]; |
---|
2513 | re[2]=intvec(bo7save,BO1[7][1]); |
---|
2514 | re[3]=intvec(b,1); |
---|
2515 | re[4]=intvec(E[2],1); |
---|
2516 | invSat[2]=intvec(1,0); |
---|
2517 | return(re); |
---|
2518 | } |
---|
2519 | kill aa,bb; |
---|
2520 | } |
---|
2521 | } |
---|
2522 | if(invSatSave<coeff_invar) |
---|
2523 | { |
---|
2524 | invSatSave=coeff_invar; |
---|
2525 | } |
---|
2526 | //--------------------------------------------------------------------------- |
---|
2527 | // Determine Center of Coeff-ideal and use it as the new center |
---|
2528 | //--------------------------------------------------------------------------- |
---|
2529 | if(!defined(templist)) |
---|
2530 | { |
---|
2531 | if(size(BO1[2])>5) |
---|
2532 | { |
---|
2533 | BO1[2]=mstd(BO1[2])[2]; |
---|
2534 | } |
---|
2535 | list templist=CenterBO(BO1,2); |
---|
2536 | //--- only a sophisticated guess of a good center computed by |
---|
2537 | //--- leaving center before intersection with the E_i. |
---|
2538 | //--- whether the guess was good, is stored in 'good'. |
---|
2539 | //--- (this variant saves charts in cases like the Whitney umbrella) |
---|
2540 | list E0,E1; |
---|
2541 | |
---|
2542 | ideal Cstd=std(radical(templist[1])); |
---|
2543 | int good=((deg(std(slocusE(Cstd))[1])==0)&&(dim(std(BO1[2]))<=2)); |
---|
2544 | //if(defined(satlist)){good=0;} |
---|
2545 | if(good) |
---|
2546 | { |
---|
2547 | for(i=1;i<=size(BO[4]);i++) |
---|
2548 | { |
---|
2549 | if((deg(BO[4][i][1])>0)&&(size(reduce(BO[4][i],Cstd))!=0)) |
---|
2550 | { |
---|
2551 | E0[size(E0)+1]=BO[4][i]+Cstd; |
---|
2552 | E1[size(E1)+1]=BO[4][i]; |
---|
2553 | } |
---|
2554 | } |
---|
2555 | good=transversalT(Cstd,E1); |
---|
2556 | if(good) |
---|
2557 | { |
---|
2558 | good=normalCross(E0); |
---|
2559 | } |
---|
2560 | } |
---|
2561 | if(good) |
---|
2562 | { |
---|
2563 | list templist2=CenterBO(BO1,1); |
---|
2564 | if(dim(std(templist2[1]))!=dim(Cstd)) |
---|
2565 | { |
---|
2566 | templist[1]=Cstd; |
---|
2567 | if(defined(shortcut)){kill shortcut;} |
---|
2568 | list shortcut=ideal(0),size(BO1[4]),templist[2]; |
---|
2569 | export(shortcut); |
---|
2570 | } |
---|
2571 | else |
---|
2572 | { |
---|
2573 | templist=templist2; |
---|
2574 | } |
---|
2575 | kill templist2; |
---|
2576 | } |
---|
2577 | else |
---|
2578 | { |
---|
2579 | //--- sophisticated guess was wrong, follow Villamayor's approach |
---|
2580 | kill templist; |
---|
2581 | list templist=CenterBO(BO1,1); |
---|
2582 | } |
---|
2583 | } |
---|
2584 | if((dim(std(templist[1]))==dim(std(BO1[1]))-1) |
---|
2585 | &&(size(templist[4])==1)) |
---|
2586 | { |
---|
2587 | if(templist[4][1]==0) |
---|
2588 | { |
---|
2589 | for(i=1;i<=size(BO1[4]);i++) |
---|
2590 | { |
---|
2591 | if(size(reduce(templist[1],std(BO1[4][i])))==0) |
---|
2592 | { |
---|
2593 | templist[4][1]=1; |
---|
2594 | break; |
---|
2595 | } |
---|
2596 | } |
---|
2597 | } |
---|
2598 | } |
---|
2599 | //!!! subsequent line should be deleted |
---|
2600 | //if(defined(satlist)){templist[3][1]=BO[3][1];} |
---|
2601 | if(debugCenter) |
---|
2602 | { |
---|
2603 | "----> In Center: Iterated Center returned:"; |
---|
2604 | templist; |
---|
2605 | } |
---|
2606 | //-------------------------------------------------------------------------- |
---|
2607 | // set up the result and return it |
---|
2608 | //-------------------------------------------------------------------------- |
---|
2609 | re[1]=templist[1]; |
---|
2610 | kill tvec; |
---|
2611 | intvec tvec; |
---|
2612 | tvec[1]=bo7save; |
---|
2613 | for(i=1;i<=size(templist[2]);i++) |
---|
2614 | { |
---|
2615 | tvec[i+1]=templist[2][i]; |
---|
2616 | } |
---|
2617 | re[2]=tvec; |
---|
2618 | if(defined(shortcut)) |
---|
2619 | { |
---|
2620 | shortcut[2]=shortcut[2]+bo7save; |
---|
2621 | shortcut[3]=tvec; |
---|
2622 | } |
---|
2623 | kill tvec; |
---|
2624 | intvec tvec; |
---|
2625 | tvec[1]=invSatSave; |
---|
2626 | for(i=1;i<=size(invSat[2]);i++) |
---|
2627 | { |
---|
2628 | tvec[i+1]=invSat[2][i]; |
---|
2629 | } |
---|
2630 | invSat[2]=tvec; |
---|
2631 | kill tvec; |
---|
2632 | intvec tvec; |
---|
2633 | tvec[1]=b; |
---|
2634 | for(i=1;i<=size(templist[3]);i++) |
---|
2635 | { |
---|
2636 | tvec[i+1]=templist[3][i]; |
---|
2637 | } |
---|
2638 | re[3]=tvec; |
---|
2639 | kill tvec; |
---|
2640 | intvec tvec; |
---|
2641 | tvec[1]=E[2]; |
---|
2642 | for(i=1;i<=size(templist[4]);i++) |
---|
2643 | { |
---|
2644 | tvec[i+1]=templist[4][i]; |
---|
2645 | } |
---|
2646 | re[4]=tvec; |
---|
2647 | |
---|
2648 | return(re); |
---|
2649 | } |
---|
2650 | example |
---|
2651 | { "EXAMPLE:"; |
---|
2652 | echo = 2; |
---|
2653 | ring R=0,(x,y),dp; |
---|
2654 | |
---|
2655 | ideal W; |
---|
2656 | ideal J=x2-y3; |
---|
2657 | intvec b=1; |
---|
2658 | list E; |
---|
2659 | ideal abb=maxideal(1); |
---|
2660 | intvec v; |
---|
2661 | intvec w=-1; |
---|
2662 | matrix M; |
---|
2663 | |
---|
2664 | list BO=W,J,b,E,abb,v,w,M,v; |
---|
2665 | |
---|
2666 | CenterBO(BO); |
---|
2667 | } |
---|
2668 | ////////////////////////////////////////////////////////////////////////////// |
---|
2669 | static |
---|
2670 | proc CoverCenter(list BO,int b, ideal Jb) |
---|
2671 | { |
---|
2672 | //---------------------------------------------------------------------------- |
---|
2673 | // Initialization |
---|
2674 | //---------------------------------------------------------------------------- |
---|
2675 | def R=basering; |
---|
2676 | int i,j,k; |
---|
2677 | intvec merk,merk2,maxv,fvec; |
---|
2678 | list L,ceList,re; |
---|
2679 | ceList[1]=ideal(0); |
---|
2680 | poly @p,@f; |
---|
2681 | ideal K,dummy; |
---|
2682 | if(!attrib(BO[2],"isSB")) |
---|
2683 | { |
---|
2684 | BO[2]=std(BO[2]); |
---|
2685 | } |
---|
2686 | for(i=1;i<=size(Jb);i++) |
---|
2687 | { |
---|
2688 | list tempmstd=mstd(slocus(Jb[i])); |
---|
2689 | if(size(tempmstd[1])>size(tempmstd[2])) |
---|
2690 | { |
---|
2691 | dummy=tempmstd[2]; |
---|
2692 | } |
---|
2693 | else |
---|
2694 | { |
---|
2695 | dummy=tempmstd[1]; |
---|
2696 | } |
---|
2697 | kill tempmstd; |
---|
2698 | L[i]=dummy; |
---|
2699 | K=K,dummy; |
---|
2700 | } |
---|
2701 | K=simplify(K,2); |
---|
2702 | //--------------------------------------------------------------------------- |
---|
2703 | // The intersection of the singular loci of the L[i] is empty. |
---|
2704 | // Find a suitable open covering of the affine chart, such that a global |
---|
2705 | // hypersurface can be found in each open set. |
---|
2706 | //--------------------------------------------------------------------------- |
---|
2707 | matrix M=lift(K,ideal(1)); |
---|
2708 | j=1; |
---|
2709 | for(i=1;i<=nrows(M);i++) |
---|
2710 | { |
---|
2711 | if(M[i,1]!=0) |
---|
2712 | { |
---|
2713 | merk[size(merk)+1]=i; |
---|
2714 | fvec[size(merk)]=j; |
---|
2715 | } |
---|
2716 | if((i-k)==size(L[j])) |
---|
2717 | { |
---|
2718 | k=i; |
---|
2719 | j++; |
---|
2720 | } |
---|
2721 | } |
---|
2722 | //-------------------------------------------------------------------------- |
---|
2723 | // Find a candidate for the center in each open set |
---|
2724 | //-------------------------------------------------------------------------- |
---|
2725 | //--- first entry of merk is 0 by construction of merk |
---|
2726 | for(i=2;i<=size(merk);i++) |
---|
2727 | { |
---|
2728 | //--- open set is D(@p) |
---|
2729 | @p=K[merk[i]]; |
---|
2730 | //--- hypersurface is V(@f) |
---|
2731 | @f=Jb[fvec[i]]; |
---|
2732 | execute("ring R1=("+charstr(R)+"),(@y,"+varstr(R)+"),dp;"); |
---|
2733 | poly p=imap(R,@p); |
---|
2734 | poly f=imap(R,@f); |
---|
2735 | list @ce; |
---|
2736 | list BO=imap(R,BO); |
---|
2737 | BO[1]=BO[1]+ideal(@y*p-1); |
---|
2738 | BO[2]=BO[2]+ideal(@y*p-1); |
---|
2739 | for(j=1;j<=size(BO[4]);j++) |
---|
2740 | { |
---|
2741 | BO[4][j]=BO[4][j]+ideal(@y*p-1); |
---|
2742 | } |
---|
2743 | //--- like usual Coeff, but hypersurface is already known |
---|
2744 | list BO1=SpecialCoeff(BO,b,f); |
---|
2745 | //--- special situation in SpecialCoeff are marked by an error code of |
---|
2746 | //--- type int |
---|
2747 | if(typeof(BO1[1])=="int") |
---|
2748 | { |
---|
2749 | if(BO1[1]==0) |
---|
2750 | { |
---|
2751 | //--- Coeff ideal was already resolved |
---|
2752 | @ce[1]=BO[2]; |
---|
2753 | @ce[2]=BO[7]; |
---|
2754 | @ce[3]=BO[3]; |
---|
2755 | } |
---|
2756 | else |
---|
2757 | { |
---|
2758 | if(BO[3]!=0) |
---|
2759 | { |
---|
2760 | //--- intersections with E do not meet conditions ==> reset |
---|
2761 | ERROR("reset in Coeff, please send the example to the autors"); |
---|
2762 | } |
---|
2763 | } |
---|
2764 | } |
---|
2765 | else |
---|
2766 | { |
---|
2767 | //--- now do the recursion as usual |
---|
2768 | @ce=CenterBO(BO1); |
---|
2769 | } |
---|
2770 | //--------------------------------------------------------------------------- |
---|
2771 | // Go back to the whole affine chart and form a suitable union of the |
---|
2772 | // candidates |
---|
2773 | //--------------------------------------------------------------------------- |
---|
2774 | //--- pass from open set to the whole affine chart by taking the closure |
---|
2775 | @ce[1]=eliminate(@ce[1],@y); |
---|
2776 | setring R; |
---|
2777 | ceList[i]=imap(R1,@ce); |
---|
2778 | //--- set up invariant vector and determine maximum value of it |
---|
2779 | if(size(ceList[i][3])==size(ceList[i][4])) |
---|
2780 | { |
---|
2781 | kill merk2,maxv; |
---|
2782 | intvec merk2,maxv; |
