1 | ///////////////////////////////////////////////////////////////////////// |
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2 | version="version resbinomial.lib 4.0.0.0 Jun_2013 "; // $Id$ |
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3 | category="Resolution of singularities"; |
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4 | info=" |
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5 | LIBRARY: resbinomial.lib Combinatorial algorithm of resolution of singularities |
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6 | of binomial ideals in arbitrary characteristic. |
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7 | Binomial resolution algorithm of Blanco |
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8 | |
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9 | AUTHORS: R. Blanco, mariarocio.blanco@uclm.es, |
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10 | @* G. Pfister, pfister@mathematik.uni-kl.de |
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11 | |
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12 | PROCEDURES: |
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13 | BINresol(J); computes a E-resolution of singularities of (J) (THE SECOND PART IS NOT IMPLEMENTED YET) |
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14 | |
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15 | Eresol(J); computes a E-resolution of singularities of (J) in char 0 |
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16 | determinecenter(L1,L2,c,n,Y,a,mb,flag,control3); computes the next blowing-up center |
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17 | Blowupcenter(L1,id,m,L2,c,n,h); makes the blowing-up |
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18 | Nonhyp(Coef,expJ,sJ,n,flag,sums); computes the ideal generated by the non hyperbolic generators of expJ |
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19 | |
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20 | identifyvar(); identifies status of variables |
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21 | Edatalist(Coef,Exp,k,n,flag); gives the E-order of each term in Exp |
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22 | EOrdlist(Coef,Exp,k,n,flag); computes the E-order of an ideal (giving in the language of lists) |
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23 | maxEord(Coef,Exp,k,n,flag); computes de maximum E-order of an ideal given by Coef and Exp |
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24 | ECoef(Coef,expP,sP,V,auxc,n,flag); Computes a simplified version of the E-Coeff ideal. The E-orders are correct, |
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25 | but tranformations of coefficients of the generators and powers of binomials |
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26 | cannot be computed easily in terms of lists. |
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27 | elimrep(L); removes repeated terms from a list |
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28 | Emaxcont(Coef,Exp,k,n,flag); computes a list of hypersurfaces of E-maximal contact |
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29 | cleanunit(mon,n,flag); clean the units in a monomial mon |
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30 | resfunction(t,auxinv,nchart,n); composes the E-resolution function |
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31 | calculateI(Coef,J,c,n,Y,a,b,D); computes the order of the non monomial part of an ideal J |
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32 | Maxord(L,n); computes the maximum exponent of an exceptional monomial ideal |
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33 | Gamma(L,c,n); computes the Gamma function for an exceptional monomial ideal given by L |
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34 | |
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35 | convertdata(C,L,n,flag); computes the ideal corresponding to C,L |
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36 | lcmofall(nchart,mobile); computes the lcm of the denominators of the E-orders for all the charts |
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37 | computemcm(Eolist); computes the lcm of the denominators of the E-orders for one chart |
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38 | |
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39 | constructH(Hhist,n,flag); construct the list of exceptional divisors accumulated at this chart |
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40 | constructblwup(blwhist,n,chy,flag); construct the ideal defining the map K[W] --> K[Wi], |
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41 | which gives the composition map of all the blowing up leading to this chart |
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42 | constructlastblwup(blwhist,n,chy,flag); construct the ideal defining the last blowup leading to this chart |
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43 | |
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44 | genoutput(chart,mobile,nchart,nsons,n,q,p); generates the output for visualization |
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45 | salida(idchart,chart,mobile,numson,previousa,n,q); generates the output for one chart |
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46 | |
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47 | iniD(n); creates a list of lists of zeros of size n |
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48 | sumlist(L1,L2); sums two lists component to component |
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49 | reslist(L1,L2); subtracts two lists component to component |
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50 | multiplylist(L,a); multiplies a list by a number, component to component |
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51 | dividelist(L1,L2); divides two lists component to component |
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52 | createlist(L1,L2); creates a list of lists of two elements |
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53 | "; |
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54 | // inidata(K,k); verifies input data, a binomial ideal K of k generators |
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55 | // data(K,k,n); transforms data on lists of length n |
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56 | // list0(n); creates a list of zeros of size n |
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57 | |
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58 | LIB "general.lib"; |
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59 | LIB "qhmoduli.lib"; |
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60 | LIB "inout.lib"; |
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61 | LIB "poly.lib"; |
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62 | LIB "resolve.lib"; |
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63 | LIB "reszeta.lib"; |
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64 | LIB "resgraph.lib"; |
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65 | //////////////////////////////////////////////////////////////////////////// |
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66 | |
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67 | static proc inidata(ideal K,int k) |
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68 | "USAGE: inidata(K,k); K any ideal, k integer (!=0) |
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69 | COMPUTE: Verifies the input data |
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70 | RETURN: flag indicating if the ideal is binomial or not |
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71 | EXAMPLE: example inidata; shows an example |
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72 | " |
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73 | { |
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74 | int i; |
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75 | for (i=1;i<=k; i++) |
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76 | { if (size(K[i])>2){return(0);} |
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77 | } |
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78 | return(1); |
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79 | } |
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80 | example |
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81 | {"EXAMPLE:"; echo = 2; |
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82 | ring r = 0,(x(1..3)),dp; |
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83 | ideal J1=x(1)^4*x(2)^2, x(1)^2+x(3)^3; |
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84 | inidata(J1,2); |
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85 | |
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86 | ideal J2=x(1)^4*x(2)^2, x(1)^2+x(2)^3+x(3)^5; |
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87 | inidata(J2,2); |
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88 | } |
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89 | ///////////////////////////////////////////////////////////////////////////////// |
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90 | |
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91 | proc changeoriginalvar() |