---|
2783 | for(j=1;j<=size(ceList[i][3]);j++) |
---|
2784 | { |
---|
2785 | merk2[2*j-1]=ceList[i][3][j]; |
---|
2786 | merk2[2*j]=ceList[i][4][j]; |
---|
2787 | ceList[i][5]=merk2; |
---|
2788 | if(maxv<merk2) |
---|
2789 | { |
---|
2790 | maxv=merk2; |
---|
2791 | } |
---|
2792 | } |
---|
2793 | } |
---|
2794 | else |
---|
2795 | { |
---|
2796 | ERROR("This situation should not occur, please send the example |
---|
2797 | to the authors."); |
---|
2798 | } |
---|
2799 | kill R1; |
---|
2800 | } |
---|
2801 | kill merk2; |
---|
2802 | intvec merk2=-2; |
---|
2803 | //--- form the union of the components of the center with maximum invariant |
---|
2804 | for(i=1;i<=size(ceList);i++) |
---|
2805 | { |
---|
2806 | if(size(reduce(ceList[i][1],BO[2]))==0) |
---|
2807 | { |
---|
2808 | //--- already resolved ==> ignore |
---|
2809 | i++; |
---|
2810 | continue; |
---|
2811 | } |
---|
2812 | if(ceList[i][5]==maxv) |
---|
2813 | { |
---|
2814 | if(merk2!=ceList[i][2]) |
---|
2815 | { |
---|
2816 | //--- E^- not of the same size as before resp. initialization |
---|
2817 | if(merk2[1]==-2) |
---|
2818 | { |
---|
2819 | //--- initialization: save size of E^- |
---|
2820 | merk2=ceList[i][2]; |
---|
2821 | re[1]=ceList[i][1]; |
---|
2822 | re[2]=ceList[i][2]; |
---|
2823 | re[3]=ceList[i][3]; |
---|
2824 | re[4]=ceList[i][4]; |
---|
2825 | } |
---|
2826 | else |
---|
2827 | { |
---|
2828 | //--- otherwise ignore |
---|
2829 | i++; |
---|
2830 | continue; |
---|
2831 | } |
---|
2832 | } |
---|
2833 | else |
---|
2834 | { |
---|
2835 | re[1]=intersect(re[1],ceList[i][1]); |
---|
2836 | } |
---|
2837 | } |
---|
2838 | } |
---|
2839 | //-------------------------------------------------------------------------- |
---|
2840 | // Perform last checks and return the result |
---|
2841 | //-------------------------------------------------------------------------- |
---|
2842 | if(size(re)!=4) |
---|
2843 | { |
---|
2844 | //--- oops: already resolved in all open sets |
---|
2845 | re[1]=BO[2]; |
---|
2846 | re[2]=-1; |
---|
2847 | re[3]=0; |
---|
2848 | re[4]=intvec(0); |
---|
2849 | } |
---|
2850 | return(re); |
---|
2851 | } |
---|
2852 | ////////////////////////////////////////////////////////////////////////////// |
---|
2853 | static |
---|
2854 | proc SpecialCoeff(list BO,int b,poly f) |
---|
2855 | { |
---|
2856 | //---------------------------------------------------------------------------- |
---|
2857 | // Coeff with given hypersurface -- no checks of the hypersurface performed |
---|
2858 | //---------------------------------------------------------------------------- |
---|
2859 | int i,ch; |
---|
2860 | ch=char(basering); |
---|
2861 | int e=int(factorial(b,ch)); |
---|
2862 | ideal C; |
---|
2863 | list L=DeltaList(BO); |
---|
2864 | int d=size(L); |
---|
2865 | //--- set up ideal |
---|
2866 | for(i=0;i<b;i++) |
---|
2867 | { |
---|
2868 | C=C+L[i+1]^(e/(b-i)); |
---|
2869 | } |
---|
2870 | //--- move to hypersurface V(Z) |
---|
2871 | ideal Z=f; |
---|
2872 | C=C,Z; |
---|
2873 | BO[1]=BO[1]+Z; |
---|
2874 | BO[2]=C; |
---|
2875 | for(i=1;i<=size(BO[4]);i++) |
---|
2876 | { |
---|
2877 | BO[6][i]=0; // reset intersection indicator |
---|
2878 | BO[4][i]=BO[4][i]+Z; // intersect the E_i |
---|
2879 | BO[2]=sat(BO[2],BO[4][i]+BO[1])[1]; |
---|
2880 | // "strict transform" of J w.r.t E, not "total" |
---|
2881 | } |
---|
2882 | return(BO); |
---|
2883 | } |
---|
2884 | |
---|
2885 | ////////////////////////////////////////////////////////////////////////////// |
---|
2886 | static |
---|
2887 | proc DropCoeff(list BO) |
---|
2888 | "Internal procedure - no help and no example available |
---|
2889 | " |
---|
2890 | { |
---|
2891 | //--- Initialization |
---|
2892 | int i,j; |
---|
2893 | list pr=minAssGTZ(BO[2]); |
---|
2894 | ideal I=BO[2]; |
---|
2895 | ideal Itemp; |
---|
2896 | ideal Idropped=1; |
---|
2897 | //--- Tests |
---|
2898 | for(i=1;i<=size(pr);i++) |
---|
2899 | { |
---|
2900 | if(i>size(pr)) |
---|
2901 | { |
---|
2902 | //--- the continue statement does not test the loop condition *sigh* |
---|
2903 | break; |
---|
2904 | } |
---|
2905 | if(deg(std(slocus(pr[i]))[1])!=0) |
---|
2906 | { |
---|
2907 | //--- this component is singular ===> we still need it |
---|
2908 | i++; |
---|
2909 | continue; |
---|
2910 | } |
---|
2911 | Itemp=sat(I,pr[i])[1]; |
---|
2912 | if(deg(std(Itemp+pr[i])[1])!=0) |
---|
2913 | { |
---|
2914 | //--- this component is not disjoint from the other ones ===> we still need it |
---|
2915 | i++; |
---|
2916 | continue; |
---|
2917 | } |
---|
2918 | if(!transversalT(pr[i],BO[4])) |
---|
2919 | { |
---|
2920 | //--- this component does not meet one of the remaining E_i transversally |
---|
2921 | //--- ===> we still need it |
---|
2922 | i++; |
---|
2923 | continue; |
---|
2924 | } |
---|
2925 | if(!normalCross(BO[4],pr[i])) |
---|
2926 | { |
---|
2927 | //--- this component is not normal crossing with the remaining E_i |
---|
2928 | //--- ===> we still need it |
---|
2929 | i++; |
---|
2930 | continue; |
---|
2931 | } |
---|
2932 | if(defined(EE)){kill EE;} |
---|
2933 | list EE; |
---|
2934 | for(j=1;j<=size(BO[4]);j++) |
---|
2935 | { |
---|
2936 | EE[j]=BO[4][j]+pr[i]; |
---|
2937 | } |
---|
2938 | if(!normalCross(EE)) |
---|
2939 | { |
---|
2940 | //--- we do not have a normal crossing situation for this component after all |
---|
2941 | //--- ===> we still need it |
---|
2942 | i++; |
---|
2943 | continue; |
---|
2944 | } |
---|
2945 | Idropped=intersect(Idropped,pr[i]); |
---|
2946 | I=Itemp; |
---|
2947 | } |
---|
2948 | return(Idropped); |
---|
2949 | } |
---|
2950 | |
---|
2951 | ////////////////////////////////////////////////////////////////////////////// |
---|
2952 | static |
---|
2953 | proc DropRedundant(list BO,list E) |
---|
2954 | "Internal procedure - no help and no example available |
---|
2955 | " |
---|
2956 | { |
---|
2957 | //--------------------------------------------------------------------------- |
---|
2958 | // Initialization and sanity checks |
---|
2959 | //--------------------------------------------------------------------------- |
---|
2960 | int ii,jj,kkdiff,nonnormal,ok; |
---|
2961 | ideal testid,dummy; |
---|
2962 | ideal center; |
---|
2963 | intvec transverse,dontdrop,zerovec; |
---|
2964 | transverse[size(BO[4])]=0; |
---|
2965 | dontdrop[size(E[4])]=0; |
---|
2966 | zerovec[size(E[4])]=0; |
---|
2967 | ideal J=BO[2]; |
---|
2968 | int dimJ=dim(std(BO[2])); |
---|
2969 | list templist; |
---|
2970 | if(size(E)<5) |
---|
2971 | { |
---|
2972 | //--- should not occur |
---|
2973 | return(E); |
---|
2974 | } |
---|
2975 | for(ii=1;ii<=BO[7][1];ii++) |
---|
2976 | { |
---|
2977 | if(BO[6][ii]==2) |
---|
2978 | { |
---|
2979 | kkdiff++; |
---|
2980 | } |
---|
2981 | } |
---|
2982 | int expDim=dimJ-E[2]+kkdiff; |
---|
2983 | if(size(E)==6) |
---|
2984 | { |
---|
2985 | nonnormal=E[6]; |
---|
2986 | } |
---|
2987 | //--------------------------------------------------------------------------- |
---|
2988 | // if dimJ were smaller than E[2], we would not have more than 3 entries in |
---|
2989 | // in the list E |
---|
2990 | // if dimJ is also at least E[3] and nonnormal is 0, we only need to test that |
---|
2991 | // * the intersection is of the expected dimension |
---|
2992 | // * the intersections of the BO[4][i] and J are normal crossing |
---|
2993 | // * the elements of E^+ have no influence (is done below) |
---|
2994 | //--------------------------------------------------------------------------- |
---|
2995 | if((E[3]<=dimJ)&&(!nonnormal)) |
---|
2996 | { |
---|
2997 | ideal bla=E[1]+BO[2]+BO[1]; |
---|
2998 | bla=radical(bla); |
---|
2999 | bla=mstd(bla)[2]; |
---|
3000 | |
---|
3001 | if(dim(std(slocusE(bla)))<0) |
---|
3002 | { |
---|
3003 | if(transversalT(J,BO[4])) |
---|
3004 | { |
---|
3005 | ok=1; |
---|
3006 | if(E[2]==E[3]) |
---|
3007 | { |
---|
3008 | //--- no further intersection with elements from E^+ |
---|
3009 | for(ii=1;ii<=size(E[4]);ii++) |
---|
3010 | { |
---|
3011 | if(dim_slocus(BO[2]+E[4][ii])!=-1) |
---|
3012 | { |
---|
3013 | dontdrop[ii]=1; |
---|
3014 | } |
---|
3015 | } |
---|
3016 | if(dontdrop==zerovec) |
---|
3017 | { |
---|
3018 | list relist; |
---|
3019 | relist[1]=std(J); |
---|
3020 | return(relist); |
---|
3021 | } |
---|
3022 | } |
---|
3023 | } |
---|
3024 | } |
---|
3025 | } |
---|
3026 | //--------------------------------------------------------------------------- |
---|
3027 | // now check whether the E_i actually occurring in the intersections meet |
---|
3028 | // J transversally (one by one) and mark those elements of E[4] where it is |
---|
3029 | // not the case |
---|
3030 | //--------------------------------------------------------------------------- |
---|
3031 | if(!ok) |
---|
3032 | { |
---|
3033 | for(ii=1;ii<=size(E[5]);ii++) |
---|
3034 | { |
---|
3035 | //--- test the ii-th tuple of E[4] resp. its indices E[5] |
---|
3036 | for(jj=1;jj<=size(E[5][ii]);jj++) |
---|
3037 | { |
---|
3038 | //--- if E[5][ii][jj]==1, E_jj is involved in E[4][ii] |
---|
3039 | if(E[5][ii][jj]==1) |
---|
3040 | { |
---|
3041 | //--- transversality not yet determined |
---|
3042 | if(transverse[jj]==0) |
---|
3043 | { |
---|
3044 | templist[1]=BO[4][jj]; |
---|
3045 | if(transversalT(BO[2],templist)) |
---|
3046 | { |
---|
3047 | transverse[jj]=1; |
---|
3048 | } |
---|
3049 | else |
---|
3050 | { |
---|
3051 | transverse[jj]=-1; |
---|
3052 | dontdrop[ii]=1; |
---|
3053 | } |
---|
3054 | } |
---|
3055 | else |
---|
3056 | { |
---|
3057 | //--- already computed transversality |
---|
3058 | if(transverse[jj]<0) |
---|
3059 | { |
---|
3060 | dontdrop[ii]=1; |
---|
3061 | } |
---|
3062 | } |
---|
3063 | } |
---|
3064 | } |
---|
3065 | } |
---|
3066 | } |
---|
3067 | //--------------------------------------------------------------------------- |
---|
3068 | // if one of the non-marked tuples from E^- in E[4] has an intersection |
---|
3069 | // of the expected dimension and does not meet any E_i from E^+ |
---|
3070 | // - except the ones which are met trivially - , it should be |
---|
3071 | // dropped from the list. |
---|
3072 | // it can also be dropped if an intersection occurs and normal crossing has |
---|
3073 | // been checked. |
---|
3074 | //--------------------------------------------------------------------------- |
---|
3075 | for(ii=1;ii<=size(E[4]);ii++) |
---|
3076 | { |
---|
3077 | //--- if E[4][ii] does not have transversal intersections, we cannot drop it |
---|
3078 | if(dontdrop[ii]==1) |
---|
3079 | { |
---|
3080 | ii++; |
---|
3081 | continue; |
---|
3082 | } |
---|
3083 | //--- testing ii-th tuple from E[4] |
---|
3084 | testid=BO[1]+BO[2]+E[4][ii]; |
---|
3085 | if(dim(std(testid))!=expDim) |
---|
3086 | { |
---|
3087 | //--- not expected dimension |
---|
3088 | dontdrop[ii]=1; |
---|
3089 | ii++; |
---|
3090 | continue; |
---|
3091 | } |
---|
3092 | testid=mstd(testid)[2]; |
---|
3093 | |
---|
3094 | if(dim(std(slocusE(testid)))>=0) |
---|
3095 | { |
---|
3096 | //--- not smooth, i.e. more than one component which intersect |
---|
3097 | dontdrop[ii]=1; |
---|
3098 | ii++; |
---|
3099 | continue; |
---|
3100 | } |
---|
3101 | //--- if E^+ is empty, we are done; otherwise check intersections with E^+ |
---|
3102 | if(BO[7][1]!=-1) |
---|
3103 | { |
---|
3104 | if(defined(pluslist)){kill pluslist;} |
---|
3105 | list pluslist; |
---|
3106 | for(jj=BO[7][1]+1;jj<=size(BO[4]);jj++) |
---|
3107 | { |
---|
3108 | dummy=BO[4][jj]+testid; |
---|
3109 | dummy=std(dummy); |
---|
3110 | if(expDim==dim(dummy)) |
---|
3111 | { |
---|
3112 | //--- intersection has wrong dimension |
---|
3113 | dontdrop[ii]=1; |
---|
3114 | break; |
---|
3115 | } |
---|
3116 | pluslist[jj-BO[7][1]]=BO[4][jj]+testid; |
---|
3117 | } |
---|
3118 | if(dontdrop[ii]==1) |
---|
3119 | { |
---|
3120 | ii++; |
---|
3121 | continue; |
---|
3122 | } |
---|
3123 | if(!normalCross(pluslist)) |
---|
3124 | { |
---|
3125 | //--- unfortunately, it is not normal crossing |
---|
3126 | dontdrop[ii]=1; |
---|
3127 | } |
---|
3128 | } |
---|
3129 | } |
---|
3130 | //--------------------------------------------------------------------------- |
---|
3131 | // The returned list should look like the truncated output of inters_E |
---|
3132 | //--------------------------------------------------------------------------- |
---|
3133 | list retlist; |
---|
3134 | for(ii=1;ii<=size(E[4]);ii++) |
---|
3135 | { |
---|
3136 | if(dontdrop[ii]==1) |
---|
3137 | { |
---|
3138 | if(size(center)>0) |
---|
3139 | { |
---|
3140 | center=intersect(center,E[4][ii]); |
---|
3141 | } |
---|
3142 | else |
---|
3143 | { |
---|
3144 | center=E[4][ii]; |
---|
3145 | } |
---|
3146 | } |
---|
3147 | } |
---|
3148 | retlist[1]=center; |
---|
3149 | retlist[2]=E[2]; |
---|
3150 | retlist[3]=E[3]; |
---|
3151 | return(retlist); |
---|
3152 | } |
---|
3153 | ////////////////////////////////////////////////////////////////////////////// |
---|
3154 | static |
---|
3155 | proc transversalT(ideal J, list E,list #) |
---|
3156 | "Internal procedure - no help and no example available |
---|
3157 | " |
---|
3158 | { |
---|
3159 | //---------------------------------------------------------------------------- |
---|
3160 | // check whether J and each element of the list E meet transversally |
---|
3161 | //---------------------------------------------------------------------------- |
---|
3162 | def R=basering; |
---|
3163 | if(size(#)>0) |
---|
3164 | { |
---|
3165 | ideal pp=#[1]; |
---|
3166 | } |
---|
3167 | int i; |
---|
3168 | ideal T,M; |
---|
3169 | ideal Jstd=std(J); |
---|
3170 | ideal Tstd; |
---|
3171 | int d=nvars(basering)-dim(Jstd)+1; // d=n-dim(V(J) \cap hypersurface) |
---|
3172 | for(i=1;i<=size(E);i++) |
---|
3173 | { |
---|
3174 | if(size(reduce(E[i],Jstd))==0) |
---|
3175 | { |
---|
3176 | //--- V(J) is contained in E[i] |
---|
3177 | return(0); |
---|
3178 | } |
---|
3179 | T=J,E[i]; |
---|
3180 | Tstd=std(T); |
---|
3181 | d=nvars(basering)-dim(Tstd); |
---|
3182 | if(deg(Tstd[1])!=0) |
---|
3183 | { |
---|
3184 | //--- intersection is non-empty |
---|
3185 | //!!! abgeklemmt, da es doch in der Praxis vorkommt und korrekt sein kann!!! |
---|
3186 | //!!! wenn ueberhaupt dann -1 zurueckgeben!!! |
---|
3187 | // if((d>=4)&&(size(T)>=10)){return(0);} |
---|
3188 | M=minor(jacob(T),d,Tstd)+T; |
---|
3189 | M=std(M); |
---|
3190 | if(deg(M[1])>0) |
---|
3191 | { |
---|
3192 | //--- intersection is not transversal |
---|
3193 | if(size(#)==0) |
---|
3194 | { |
---|
3195 | return(0); |
---|
3196 | } |
---|
3197 | M=std(radical(M)); |
---|
3198 | if(size(reduce(pp,M))>0){return(0);} |
---|
3199 | } |
---|
3200 | } |
---|
3201 | } |
---|
3202 | //--- passed all tests |
---|
3203 | return(1); |
---|
3204 | } |
---|
3205 | /////////////////////////////////////////////////////////////////////////////// |
---|
3206 | static |
---|
3207 | proc transversalTB(ideal J, list E,ideal V) |
---|
3208 | "Internal procedure - no help and no example available |
---|
3209 | " |
---|
3210 | { |
---|
3211 | //---------------------------------------------------------------------------- |
---|
3212 | // check whether J and each element of the list E meet transversally |
---|
3213 | //---------------------------------------------------------------------------- |
---|
3214 | def R=basering; |
---|
3215 | |
---|
3216 | int i; |
---|
3217 | ideal T,M; |
---|
3218 | ideal Jstd=std(J); |
---|
3219 | ideal Tstd; |
---|
3220 | int d=nvars(basering)-dim(Jstd)+1; // d=n-dim(V(J) \cap hypersurface) |
---|
3221 | for(i=1;i<=size(E);i++) |
---|
3222 | { |
---|
3223 | if(size(reduce(E[i],Jstd))==0) |
---|
3224 | { |
---|
3225 | //--- V(J) is contained in E[i] |
---|
3226 | return(0); |
---|
3227 | } |
---|
3228 | T=J,E[i]; |
---|
3229 | Tstd=std(T); |
---|
3230 | d=nvars(basering)-dim(Tstd); |
---|
3231 | if(deg(Tstd[1])!=0) |
---|
3232 | { |
---|
3233 | //--- intersection is non-empty |
---|
3234 | if((d>=4)&&(size(T)>=10)){return(0);} |
---|
3235 | M=minor(jacob(T),d,Tstd)+T; |
---|
3236 | M=std(M+V); |
---|
3237 | if(deg(M[1])>0) |
---|
3238 | { |
---|
3239 | return(0); |
---|
3240 | } |
---|
3241 | } |
---|
3242 | } |
---|
3243 | //--- passed all tests |
---|
3244 | return(1); |
---|
3245 | } |
---|
3246 | /////////////////////////////////////////////////////////////////////////////// |
---|
3247 | static |
---|
3248 | proc powerI(ideal I,int n,int m) |
---|
3249 | { |
---|
3250 | //--- compute (n!/m)-th power of I, more efficient variant |
---|
3251 | int i; |
---|
3252 | int mon=1; |
---|
3253 | int ch=char(basering); |
---|
3254 | for(i=1;i<=size(I);i++) |
---|
3255 | { |
---|
3256 | if(size(I[i])>1){mon=0;break;} |
---|
3257 | } |
---|
3258 | if(mon) |
---|
3259 | { |
---|
3260 | if(size(reduce(I,std(radical(I[1]))))<size(I)-1){mon=0;} |
---|
3261 | } |
---|
3262 | if((mon)&&(size(I)>3)) |
---|
3263 | { |
---|
3264 | int e=int(factorial(n,ch))/m; |
---|
3265 | ideal J=1; |
---|
3266 | poly p=I[1]; |
---|
3267 | I=I[2..size(I)]; |
---|
3268 | ideal K=p^e; |
---|
3269 | for(i=1;i<=e;i++) |
---|
3270 | { |
---|
3271 | J=interred(J*I); |
---|
3272 | K=K,(p^(e-i))*J; |
---|
3273 | } |
---|
3274 | return(K); |
---|
3275 | } |
---|
3276 | for(i=n;i>1;i--) |
---|
3277 | { |
---|
3278 | if(i!=m) |
---|
3279 | { |
---|
3280 | I=I^i; |
---|
3281 | } |
---|
3282 | } |
---|
3283 | return(I); |
---|
3284 | } |
---|
3285 | |
---|
3286 | /////////////////////////////////////////////////////////////////////////////// |
---|
3287 | static |
---|
3288 | proc Coeff(list BO, int b, list #) |
---|
3289 | "USAGE: Coeff (BO); |
---|
3290 | @* BO = basic object, a list: ideal W, |
---|
3291 | @* ideal J, |
---|
3292 | @* intvec b (already truncated for Coeff), |
---|
3293 | @* list Ex (already truncated for Coeff), |
---|
3294 | @* ideal ab, |
---|
3295 | @* intvec v, |
---|
3296 | @* intvec w (already truncated for Coeff), |
---|
3297 | @* matrix M |
---|
3298 | @* b = integer indication bmax(BO) |
---|
3299 | ASSUME: R = basering, a polynomial ring, W an ideal of R, |
---|
3300 | @* J = ideal containing W |
---|
3301 | COMPUTE: Coeff-Ideal of BO as defined in [Bravo,Encinas,Villamayor] |
---|
3302 | RETURN: basic object of the Coeff-Ideal |
---|
3303 | EXAMPLE: example Coeff; shows an example |
---|
3304 | " |
---|
3305 | { |
---|
3306 | //!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! |
---|
3307 | //!!! TASK: lower dimension by more than one in a single step if possible !!! |
---|
3308 | //!!! (improve bookkeeping of invariants in Coeff and Center) !!! |
---|
3309 | //!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! |
---|
3310 | //--------------------------------------------------------------------------- |
---|
3311 | // Initialization and sanity checks |
---|
3312 | //--------------------------------------------------------------------------- |
---|
3313 | int ch=char(basering); |
---|
3314 | int i,k,dummy,errtype; |
---|
3315 | int ma=size(BO[4]); |
---|
3316 | intvec merk; |
---|
3317 | if(!defined(debugCoeff)) |
---|
3318 | { |
---|
3319 | int debugCoeff; |
---|
3320 | } |
---|
3321 | ideal C; |
---|
3322 | list L; |
---|
3323 | if(size(#)!=0) |
---|
3324 | { |
---|
3325 | if(typeof(#[1])=="ideal") |
---|
3326 | { |
---|
3327 | L=#; |
---|
3328 | } |
---|
3329 | else |
---|
3330 | { |
---|
3331 | ma=#[1]; |
---|
3332 | L=DeltaList(BO); |
---|
3333 | } |
---|
3334 | } |
---|
3335 | else |
---|
3336 | { |
---|
3337 | L=DeltaList(BO); |
---|
3338 | } |
---|
3339 | |
---|
3340 | if(debugCoeff) |
---|
3341 | { |
---|
3342 | "----> In Coeff: result of DeltaList:"; |
---|
3343 | L; |
---|
3344 | } |
---|
3345 | int d=size(L); // bmax of BO |
---|
3346 | if((debugCoeff)&&(d!=b)) |
---|
3347 | { |
---|
3348 | "!!!!!! Length of DeltaList does not equal second argument !!!!!!"; |
---|
3349 | "!!!!!! BO might not have been ord ~ 1 or wrong b !!!!!!"; |
---|
3350 | } |
---|
3351 | if(b>=6){return(0);} // b is too big |
---|
3352 | int e=int(factorial(b,ch)); // b of Coeff-Ideal |
---|
3353 | if(e==0) |
---|
3354 | { |
---|
3355 | ERROR( "// characteristic too small for forming b! ."); |
---|
3356 | } |
---|
3357 | if(b==0) |
---|
3358 | { |
---|
3359 | ERROR( "// second argument to Coeff should never be zero." ); |
---|
3360 | } |
---|
3361 | //---------------------------------------------------------------------------- |
---|
3362 | // Form the Coeff-Ideal |
---|
3363 | // Step 1: choose hypersurface |
---|
3364 | // Step 2: sum over correct powers of Delta^i(BO[2]) |
---|
3365 | // Step 3: do the intersection |
---|
3366 | //---------------------------------------------------------------------------- |
---|
3367 | //--- Step 1 |
---|
3368 | ideal Z; |
---|
3369 | poly p; |
---|
3370 | for(i=1;i<=ncols(L[d]);i++) |
---|
3371 | { |
---|
3372 | //--- Look for smooth hypersurface in generators of Delta^(bmax-1)(BO[2]) |
---|
3373 | dummy=goodChoice(BO,L[d][i]); |
---|
3374 | if(!dummy) |
---|
3375 | { |
---|
3376 | Z= L[d][i]; |
---|
3377 | break; |
---|
3378 | } |
---|
3379 | else |
---|
3380 | { |
---|
3381 | if(dummy>1) |
---|
3382 | { |
---|
3383 | merk[size(merk)+1]=i; |
---|
3384 | } |
---|
3385 | if(dummy>errtype) |
---|
3386 | { |
---|
3387 | errtype=dummy; |
---|
3388 | } |
---|
3389 | } |
---|
3390 | } |
---|
3391 | if(size(Z)==0) |
---|
3392 | { |
---|
3393 | //--- no suitable element in generators of Delta^(bmax-1)(BO[2]) |
---|
3394 | //--- try random linear combination |
---|
3395 | for(k=1;k<=10;k++) |
---|
3396 | { |
---|
3397 | for(i=2;i<=size(merk);i++) |
---|
3398 | { |
---|
3399 | p=p+random(-100,100)*L[d][merk[i]]; |
---|
3400 | } |
---|
3401 | dummy=goodChoice(BO,p); |
---|
3402 | if(!dummy) |
---|
3403 | { |
---|
3404 | Z=p; |
---|
3405 | break; |
---|
3406 | } |
---|
3407 | else |
---|
3408 | { |
---|
3409 | p=0; |
---|
3410 | } |
---|
3411 | } |
---|
3412 | if(dummy) |
---|
3413 | { |
---|
3414 | for(i=1;i<=size(L[d]);i++) |
---|
3415 | { |
---|
3416 | p=p+random(-100,100)*L[d][i]; |
---|
3417 | } |
---|
3418 | dummy=goodChoice(BO,p); |
---|
3419 | if(!dummy) |
---|
3420 | { |
---|
3421 | //--- found a suitable one |
---|
3422 | Z=p; |
---|
3423 | } |
---|
3424 | } |
---|
3425 | if(dummy) |
---|
3426 | { |
---|
3427 | //--- did not find a suitable random linear combination either |
---|
3428 | if(dummy>errtype) |
---|
3429 | { |
---|
3430 | errtype=dummy; |
---|
3431 | } |
---|
3432 | list retlist=errtype,L[d]; |
---|
3433 | return(retlist); |
---|
3434 | } |
---|
3435 | } |
---|
3436 | if(debugCoeff) |
---|
3437 | { |
---|
3438 | "----> In Coeff: Chosen hypersurface"; |
---|
3439 | Z; |
---|
3440 | } |
---|
3441 | //--- Step 2 |
---|
3442 | C=Z; |
---|
3443 | for(i=0;i<b;i++) |
---|
3444 | { |
---|
3445 | C=C,powerI(simplify(reduce(L[i+1],std(Z)),2),b,b-i); |
---|
3446 | } |
---|
3447 | C=interred(C); |
---|
3448 | |
---|
3449 | if(debugCoeff) |
---|
3450 | { |
---|
3451 | "----> In Coeff: J before saturation"; |
---|
3452 | C; |
---|
3453 | } |
---|
3454 | |
---|
3455 | //--- Step 3 |
---|
3456 | BO[1]=BO[1]+Z; |
---|
3457 | BO[2]=C; |
---|
3458 | for(i=1;i<=size(BO[4]);i++) |
---|
3459 | { |
---|
3460 | BO[6][i]=0; // reset intersection indicator |
---|
3461 | BO[4][i]=BO[4][i]+Z; // intersect the E_i |
---|
3462 | if(i<=ma) |
---|
3463 | { |
---|
3464 | BO[2]=sat(BO[2],BO[4][i]+BO[1])[1]; |
---|
3465 | // "strict transform" of J w.r.t E, not "total" |
---|
3466 | } |
---|
3467 | } |
---|
3468 | if(debugCoeff) |
---|
3469 | { |
---|
3470 | "----> In Coeff:"; |
---|
3471 | " J after saturation:"; |
---|
3472 | BO[2]; |
---|
3473 | } |
---|
3474 | return(BO); |
---|
3475 | } |
---|
3476 | example |
---|
3477 | {"EXAMPLE:"; |
---|
3478 | echo = 2; |
---|
3479 | ring R=0,(x,y,z),dp; |
---|
3480 | |
---|
3481 | ideal W; |
---|
3482 | ideal J=z^2+x^2*y^2; |
---|
3483 | intvec b=0; |
---|
3484 | list E; |
---|
3485 | ideal abb=maxideal(1); |
---|
3486 | intvec v; |
---|
3487 | intvec w=-1; |
---|
3488 | matrix M; |
---|
3489 | |
---|
3490 | list BO=W,J,b,E,abb,v,w,M; |
---|
3491 | |
---|
3492 | Coeff(BO,2); |
---|
3493 | } |
---|
3494 | ////////////////////////////////////////////////////////////////////////////// |
---|
3495 | static |
---|
3496 | proc goodChoice(list BO, poly p) |
---|
3497 | "Internal procedure - no help and no example available |
---|
3498 | " |
---|
3499 | { |
---|
3500 | //--------------------------------------------------------------------------- |
---|
3501 | // test whether new W is smooth |
---|
3502 | //--------------------------------------------------------------------------- |
---|
3503 | ideal W=BO[1]+ideal(p); |
---|
3504 | if(size(reduce(p,std(BO[1])))==0) |
---|
3505 | { |
---|
3506 | //--- p is already in BO[1], i.e. does not define a hypersurface in W |
---|
3507 | return(1); |
---|
3508 | } |
---|
3509 | if(dim(std(slocusE(W)))>=0) |
---|
3510 | // if(dim(timeStd(slocusE(W),20))>=0) |
---|
3511 | { |
---|
3512 | //--- new W would not be smooth |
---|
3513 | return(1); |
---|
3514 | } |
---|
3515 | if(size(BO[4])==0) |
---|
3516 | { |
---|
3517 | //--- E is empty, no further tests necessary |
---|
3518 | return(0); |
---|
3519 | } |
---|
3520 | //-------------------------------------------------------------------------- |
---|
3521 | // test whether the hypersurface meets the E_i transversally |
---|
3522 | //-------------------------------------------------------------------------- |
---|
3523 | list E=BO[4]; |
---|
3524 | int i,d; |
---|
3525 | ideal T=W; |
---|
3526 | ideal Tstd=std(T); |
---|
3527 | d=nvars(basering)-dim(Tstd)+1; |
---|
3528 | ideal M; |
---|
3529 | for(i=1;i<=size(E);i++) |
---|
3530 | { |
---|
3531 | T=W,E[i]; |
---|
3532 | M=minor(jacob(T),d,Tstd)+T; |
---|
3533 | M=std(M); |
---|
3534 | if(deg(M[1])>0) |
---|
3535 | { |
---|
3536 | //--- intersection not transversal |
---|
3537 | return(2); |
---|
3538 | } |
---|
3539 | } |
---|
3540 | //-------------------------------------------------------------------------- |
---|
3541 | // test whether the new E_i have normal crossings |
---|
3542 | //-------------------------------------------------------------------------- |
---|
3543 | for(i=1;i<=size(E);i++) |
---|
3544 | { |
---|
3545 | E[i]=E[i],p; |
---|
3546 | } |
---|
3547 | if(normalCross(E)) |
---|
3548 | { |
---|
3549 | return(0); |
---|
3550 | } |
---|
3551 | else |
---|
3552 | { |
---|
3553 | return(2); |
---|
3554 | } |
---|
3555 | } |
---|
3556 | ////////////////////////////////////////////////////////////////////////////// |
---|
3557 | |
---|
3558 | proc showBO(list BO) |
---|
3559 | "USAGE: showBO(BO); |
---|
3560 | @* BO=basic object, a list: ideal W, |
---|
3561 | @* ideal J, |
---|
3562 | @* intvec b (already truncated for Coeff), |
---|
3563 | @* list Ex (already truncated for Coeff), |
---|
3564 | @* ideal ab, |
---|
3565 | @* intvec v, |
---|
3566 | @* intvec w (already truncated for Coeff), |
---|
3567 | @* matrix M |
---|
3568 | RETURN: nothing, only pretty printing |
---|
3569 | EXAMPLE: none |
---|
3570 | " |
---|
3571 | { |
---|
3572 | " "; |
---|
3573 | "==== W: ";BO[1];" "; |
---|
3574 | "==== J: ";BO[2];" "; |
---|
3575 | int i; |
---|
3576 | list M; |
---|
3577 | for(i=1;i<=size(BO[4]);i++) |
---|
3578 | { |
---|
3579 | M[i]=ideal(BO[4][i]); |
---|
3580 | } |
---|
3581 | "==== E: ";print(M);" "; |
---|
3582 | "==== Intersection"; print(BO[8]);" "; |
---|
3583 | } |
---|
3584 | ////////////////////////////////////////////////////////////////////////////// |
---|
3585 | static |
---|
3586 | proc max(int i,int j) |
---|
3587 | "Internal procedure - no help and no example available |
---|
3588 | " |
---|
3589 | { |
---|
3590 | //--- why is there no proc for max in general.lib? |
---|
3591 | if(i>j){return(i);} |
---|
3592 | return(j); |
---|
3593 | } |
---|
3594 | ////////////////////////////////////////////////////////////////////////////// |
---|
3595 | static |
---|
3596 | proc min(int i,int j) |
---|
3597 | "Internal procedure - no help and no example available |
---|
3598 | " |
---|
3599 | { |
---|
3600 | //--- why is there no proc for max in general.lib? |
---|
3601 | if(i<j){return(i);} |
---|
3602 | return(j); |
---|
3603 | } |
---|
3604 | ////////////////////////////////////////////////////////////////////////////// |
---|
3605 | //////////////////////// main procedure //////////////////////////////// |
---|
3606 | ////////////////////////////////////////////////////////////////////////////// |
---|
3607 | proc resolve(ideal J, list #) |
---|
3608 | "USAGE: resolve (J); or resolve (J,i[,k]); |
---|
3609 | @* J ideal |
---|
3610 | @* i,k int |
---|
3611 | COMPUTE: a resolution of J, |
---|
3612 | @* if i > 0 debugging is turned on according to the following switches: |
---|
3613 | @* j1: value 0 or 1; turn off or on correctness checks in all steps |
---|
3614 | @* j2: value 0 or 2; turn off or on debugCenter |
---|
3615 | @* j3: value 0 or 4; turn off or on debugBlowUp |
---|
3616 | @* j4: value 0 or 8; turn off or on debugCoeff |
---|
3617 | @* j5: value 0 or 16:turn off or on debugging of Intersection with E^- |
---|
3618 | @* j6: value 0 or 32:turn off or on stop after pass throught the loop |
---|
3619 | @* i=j1+j2+j3+j4+j5+j6 |
---|
3620 | RETURN: a list l of 2 lists of rings |
---|
3621 | l[1][i] is a ring containing a basic object BO, the result of the |
---|
3622 | resolution. |
---|
3623 | l[2] contains all rings which occured during the resolution process |
---|
3624 | EXAMPLE: example resolve; shows an example |
---|
3625 | " |
---|
3626 | { |
---|
3627 | //---------------------------------------------------------------------------- |
---|
3628 | // Initialization and sanity checks |
---|
3629 | //---------------------------------------------------------------------------- |
---|
3630 | def R=basering; |
---|
3631 | list allRings; |
---|
3632 | allRings[1]=R; |
---|
3633 | list endRings; |
---|
3634 | module path=[0,-1]; |
---|
3635 | ideal W; |
---|
3636 | list E; |
---|
3637 | ideal abb=maxideal(1); |
---|
3638 | intvec v; |
---|
3639 | intvec bvec; |
---|
3640 | intvec w=-1; |
---|
3641 | matrix intE; |
---|
3642 | int extra,bm; |
---|
3643 | if(defined(BO)){kill BO;} |
---|
3644 | if(defined(cent)){kill cent;} |
---|
3645 | |
---|
3646 | ideal Jrad=equiRadical(J); |
---|
3647 | if(size(reduce(Jrad,std(J)))!=0) |
---|
3648 | { |
---|
3649 | "WARNING! The input is not reduced or not equidimensional!"; |
---|
3650 | "We will continue with the reduced top-dimensional part of input"; |
---|
3651 | J=Jrad; |
---|
3652 | } |
---|
3653 | |
---|
3654 | int i,j,debu,loca,locaT,ftemp,debugResolve,smooth; |
---|
3655 | //--- switches for local and for debugging may occur in any order |
---|
3656 | i=size(#); |
---|
3657 | extra=3; |
---|
3658 | for(j=1;j<=i;j++) |
---|
3659 | { |
---|
3660 | if(typeof(#[j])=="int") |
---|
3661 | { |
---|
3662 | debugResolve=#[j]; |
---|
3663 | //--- debu: debug switch for resolve, smallest bit in debugResolve |
---|
3664 | debu=debugResolve mod 2; |
---|
3665 | } |
---|
3666 | else |
---|
3667 | { |
---|
3668 | if(#[j]=="M") |
---|
3669 | { |
---|
3670 | bm=1; |
---|
3671 | ERROR("Not implemented yet"); |
---|
3672 | } |
---|
3673 | if(#[j]=="E"){extra=0;} |
---|
3674 | if(#[j]=="A"){extra=2;} |
---|
3675 | if(#[j]=="K"){extra=3;} |
---|
3676 | if(#[j]=="L"){loca=1;} |
---|
3677 | } |
---|
3678 | } |
---|
3679 | if(loca) |
---|
3680 | { |
---|
3681 | list qs=minAssGTZ(J); |
---|
3682 | ideal K=ideal(1); |
---|
3683 | for(j=1;j<=size(qs);j++) |
---|
3684 | { |
---|
3685 | if(size(reduce(qs[j],std(maxideal(1))))==0) |
---|
3686 | { |
---|
3687 | K=intersect(K,qs[j]); |
---|
3688 | } |
---|
3689 | } |
---|
3690 | J=K; |
---|
3691 | list qr=minAssGTZ(slocus(J)); |
---|
3692 | K=ideal(1); |
---|
3693 | for(j=1;j<=size(qr);j++) |
---|
3694 | { |
---|
3695 | if(size(reduce(qr[j],std(maxideal(1))))!=0) |
---|
3696 | { |
---|
3697 | K=intersect(K,qr[j]); |
---|
3698 | smooth++; |
---|
3699 | } |
---|
3700 | else |
---|
3701 | { |
---|
3702 | if(dim(std(qr[j]))>0){loca=0;} |
---|
3703 | //---- test for isolated singularity at 0 |
---|
3704 | } |
---|
3705 | } |
---|
3706 | K=std(K); |
---|
3707 | //---- if deg(K[1])==0 the point 0 is on all components of the singular |
---|
3708 | //---- locus and we can work globally |
---|
3709 | if(smooth==size(qr)){smooth=-1;} |
---|
3710 | //---- the point 0 is not on the singular locus |
---|
3711 | if((deg(K[1])>0)&&(smooth>=0)&&(!loca)) |
---|
3712 | { |
---|
3713 | locaT=1; |
---|
3714 | poly @p; |
---|
3715 | for(j=1;j<=size(K);j++) |
---|
3716 | { |
---|
3717 | if(jet(K[j],0)!=0) |
---|
3718 | { |
---|
3719 | @p=K[j]; |
---|
3720 | break; |
---|
3721 | } |
---|
3722 | } |
---|
3723 | export(@p); |
---|
3724 | } |
---|
3725 | if((loca)&&(!smooth)){loca=0;} |
---|
3726 | //---- the case that 0 is isolated singularity and the only singular point |
---|
3727 | } |
---|
3728 | export(locaT); |
---|
3729 | //---In case of option "L" the following holds |
---|
3730 | //---loca=0 and locaT=0 we perform the global case |
---|
3731 | //---loca !=0: 0 is isolated singular point, but there are other singularities |
---|
3732 | //---locaT!=0: O is singular point, but not isolated, and there is a componente//--- of the singular locus not containing 0 |
---|
3733 | |
---|
3734 | //--- if necessary, set the corresponding debugFlags |
---|
3735 | if(defined(debugResolve)) |
---|
3736 | { |
---|
3737 | //--- 2nd bit from the right |
---|
3738 | int debugCenter=(debugResolve div 2) mod 2; |
---|
3739 | export debugCenter; |
---|
3740 | //--- 3rd bit from the right |
---|
3741 | int debugBlowUp=(debugResolve div 4) mod 2; |
---|
3742 | export debugBlowUp; |
---|
3743 | //--- 4th bit from the right |
---|
3744 | int debugCoeff=(debugResolve div 8) mod 2; |
---|
3745 | export debugCoeff; |
---|
3746 | //--- 5th bit from the right |
---|
3747 | int debug_Inters_E=(debugResolve div 16) mod 2; |
---|
3748 | export debug_Inters_E; |
---|
3749 | //--- 6th bit from the right |
---|
3750 | int praes_stop=(debugResolve div 32) mod 2; |
---|
3751 | } |
---|
3752 | //--- set the correct attributes to J for speed ups |
---|
3753 | if( typeof(attrib(J,"isEqui"))!="int" ) |
---|
3754 | { |
---|
3755 | if(size(J)==1) |
---|
3756 | { |
---|
3757 | attrib(J,"isEqui",1); |
---|
3758 | } |
---|
3759 | else |
---|
3760 | { |
---|
3761 | attrib(J,"isEqui",0); |
---|
3762 | } |
---|
3763 | } |
---|
3764 | if(size(J)==1) |
---|
3765 | { |
---|
3766 | attrib(J,"isHy",1); |
---|
3767 | } |
---|
3768 | else |
---|
3769 | { |
---|
3770 | attrib(J,"isHy",0); |
---|
3771 | } |
---|
3772 | //--- create the BO |
---|
3773 | list BO=W,J,bvec,E,abb,v,w,intE; |
---|
3774 | if(defined(invSat)){kill invSat;} |
---|
3775 | list invSat=ideal(0),intvec(0); |
---|
3776 | export(invSat); |
---|
3777 | if(bm) |
---|
3778 | { |
---|
3779 | intmat invmat[2][1]=0,-1; |
---|
3780 | BO[9]=invmat; |
---|
3781 | } |
---|
3782 | else |
---|
3783 | { |
---|
3784 | BO[9]=intvec(0); |
---|
3785 | } |
---|
3786 | export BO; |
---|
3787 | list tmpList; |
---|
3788 | int blo; |
---|
3789 | int k,Ecount,tmpPtr; |
---|
3790 | i=0; |
---|
3791 | if(smooth==-1) |
---|
3792 | { |
---|
3793 | endRings[1]=R; |
---|
3794 | list result=endRings,allRings; |
---|
3795 | if(debu) |
---|
3796 | { |
---|
3797 | "============= result will be tested =========="; |
---|
3798 | " "; |
---|
3799 | "the number of charts obtained:",size(endRings); |
---|
3800 | "============= result is o.k. =========="; |
---|
3801 | } |
---|
3802 | kill debugCenter,debugBlowUp,debugCoeff,debug_Inters_E; |
---|
3803 | return(result); |
---|
3804 | } |
---|
3805 | //----------------------------------------------------------------------------- |
---|
3806 | // While there are rings to be considered, determine center and blow up |
---|
3807 | //----------------------------------------------------------------------------- |
---|
3808 | while(i<size(allRings)) |
---|
3809 | { |
---|
3810 | i++; |
---|
3811 | def S=allRings[i]; |
---|
3812 | setring S; |
---|
3813 | list pr; |
---|
3814 | ideal Jstd=std(BO[2]); |
---|
3815 | //----------------------------------------------------------------------------- |
---|
3816 | // Determine Center |
---|
3817 | //----------------------------------------------------------------------------- |
---|
3818 | if(i==1) |
---|
3819 | { |
---|
3820 | list deltaL=DeltaList(BO); |
---|
3821 | ideal sL=radical(deltaL[size(deltaL)]); |
---|
3822 | if((deg(std(slocus(sL))[1])==0)&&(size(minAssGTZ(sL))==1)) |
---|
3823 | { |
---|
3824 | list @ce=sL,intvec(-1),intvec(0),intvec(0); |
---|
3825 | ideal cent=@ce[1]; |
---|
3826 | } |
---|
3827 | } |
---|
3828 | //--- before computing a center, check whether we have a stored one |
---|
3829 | if(size(BO)>9) |
---|
3830 | { |
---|
3831 | while(size(BO[10])>0) |
---|
3832 | { |
---|
3833 | list @ce=BO[10][1]; |
---|
3834 | //--- check of the center |
---|
3835 | // @ce=correctC(BO,@ce,bm); |
---|
3836 | //--- use stored center |
---|
3837 | BO[10]=delete(BO[10],1); |
---|
3838 | if(size(@ce[1])==0) |
---|
3839 | { |
---|
3840 | //--- stored center was not ok |
---|
3841 | continue; |
---|
3842 | } |
---|
3843 | tmpPtr=0; |
---|
3844 | for(Ecount=1;Ecount <= size(@ce[2]); Ecount++) |
---|
3845 | { |
---|
3846 | if(@ce[2][Ecount]>-1) |
---|
3847 | { |
---|
3848 | tmpPtr=tmpPtr+@ce[2][Ecount]; |
---|
3849 | } |
---|
3850 | else |
---|
3851 | { |
---|
3852 | @ce[2][Ecount]=size(BO[4])-tmpPtr-1; |
---|
3853 | for(int cnthlp=1;cnthlp<=size(BO[10]);cnthlp++) |
---|
3854 | { |
---|
3855 | BO[10][cnthlp][2][Ecount]=@ce[2][Ecount]; |
---|
3856 | } |
---|
3857 | kill cnthlp; |
---|
3858 | break; |
---|
3859 | } |
---|
3860 | } |
---|
3861 | if(Ecount<size(@ce[2])) |
---|
3862 | { |
---|
3863 | for(tmpPtr=Ecount+1;tmpPtr<=size(@ce[2]);tmpPtr++) |
---|
3864 | { |
---|
3865 | @ce[2][tmpPtr]=0; |
---|
3866 | for(int cnthlp=1;cnthlp<=size(BO[10]);cnthlp++) |
---|
3867 | { |
---|
3868 | BO[10][cnthlp][2][tmpPtr]=@ce[2][tmpPtr]; |
---|
3869 | } |
---|
3870 | kill cnthlp; |
---|
3871 | } |
---|
3872 | } |
---|
3873 | break; |
---|
3874 | } |
---|
3875 | if(defined(@ce)) |
---|
3876 | { |
---|
3877 | if(size(@ce[1])==0) |
---|
3878 | { |
---|
3879 | kill @ce; |
---|
3880 | } |
---|
3881 | else |
---|
3882 | { |
---|
3883 | ideal cent=@ce[1]; |
---|
3884 | } |
---|
3885 | } |
---|
3886 | if(size(BO[10])==0) |
---|
3887 | { |
---|
3888 | //--- previously had stored centers, all have been used; we need to clean up |
---|
3889 | BO=delete(BO,10); |
---|
3890 | } |
---|
3891 | } |
---|
3892 | if((loca)&&(i==1)) |
---|
3893 | { |
---|
3894 | //--- local case: initial step is blow-up in origin |
---|
3895 | if(defined(@ce)){kill @ce;} |
---|
3896 | if(defined(cent)){kill cent;} |
---|
3897 | if(size(reduce(slocusE(BO[2]),std(maxideal(1))))==0) |
---|
3898 | { |
---|
3899 | list @ce=maxideal(1),intvec(-1),intvec(0),intvec(0); |
---|
3900 | } |
---|
3901 | else |
---|
3902 | { |
---|
3903 | list @ce=BO[2],intvec(-1),intvec(1),intvec(0); |
---|
3904 | } |
---|
3905 | ideal cent=@ce[1]; |
---|
3906 | } |
---|
3907 | if(((loca)||(locaT))&&(i!=1)) |
---|
3908 | { |
---|
3909 | int JmeetsE; |
---|
3910 | for(j=1;j<=size(BO[4]);j++) |
---|
3911 | { |
---|
3912 | if(deg(std(BO[2]+BO[4][j])[1])!=0) |
---|
3913 | { |
---|
3914 | JmeetsE=1; |
---|
3915 | break; |
---|
3916 | } |
---|
3917 | } |
---|
3918 | if(!JmeetsE) |
---|
3919 | { |
---|
3920 | list @ce=BO[2],intvec(-1),intvec(1),intvec(0); |
---|
3921 | ideal cent=@ce[1]; |
---|
3922 | } |
---|
3923 | kill JmeetsE; |
---|
3924 | } |
---|
3925 | if((locaT)&&(!defined(@ce))) |
---|
3926 | { |
---|
3927 | if(@p!=1) |
---|
3928 | { |
---|
3929 | list tr=minAssGTZ(slocusE(BO[2])); |
---|
3930 | ideal L=ideal(1); |
---|
3931 | for(j=1;j<=size(tr);j++) |
---|
3932 | { |
---|
3933 | if(size(reduce(ideal(@p),std(tr[j])))==0) |
---|
3934 | { |
---|
3935 | L=intersect(L,tr[j]); |
---|
3936 | } |
---|
3937 | } |
---|
3938 | L=std(L); |
---|
3939 | if(deg(L[1])==0) |
---|
3940 | { |
---|
3941 | @p=1; |
---|
3942 | } |
---|
3943 | else |
---|
3944 | { |
---|
3945 | ideal fac=factorize(@p,1); |
---|
3946 | if(size(fac)==1) |
---|
3947 | { |
---|
3948 | @p=fac[1]; |
---|
3949 | } |
---|
3950 | else |
---|
3951 | { |
---|
3952 | for(j=1;j<=size(fac);j++) |
---|
3953 | { |
---|
3954 | if(reduce(fac[j],L)==0) |
---|
3955 | { |
---|
3956 | @p=fac[j]; |
---|
3957 | break; |
---|
3958 | } |
---|
3959 | } |
---|
3960 | } |
---|
3961 | } |
---|
3962 | kill tr,L; |
---|
3963 | } |
---|
3964 | execute("ring R1=("+charstr(S)+"),(@z,"+varstr(S)+"),dp;"); |
---|
3965 | poly p=imap(S,@p); |
---|
3966 | list BO=imap(S,BO); |
---|
3967 | list invSat=imap(S,invSat); |
---|
3968 | export(invSat); |
---|
3969 | ideal te=std(BO[2]); |
---|
3970 | BO[1]=BO[1]+ideal(@z*p-1); |
---|
3971 | BO[2]=BO[2]+ideal(@z*p-1); |
---|
3972 | for(j=1;j<=size(BO[4]);j++) |
---|
3973 | { |
---|
3974 | BO[4][j]=BO[4][j]+ideal(@z*p-1); |
---|
3975 | } |
---|
3976 | //--- for computation of center: drop components not meeting the Ei |
---|
3977 | def BO2=BO; |
---|
3978 | list qs=minAssGTZ(BO2[2]); |
---|
3979 | ideal K=ideal(1); |
---|
3980 | for(j=1;j<=size(qs);j++) |
---|
3981 | { |
---|
3982 | if(CompMeetsE(qs[j],BO2[4])) |
---|
3983 | { |
---|
3984 | K=intersect(K,qs[j]); |
---|
3985 | } |
---|
3986 | } |
---|
3987 | BO2[2]=K; |
---|
3988 | //--- check whether we are done |
---|
3989 | if(deg(std(BO2[2])[1])==0) |
---|
3990 | { |
---|
3991 | list @ce=BO[2],intvec(-1),intvec(1),intvec(0); |
---|
3992 | } |
---|
3993 | if(!defined(@ce)) |
---|
3994 | { |
---|
3995 | if(bm) |
---|
3996 | { |
---|
3997 | list @ce=CenterBM(BO2); |
---|
3998 | } |
---|
3999 | else |
---|
4000 | { |
---|
4001 | list @ce=CenterBO(BO2); |
---|
4002 | } |
---|
4003 | } |
---|
4004 | //--- if computation of center returned BO2[2], we are done |
---|
4005 | //--- ==> set @ce to BO[2], because later checks work with BO instead of BO2 |
---|
4006 | if((size(reduce(@ce[1],std(BO2[2])))==0)&& |
---|
4007 | (size(reduce(BO2[2],std(@ce[1])))==0)) |
---|
4008 | { |
---|
4009 | @ce[1]=BO[2]; |
---|
4010 | } |
---|
4011 | if(size(specialReduce(@ce[1],te,p))==0) |
---|
4012 | { |
---|
4013 | BO=imap(S,BO); |
---|
4014 | @ce[1]=BO[2]; |
---|
4015 | } |
---|
4016 | else |
---|
4017 | { |
---|
4018 | //@ce=correctC(BO,@ce,bm); |
---|
4019 | @ce[1]=eliminate(@ce[1],@z); |
---|
4020 | } |
---|
4021 | setring S; |
---|
4022 | list @ce=imap(R1,@ce); |
---|
4023 | kill R1; |
---|
4024 | |
---|
4025 | if((size(reduce(BO[2],std(@ce[1])))==0) |
---|
4026 | &&(size(reduce(@ce[1],Jstd))==0)) |
---|
4027 | { |
---|
4028 | //--- J and center coincide |
---|
4029 | pr[1]=@ce[1]; |
---|
4030 | ideal cent=@ce[1]; |
---|
4031 | } |
---|
4032 | else |
---|
4033 | { |
---|
4034 | //--- decompose center and use first component |
---|
4035 | pr=minAssGTZ(@ce[1]); |
---|
4036 | if(size(reduce(@p,std(pr[1])))==0){"Achtung";~;} |
---|
4037 | if(deg(std(slocus(pr[1]))[1])>0){"singulaer";~;} |
---|
4038 | ideal cent=pr[1]; |
---|
4039 | } |
---|
4040 | if(size(pr)>1) |
---|
4041 | { |
---|
4042 | //--- store the other components |
---|
4043 | for(k=2;k<=size(pr);k++) |
---|
4044 | { |
---|
4045 | if(size(reduce(@p,std(pr[k])))==0){"Achtung";~;} |
---|
4046 | if(deg(std(slocus(pr[k]))[1])>0){"singulaer";~;} |
---|
4047 | if(size(reduce(@p,std(pr[k])))!=0) |
---|
4048 | { |
---|
4049 | tmpList[size(tmpList)+1]=list(pr[k],@ce[2],@ce[3],@ce[4]); |
---|
4050 | } |
---|
4051 | } |
---|
4052 | BO[10]=tmpList; |
---|
4053 | kill tmpList; |
---|
4054 | list tmpList; |
---|
4055 | } |
---|
4056 | } |
---|
4057 | if(!defined(@ce)) |
---|
4058 | { |
---|
4059 | //--- no previously determined center, we need to compute one |
---|
4060 | if(loca) |
---|
4061 | { |
---|
4062 | //--- local case: center should be inside exceptional locus |
---|
4063 | ideal Ex=ideal(1); |
---|
4064 | k=0; |
---|
4065 | for(j=1;j<=size(BO[4]);j++) |
---|
4066 | { |
---|
4067 | if(deg(BO[4][j][1])!=0) |
---|
4068 | { |
---|
4069 | Ex=Ex*BO[4][j]; //----!!!!hier evtl. Durchschnitt??? |
---|
4070 | k++; |
---|
4071 | } |
---|
4072 | } |
---|
4073 | //--- for computation of center: drop components not meeting the Ei |
---|
4074 | list BOloc=BO; |
---|
4075 | list qs=minAssGTZ(BOloc[2]); |
---|
4076 | ideal K=ideal(1); |
---|
4077 | for(j=1;j<=size(qs);j++) |
---|
4078 | { |
---|
4079 | if(CompMeetsE(qs[j],BOloc[4])) |
---|
4080 | { |
---|
4081 | K=intersect(K,qs[j]); |
---|
4082 | } |
---|
4083 | } |
---|
4084 | BOloc[2]=K; |
---|
4085 | //--- check whether we are done |
---|
4086 | if(deg(std(BOloc[2])[1])==0) |
---|
4087 | { |
---|
4088 | list @ce=BO[2],intvec(-1),intvec(1),intvec(0); |
---|
4089 | } |
---|
4090 | if(!defined(@ce)) |
---|
4091 | { |
---|
4092 | if(BO[3][1]!=0) |
---|
4093 | { |
---|
4094 | BOloc[2]=BO[2]+Ex^((BO[3][1] div k)+1);//!!!!Vereinfachen??? |
---|
4095 | } |
---|
4096 | else |
---|
4097 | { |
---|
4098 | BOloc[2]=BO[2]+Ex^((size(DeltaList(BO)) div k)+1); |
---|
4099 | } |
---|
4100 | if(bm) |
---|
4101 | { |
---|
4102 | list @ce=CenterBM(BOloc); |
---|
4103 | } |
---|
4104 | else |
---|
4105 | { |
---|
4106 | list @ce=CenterBO(BOloc); |
---|
4107 | } |
---|
4108 | if(size(reduce(Ex,std(@ce[1])))!=0) |
---|
4109 | { |
---|
4110 | list tempPr=minAssGTZ(@ce[1]); |
---|
4111 | for(k=size(tempPr);k>=1;k--) |
---|
4112 | { |
---|
4113 | if(size(reduce(Ex,std(tempPr[k])))!=0) |
---|
4114 | { |
---|
4115 | tempPr=delete(tempPr,k); |
---|
4116 | } |
---|
4117 | } |
---|
4118 | @ce[1]=1; |
---|
4119 | for(k=1;k<=size(tempPr);k++) |
---|
4120 | { |
---|
4121 | @ce[1]=intersect(@ce[1],tempPr[k]); |
---|
4122 | } |
---|
4123 | if(deg(std(@ce[1])[1])==0) |
---|
4124 | { |
---|
4125 | @ce[1]=BO[2]; |
---|
4126 | } |
---|
4127 | } |
---|
4128 | } |
---|
4129 | //--- test whether we are done |
---|
4130 | if(size(reduce(slocusE(BO[2]),std(@ce[1])))!=0) |
---|
4131 | { |
---|
4132 | if(transversalT(BO[2],BO[4])) |
---|
4133 | { |
---|
4134 | if(defined(E)){kill E;} |
---|
4135 | list E=BO[4]; |
---|
4136 | for(j=1;j<=size(E);j++){if(deg(E[j][1])>0){E[j]=E[j]+BO[2];}} |
---|
4137 | if(normalCross(E)) |
---|
4138 | { |
---|
4139 | @ce[1]=BO[2]; |
---|
4140 | } |
---|
4141 | kill E; |
---|
4142 | } |
---|
4143 | } |
---|
4144 | } |
---|
4145 | else |
---|
4146 | { |
---|
4147 | //--- non-local |
---|
4148 | if(bm) |
---|
4149 | { |
---|
4150 | list @ce=CenterBM(BO); |
---|
4151 | } |
---|
4152 | else |
---|
4153 | { |
---|
4154 | list @ce=CenterBO(BO); |
---|
4155 | } |
---|
4156 | //--- check of the center |
---|
4157 | //@ce=correctC(BO,@ce,bm); |
---|
4158 | if((size(@ce[1])==0)&&(size(@ce[4])<(size(@ce[3])-1))) |
---|
4159 | { |
---|
4160 | intvec xxx=@ce[3]; |
---|
4161 | xxx=xxx[1..size(@ce[4])]; |
---|
4162 | @ce[3]=xxx; |
---|
4163 | xxx=@ce[2]; |
---|
4164 | xxx=xxx[1..size(@ce[4])]; |
---|
4165 | @ce[2]=xxx; |
---|
4166 | kill xxx; |
---|
4167 | } |
---|
4168 | } |
---|
4169 | if((size(reduce(BO[2],std(@ce[1])))==0) |
---|
4170 | &&(size(reduce(@ce[1],Jstd))==0)) |
---|
4171 | { |
---|
4172 | //--- J and center coincide |
---|
4173 | pr[1]=@ce[1]; |
---|
4174 | ideal cent=@ce[1]; |
---|
4175 | } |
---|
4176 | else |
---|
4177 | { |
---|
4178 | //--- decompose center and use first component |
---|
4179 | pr=minAssGTZ(@ce[1]); |
---|
4180 | ideal cent=pr[1]; |
---|
4181 | } |
---|
4182 | if(size(pr)>1) |
---|
4183 | { |
---|
4184 | //--- store the other components |
---|
4185 | for(k=2;k<=size(pr);k++) |
---|
4186 | { |
---|
4187 | tmpList[k-1]=list(pr[k],@ce[2],@ce[3],@ce[4]); |
---|
4188 | } |
---|
4189 | BO[10]=tmpList; |
---|
4190 | kill tmpList; |
---|
4191 | list tmpList; |
---|
4192 | } |
---|
4193 | } |
---|
4194 | //--- do not forget to update BO[7] and BO[3] |
---|
4195 | export cent; |
---|
4196 | BO[7]=@ce[2]; |
---|
4197 | BO[3]=@ce[3]; |
---|
4198 | if((loca||locaT)&&(size(@ce)<4)){@ce[4]=0;} //Provisorium !!! |
---|
4199 | if((size(@ce[4])<size(@ce[2])-1)||(size(@ce[4])<size(@ce[3])-1)) |
---|
4200 | { |
---|
4201 | if((deg(std(@ce[1])[1])==0)&&(deg(std(BO[2])[1])==0)) |
---|
4202 | { |
---|
4203 | intvec nullvec; |
---|
4204 | nullvec[size(@ce[2])-1]=0; |
---|
4205 | @ce[4]=nullvec; |
---|
4206 | kill nullvec; |
---|
4207 | } |
---|
4208 | else |
---|
4209 | { |
---|
4210 | "ERROR:@ce[4] hat falsche Laenge - nicht-trivialer Fall"; |
---|
4211 | ~; |
---|
4212 | } |
---|
4213 | } |
---|
4214 | if((typeof(@ce[4])=="intvec") || (typeof(@ce[4])=="intmat")) |
---|
4215 | { |
---|
4216 | BO[9]=@ce[4]; |
---|
4217 | } |
---|
4218 | //--------------------------------------------------------------------------- |
---|
4219 | // various checks and debug output |
---|
4220 | //--------------------------------------------------------------------------- |
---|
4221 | if((debu) || (praes_stop)) |
---|
4222 | { |
---|
4223 | //--- Show BO of this step |
---|
4224 | "++++++++++++++ BO +++++++++++++++++++++++"; |
---|
4225 | size(endRings); |
---|
4226 | size(allRings); |
---|
4227 | i; |
---|
4228 | "+++++++++++++++++++++++++++++++++++++++++++++++"; |
---|
4229 | showBO(BO); |
---|
4230 | "------- Center ------------"; |
---|
4231 | interred(cent); |
---|
4232 | "----------------------------"; |
---|
4233 | } |
---|
4234 | if(praes_stop) |
---|
4235 | { |
---|
4236 | ~; |
---|
4237 | } |
---|
4238 | if(debu) |
---|
4239 | { |
---|
4240 | //--- various checks, see output for comments |
---|
4241 | if(size(BO[1])>0) |
---|
4242 | { |
---|
4243 | if(deg(BO[1][1])==0) |
---|
4244 | { |
---|
4245 | "!!! W is empty !!!"; |
---|
4246 | path; |
---|
4247 | setring R; |
---|
4248 | kill S; |
---|
4249 | list result=endRings,allRings; |
---|
4250 | return(result); |
---|
4251 | } |
---|
4252 | if(deg(std(slocusE(BO[1]))[1])>0) |
---|
4253 | { |
---|
4254 | "!!! W not smooth !!!"; |
---|
4255 | path; |
---|
4256 | setring R; |
---|
4257 | kill S; |
---|
4258 | list result=endRings,allRings; |
---|
4259 | return(result); |
---|
4260 | } |
---|
4261 | } |
---|
4262 | if((!loca)&&(!locaT)) |
---|
4263 | { |
---|
4264 | if(deg(std(slocusE(cent+BO[1]))[1])>0) |
---|
4265 | { |
---|
4266 | "!!! Center not smooth !!!"; |
---|
4267 | path; |
---|
4268 | std(cent+BO[1]); |
---|
4269 | ~; |
---|
4270 | setring R; |
---|
4271 | kill S; |
---|
4272 | list result=endRings,allRings; |
---|
4273 | return(result); |
---|
4274 | } |
---|
4275 | } |
---|
4276 | for(j=1;j<=size(BO[4]);j++) |
---|
4277 | { |
---|
4278 | if(deg(BO[4][j][1])>0) |
---|
4279 | { |
---|
4280 | if(deg(std(slocusE(BO[4][j]+BO[1]))[1])>0) |
---|
4281 | { |
---|
4282 | "!!! exceptional divisor is not smooth !!!"; |
---|
4283 | path; |
---|
4284 | setring R; |
---|
4285 | kill S; |
---|
4286 | list result=endRings,allRings; |
---|
4287 | return(result); |
---|
4288 | } |
---|
4289 | } |
---|
4290 | } |
---|
4291 | if((!loca)&&(!locaT)) |
---|
4292 | { |
---|
4293 | if((norC(BO,cent))&&(size(reduce(cent,Jstd))!=0)) |
---|
4294 | { |
---|
4295 | "!!! this chart is already finished !!!"; |
---|
4296 | cent=BO[2]; |
---|
4297 | ~; |
---|
4298 | } |
---|
4299 | } |
---|
4300 | } |
---|
4301 | //---------------------------------------------------------------------------- |
---|
4302 | // Do the blow up |
---|
4303 | //---------------------------------------------------------------------------- |
---|
4304 | //!!!! Change this as soon as there is time!!! |
---|
4305 | //!!!! quick and dirty bug fix for old shortcut which has not yet been killed |
---|
4306 | if((dim(std(cent))==0)&&defined(shortcut)) {kill shortcut;} |
---|
4307 | //!!! end of bugfix |
---|
4308 | if(size(reduce(cent,Jstd))!=0) |
---|
4309 | { |
---|
4310 | //--- center does not equal J |
---|
4311 | tmpList=blowUpBO(BO,cent,extra); |
---|
4312 | if((debu)&&(!loca)&&(!locaT)) |
---|
4313 | { |
---|
4314 | //--- test it, if debu is set |
---|
4315 | if(!testBlowUp(BO,cent,tmpList,i,extra)) |
---|
4316 | { |
---|
4317 | "non-redundant chart has been killed!"; |
---|
4318 | ~; |
---|
4319 | } |
---|
4320 | } |
---|
4321 | //--- extend the list of all rings |
---|
4322 | allRings[size(allRings)+1..size(allRings)+size(tmpList)]= |
---|
4323 | tmpList[1..size(tmpList)]; |
---|
4324 | for(j=1;j<=size(tmpList);j++) |
---|
4325 | { |
---|
4326 | def Q=allRings[size(allRings)-j+1]; |
---|
4327 | setring Q; |
---|
4328 | def path=imap(S,path); |
---|
4329 | path=path,[i,size(tmpList)-j+1]; |
---|
4330 | export path; |
---|
4331 | setring S; |
---|
4332 | kill Q; |
---|
4333 | } |
---|
4334 | kill tmpList; |
---|
4335 | list tmpList; |
---|
4336 | } |
---|
4337 | else |
---|
4338 | { |
---|
4339 | //--- center equals J |
---|
4340 | k=0; |
---|
4341 | for(j=1;j<=size(BO[6]);j++) |
---|
4342 | { |
---|
4343 | if(BO[6][j]!=1) |
---|
4344 | { |
---|
4345 | //--- there is an E_i which meets J in this chart |
---|
4346 | k=1; |
---|
4347 | break; |
---|
4348 | } |
---|
4349 | } |
---|
4350 | if(k) |
---|
4351 | { |
---|
4352 | //--- chart finished, non-redundant |
---|
4353 | endRings[size(endRings)+1]=S; |
---|
4354 | } |
---|
4355 | } |
---|
4356 | kill pr; |
---|
4357 | setring R; |
---|
4358 | kill S; |
---|
4359 | } |
---|
4360 | //--------------------------------------------------------------------------- |
---|
4361 | // set up the result, test it (if debu is set) and return it |
---|
4362 | //--------------------------------------------------------------------------- |
---|
4363 | list result=endRings,allRings; |
---|
4364 | if(debu) |
---|
4365 | { |
---|
4366 | "============= result will be tested =========="; |
---|
4367 | " "; |
---|
4368 | "the number of charts obtained:",size(endRings); |
---|
4369 | if(locaT){loca=2;} |
---|
4370 | int tes=testRes(J,endRings,loca); |
---|
4371 | if(tes) |
---|
4372 | { |
---|
4373 | "============= result is o.k. =========="; |
---|
4374 | } |
---|
4375 | else |
---|
4376 | { |
---|
4377 | "============ result is wrong =========="; ~; |
---|
4378 | } |
---|
4379 | } |
---|
4380 | kill debugCenter,debugBlowUp,debugCoeff,debug_Inters_E; |
---|
4381 | if(locaT){kill @p;} |
---|
4382 | kill locaT; |
---|
4383 | return(result); |
---|
4384 | } |
---|
4385 | example |
---|
4386 | {"EXAMPLE:"; |
---|
4387 | echo = 2; |
---|
4388 | ring R=0,(x,y,z),dp; |
---|
4389 | ideal J=x3+y5+yz2+xy4; |
---|
4390 | list L=resolve(J,1); |
---|
4391 | def Q=L[1][7]; |
---|
4392 | setring Q; |
---|
4393 | showBO(BO); |
---|
4394 | } |
---|
4395 | ////////////////////////////////////////////////////////////////////////// |
---|
4396 | //static |
---|
4397 | proc CompMeetsE(ideal J, list E) |
---|
4398 | "Internal procedure - no help and no example available |
---|
4399 | " |
---|
4400 | { |
---|
4401 | int i; |
---|
4402 | for(i=1;i<=size(E);i++) |
---|
4403 | { |
---|
4404 | if(deg(std(E[i])[1])!=0) |
---|
4405 | { |
---|
4406 | if(deg(std(J+E[i])[1])!=0) |
---|
4407 | { |
---|
4408 | return(1); |
---|
4409 | } |
---|
4410 | } |
---|
4411 | } |
---|
4412 | return(0); |
---|
4413 | } |
---|
4414 | |
---|
4415 | //======================================================================== |
---|
4416 | //-------------- procedures for testing the result ---------------------- |
---|
4417 | // (not yet commented) |
---|
4418 | //======================================================================== |
---|
4419 | |
---|
4420 | ////////////////////////////////////////////////////////////////////////// |
---|
4421 | static |
---|
4422 | proc testRes(ideal J,list L,int loca) |
---|
4423 | "Internal procedure - no help and no example available |
---|
4424 | " |
---|
4425 | { |
---|
4426 | int loc; |
---|
4427 | if(defined(locaT)){loc=locaT;} |
---|
4428 | if(loc){loca=0;} |
---|
4429 | def R=basering; |
---|
4430 | ideal M=maxideal(1); |
---|
4431 | int i,j,tr; |
---|
4432 | for(i=1;i<=size(L);i++) |
---|
4433 | { |
---|
4434 | def Q=L[i]; |
---|
4435 | setring Q; |
---|
4436 | ideal J=BO[2]; |
---|
4437 | list E=BO[4]; |
---|
4438 | map phi=R,BO[5]; |
---|
4439 | ideal K=phi(J)+BO[1]; |
---|
4440 | ideal stTK=std(K); |
---|
4441 | |
---|
4442 | if(loca) |
---|
4443 | { |
---|
4444 | ideal M=phi(M)+BO[1]; |
---|
4445 | ideal stTM=std(M); |
---|
4446 | } |
---|
4447 | for(j=1;j<=size(E);j++) |
---|
4448 | { |
---|
4449 | if(deg(E[j][1])>0) |
---|
4450 | { |
---|
4451 | stTK=sat(stTK,E[j])[1]; |
---|
4452 | } |
---|
4453 | if(loca) |
---|
4454 | { |
---|
4455 | stTM=sat(stTM,E[j])[1]; |
---|
4456 | } |
---|
4457 | } |
---|
4458 | ideal sL=slocusE(J); |
---|
4459 | if(loca){sL=sL+stTM;} |
---|
4460 | ideal sLstd=std(sL); |
---|
4461 | if(deg(sLstd[1])>0) |
---|
4462 | { |
---|
4463 | if(!loc) |
---|
4464 | { |
---|
4465 | "J is not smooth";i; |
---|
4466 | setring R; |
---|
4467 | return(0); |
---|
4468 | } |
---|
4469 | if(size(reduce(@p,std(radical(sLstd))))>0) |
---|
4470 | { |
---|
4471 | "J is not smooth";i; |
---|
4472 | setring R; |
---|
4473 | return(0); |
---|
4474 | } |
---|
4475 | } |
---|
4476 | if(!((size(reduce(J,std(stTK)))==0) |
---|
4477 | &&(size(reduce(stTK,std(J)))==0))) |
---|
4478 | { |
---|
4479 | "map is wrong";i; |
---|
4480 | setring R; |
---|
4481 | return(0); |
---|
4482 | } |
---|
4483 | if(loc){tr=transversalT(J,E,@p);} |
---|
4484 | else{tr=transversalT(J,E);} |
---|
4485 | if(!tr) |
---|
4486 | { |
---|
4487 | "E not transversal with J";i; |
---|
4488 | setring R; |
---|
4489 | return(0); |
---|
4490 | } |
---|
4491 | if(!normalCross(E)) |
---|
4492 | { |
---|
4493 | "E not normal crossings";i; |
---|
4494 | setring R; |
---|
4495 | return(0); |
---|
4496 | } |
---|
4497 | for(j=1;j<=size(E);j++) |
---|
4498 | { |
---|
4499 | if(deg(E[j][1])>0){E[j]=E[j]+J;} |
---|
4500 | } |
---|
4501 | if(!normalCross(E)) |
---|
4502 | { |
---|
4503 | "E not normal crossings with J";i; |
---|
4504 | setring R; |
---|
4505 | return(0); |
---|
4506 | } |
---|
4507 | kill J,E,phi,K,stTK; |
---|
4508 | if(loca){kill M,stTM;} |
---|
4509 | setring R; |
---|
4510 | kill Q; |
---|
4511 | } |
---|
4512 | return(1); |
---|
4513 | } |
---|
4514 | ////////////////////////////////////////////////////////////////////////////// |
---|
4515 | static |
---|
4516 | proc testBlowUp(list BO,ideal cent,list tmpList, int j, int extra) |
---|
4517 | { |
---|
4518 | def R=basering; |
---|
4519 | int n=nvars(basering); |
---|
4520 | int i; |
---|
4521 | if((extra!=3)&&(extra!=2)) |
---|
4522 | { |
---|
4523 | ideal K=BO[1],BO[2],cent; |
---|
4524 | for(i=1;i<=size(BO[4]);i++) |
---|
4525 | { |
---|
4526 | K=K,BO[4][i]; |
---|
4527 | } |
---|
4528 | list N=findvars(K,0); |
---|
4529 | //list N=findvars(BO[2],0); |
---|
4530 | if(size(N[1])<n) |
---|
4531 | { |
---|
4532 | string newvar=string(N[1]); |
---|
4533 | execute("ring R1=("+charstr(R)+"),("+newvar+"),dp;"); |
---|
4534 | list BO=imap(R,BO); |
---|
4535 | ideal cent=imap(R,cent); |
---|
4536 | n=nvars(R1); |
---|
4537 | } |
---|
4538 | else |
---|
4539 | { |
---|
4540 | def R1=basering; |
---|
4541 | } |
---|
4542 | } |
---|
4543 | else |
---|
4544 | { |
---|
4545 | def R1=basering; |
---|
4546 | } |
---|
4547 | |
---|
4548 | i=0; |
---|
4549 | ideal T=cent; |
---|
4550 | ideal TW; |
---|
4551 | for(i=1;i<=size(tmpList);i++) |
---|
4552 | { |
---|
4553 | def Q=tmpList[i]; |
---|
4554 | setring Q; |
---|
4555 | map phi=R1,lastMap; |
---|
4556 | ideal TE=radical(slocusE(BO[2])); |
---|
4557 | setring R1; |
---|
4558 | TW=preimage(Q,phi,TE); |
---|
4559 | T=intersect(T,TW); |
---|
4560 | kill Q; |
---|
4561 | } |
---|
4562 | ideal sL=intersect(slocusE(BO[2]),cent); |
---|
4563 | if(size(reduce(sL,std(radical(T))))>0){setring R;return(0);} |
---|
4564 | if(size(reduce(T,std(radical(sL))))>0){setring R;return(0);} |
---|
4565 | setring R; |
---|
4566 | return(1); |
---|
4567 | } |
---|
4568 | ////////////////////////////////////////////////////////////////////////////// |
---|
4569 | static |
---|
4570 | proc normalCross(list E,list #) |
---|
4571 | "Internal procedure - no help and no example available |
---|
4572 | " |
---|
4573 | { |
---|
4574 | int loc; |
---|
4575 | if((defined(locaT))&&(defined(@p))) |
---|
4576 | { |
---|
4577 | loc=1; |
---|
4578 | ideal pp=@p; |
---|
4579 | } |
---|
4580 | int i,d,j; |
---|
4581 | int n=nvars(basering); |
---|
4582 | list E1,E2; |
---|
4583 | ideal K,M,Estd,cent; |
---|
4584 | intvec v,w; |
---|
4585 | if(size(#)>0){cent=#[1];} |
---|
4586 | |
---|
4587 | for(i=1;i<=size(E);i++) |
---|
4588 | { |
---|
4589 | Estd=std(E[i]); |
---|
4590 | if(deg(Estd[1])>0) |
---|
4591 | { |
---|
4592 | E1[size(E1)+1]=Estd; |
---|
4593 | } |
---|
4594 | } |
---|
4595 | E=E1; |
---|
4596 | for(i=1;i<=size(E);i++) |
---|
4597 | { |
---|
4598 | v=i; |
---|
4599 | E1[i]=list(E[i],v); |
---|
4600 | } |
---|
4601 | list ll; |
---|
4602 | int re=1; |
---|
4603 | int ok; |
---|
4604 | while(size(E1)>0) |
---|
4605 | { |
---|
4606 | K=E1[1][1]; |
---|
4607 | v=E1[1][2]; |
---|
4608 | attrib(K,"isSB",1); |
---|
4609 | E1=delete(E1,1); |
---|
4610 | d=n-dim(K); |
---|
4611 | M=minor(jacob(K),d)+K; |
---|
4612 | if(deg(std(M)[1])>0) |
---|
4613 | { |
---|
4614 | if(size(#)>0) |
---|
4615 | { |
---|
4616 | if(size(reduce(M,std(cent)))>0) |
---|
4617 | { |
---|
4618 | ll[size(ll)+1]=std(M); |
---|
4619 | } |
---|
4620 | else |
---|
4621 | { |
---|
4622 | ok=1; |
---|
4623 | } |
---|
4624 | } |
---|
4625 | if(!loc) |
---|
4626 | { |
---|
4627 | re=0; |
---|
4628 | } |
---|
4629 | else |
---|
4630 | { |
---|
4631 | if(size(reduce(pp,std(radical(M))))>0){re=0;} |
---|
4632 | } |
---|
4633 | } |
---|
4634 | for(i=1;i<=size(E);i++) |
---|
4635 | { |
---|
4636 | for(j=1;j<=size(v);j++){if(v[j]==i){break;}} |
---|
4637 | if(j<=size(v)){if(v[j]==i){i++;continue;}} |
---|
4638 | Estd=std(K+E[i]); |
---|
4639 | w=v; |
---|
4640 | if(deg(Estd[1])==0){i++;continue;} |
---|
4641 | if(d==n-dim(Estd)) |
---|
4642 | { |
---|
4643 | if(size(#)>0) |
---|
4644 | { |
---|
4645 | if(size(reduce(Estd,std(cent)))>0) |
---|
4646 | { |
---|
4647 | ll[size(ll)+1]=Estd; |
---|
4648 | } |
---|
4649 | else |
---|
4650 | { |
---|
4651 | ok=1; |
---|
4652 | } |
---|
4653 | } |
---|
4654 | if(!loc) |
---|
4655 | { |
---|
4656 | re=0; |
---|
4657 | } |
---|
4658 | else |
---|
4659 | { |
---|
4660 | if(size(reduce(pp,std(radical(M))))>0){re=0;} |
---|
4661 | } |
---|
4662 | } |
---|
4663 | w[size(w)+1]=i; |
---|
4664 | E2[size(E2)+1]=list(Estd,w); |
---|
4665 | } |
---|
4666 | if(size(E2)>0) |
---|
4667 | { |
---|
4668 | if(size(E1)>0) |
---|
4669 | { |
---|
4670 | E1[size(E1)+1..size(E1)+size(E2)]=E2[1..size(E2)]; |
---|
4671 | } |
---|
4672 | else |
---|
4673 | { |
---|
4674 | E1=E2; |
---|
4675 | } |
---|
4676 | } |
---|
4677 | kill E2; |
---|
4678 | list E2; |
---|
4679 | } |
---|
4680 | /* |
---|
4681 | if((!ok)&&(!re)&&(size(#)==1)) |
---|
4682 | { |
---|
4683 | |
---|
4684 | "the center is wrong"; |
---|
4685 | "it could be one of the following list"; |
---|
4686 | ll; |
---|
4687 | ~; |
---|
4688 | } |
---|
4689 | */ |
---|
4690 | if((!ok)&&(!re)&&(size(#)==2)) |
---|
4691 | { |
---|
4692 | return(2); //for Center correction |
---|
4693 | } |
---|
4694 | return(re); |
---|
4695 | } |
---|
4696 | ////////////////////////////////////////////////////////////////////////////// |
---|
4697 | static |
---|
4698 | proc normalCrossB(ideal J,list E,ideal V) |
---|
4699 | "Internal procedure - no help and no example available |
---|
4700 | " |
---|
4701 | { |
---|
4702 | int i,d,j; |
---|
4703 | int n=nvars(basering); |
---|
4704 | list E1,E2; |
---|
4705 | ideal K,M,Estd; |
---|
4706 | intvec v,w; |
---|
4707 | |
---|
4708 | for(i=1;i<=size(E);i++) |
---|
4709 | { |
---|
4710 | Estd=std(E[i]+J); |
---|
4711 | if(deg(Estd[1])>0) |
---|
4712 | { |
---|
4713 | E1[size(E1)+1]=Estd; |
---|
4714 | } |
---|
4715 | } |
---|
4716 | E=E1; |
---|
4717 | for(i=1;i<=size(E);i++) |
---|
4718 | { |
---|
4719 | v=i; |
---|
4720 | E1[i]=list(E[i],v); |
---|
4721 | } |
---|
4722 | list ll; |
---|
4723 | int re=1; |
---|
4724 | |
---|
4725 | while((size(E1)>0)&&(re==1)) |
---|
4726 | { |
---|
4727 | K=E1[1][1]; |
---|
4728 | v=E1[1][2]; |
---|
4729 | attrib(K,"isSB",1); |
---|
4730 | E1=delete(E1,1); |
---|
4731 | d=n-dim(K); |
---|
4732 | M=minor(jacob(K),d)+K; |
---|
4733 | if(deg(std(M+V)[1])>0) |
---|
4734 | { |
---|
4735 | re=0; |
---|
4736 | break; |
---|
4737 | } |
---|
4738 | for(i=1;i<=size(E);i++) |
---|
4739 | { |
---|
4740 | for(j=1;j<=size(v);j++){if(v[j]==i){break;}} |
---|
4741 | if(j<=size(v)){if(v[j]==i){i++;continue;}} |
---|
4742 | Estd=std(K+E[i]); |
---|
4743 | w=v; |
---|
4744 | if(deg(Estd[1])==0){i++;continue;} |
---|
4745 | if(d==n-dim(Estd)) |
---|
4746 | { |
---|
4747 | if(deg(std(Estd+V)[1])>0) |
---|
4748 | { |
---|
4749 | re=0; |
---|
4750 | break; |
---|
4751 | } |
---|
4752 | } |
---|
4753 | w[size(w)+1]=i; |
---|
4754 | E2[size(E2)+1]=list(Estd,w); |
---|
4755 | } |
---|
4756 | if(size(E2)>0) |
---|
4757 | { |
---|
4758 | if(size(E1)>0) |
---|
4759 | { |
---|
4760 | E1[size(E1)+1..size(E1)+size(E2)]=E2[1..size(E2)]; |
---|
4761 | } |
---|
4762 | else |
---|
4763 | { |
---|
4764 | E1=E2; |
---|
4765 | } |
---|
4766 | } |
---|
4767 | kill E2; |
---|
4768 | list E2; |
---|
4769 | } |
---|
4770 | return(re); |
---|
4771 | } |
---|
4772 | ////////////////////////////////////////////////////////////////////////////// |
---|
4773 | static |
---|
4774 | proc norC(list BO,ideal cent) |
---|
4775 | "Internal procedure - no help and no example available |
---|
4776 | " |
---|
4777 | { |
---|
4778 | int j; |
---|
4779 | list E=BO[4]; |
---|
4780 | ideal N=BO[2]; |
---|
4781 | if(BO[3][1]>1){return(0);} |
---|
4782 | if(deg(std(slocusE(BO[2]))[1])>0){return(0);} |
---|
4783 | if(!transversalT(N,E)){return(0);} |
---|
4784 | for(j=1;j<=size(E);j++){if(deg(E[j][1])>0){E[j]=E[j]+N;}} |
---|
4785 | if(!normalCross(E,cent)){return(0);} |
---|
4786 | return(1); |
---|
4787 | } |
---|
4788 | ////////////////////////////////////////////////////////////////////////////// |
---|
4789 | static |
---|
4790 | proc specialReduce(ideal I,ideal J,poly p) |
---|
4791 | { |
---|
4792 | matrix M; |
---|
4793 | int i,j; |
---|
4794 | for(i=1;i<=ncols(I);i++) |
---|
4795 | { |
---|
4796 | M=coeffs(I[i],@z); |
---|
4797 | I[i]=0; |
---|
4798 | for(j=1;j<=nrows(M);j++) |
---|
4799 | { |
---|
4800 | I[i]=I[i]+M[j,1]*p^(nrows(M)-j); |
---|
4801 | } |
---|
4802 | I[i]=reduce(I[i],J); |
---|
4803 | } |
---|
4804 | return(I); |
---|
4805 | } |
---|