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92 | "USAGE: changeoriginalvar(); |
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93 | COMPUTE: Change the name of the variables to x(1...n), only necessary at the beginning |
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94 | RETURN: the new ring with the suitable names |
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95 | EXAMPLE: example changeoriginalvar; shows an example |
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96 | " |
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97 | { |
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98 | int i,n,cont; |
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99 | |
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100 | n=nvars(basering); |
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101 | cont=0; |
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102 | def r=basering; |
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103 | |
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104 | // check the name of the variables |
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105 | |
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106 | for (i=1;i<=n; i++){if (varstr(i)[1]=="x" or varstr(i)[1]=="y"){cont=cont+1;}} |
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107 | |
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108 | // change them if there exists some variable different from x(i) or y(i) |
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109 | |
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110 | if (cont!=n or n<=2){ |
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111 | // making the change |
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112 | def Rnew=changevar ("x()"); |
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113 | setring Rnew; |
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114 | // print("INVERTIBLE VARIABLES NOT CONSIDERED AT THE BEGINNING"); |
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115 | return(Rnew,1); |
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116 | } |
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117 | else{ // print("INVERTIBLE VARIABLES ALREADY CONSIDERED AT THE BEGINNING"); |
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118 | return(r,0); |
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119 | } |
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120 | } |
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121 | example |
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122 | {"EXAMPLE:"; echo = 2; |
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123 | ring r = 0,(x(1),y(2),x(3),y(4),x(5..7),y(8)),dp; |
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124 | changeoriginalvar(); |
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125 | |
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126 | ring r = 0,(x,y,z,w),dp; |
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127 | changeoriginalvar(); |
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128 | } |
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129 | |
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130 | ///////////////////////////////////////////////////////////////////////////////// |
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131 | |
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132 | proc identifyvar() |
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133 | "USAGE: identifyvar(); |
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134 | COMPUTE: Asign 0 to variables x and 1 to variables y, only necessary at the beginning |
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135 | RETURN: list, say l, of size the dimension of the basering |
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136 | l[i] is: 0 if the i-th variable is x(i), |
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137 | 1 if the i-th variable is y(i) |
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138 | EXAMPLE: example identifyvar; shows an example |
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139 | " |
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140 | { |
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141 | int i,n; |
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142 | list flaglist; |
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143 | |
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144 | n=nvars(basering); |
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145 | flaglist=list0(n); |
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146 | |
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147 | for (i=1;i<=n; i++){if (varstr(i)[1]=="y"){flaglist[i]=1;}} |
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148 | |
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149 | return(flaglist); |
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150 | } |
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151 | example |
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152 | {"EXAMPLE:"; echo = 2; |
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153 | ring r = 0,(x(1),y(2),x(3),y(4),x(5..7),y(8)),dp; |
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154 | identifyvar(); |
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155 | } |
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156 | ///////////////////////////////////////////////////////////////////////////////// |
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157 | |
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158 | proc data(ideal K,int k,int n) |
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159 | "USAGE: data(K,k,n); K any ideal, k integer (!=0), n integer (!=0) |
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160 | COMPUTE: Construcs a list with the coefficients and exponents of one ideal |
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161 | RETURN: lists of coefficients and exponents of K |
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162 | EXAMPLE: example data; shows an example |
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163 | " |
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164 | {int i,j,lon; |
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165 | number aa; |
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166 | intvec cc; |
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167 | list bb,dd,aux,ddaux,Coef,Exp; |
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168 | |
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169 | for (i=1;i<=k; i++) |
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170 | { lon=size(K[i]); |
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171 | |
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172 | // binomial |
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173 | if (lon==2){aa=leadcoef(K[i][1]); |
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174 | bb=aa; |
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175 | Coef[i]=bb; // coefficients |
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176 | cc=leadexp(K[i][1]); // exponents |
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177 | |
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178 | // cc is an intvec, transform cc in dd, a list of lists |
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179 | dd=cc[1..n]; |
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180 | aux[1]=dd; |
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181 | // the same for the second term |
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182 | |
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183 | aa=leadcoef(K[i][2]); |
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184 | bb=aa; |
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185 | Coef[i]=Coef[i] + bb; // all the coefficients of i-th generator of K |
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186 | cc=leadexp(K[i][2]); |
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187 | |
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188 | dd=cc[1..n]; |
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189 | aux[2]=dd; |
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190 | Exp[i]=aux;} |
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191 | |
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192 | // monomial |
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193 | if (lon==1){aux=list(); |
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194 | aa=leadcoef(K[i][1]); |
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195 | bb=aa; |
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196 | Coef[i]=bb; |
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197 | cc=leadexp(K[i][1]); |
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198 | dd=cc[1..n]; |
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199 | aux[1]=dd; |
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200 | Exp[i]=aux;} |
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201 | } //end for |
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202 | return(Coef,Exp); |
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203 | } |
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204 | example |
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205 | {"EXAMPLE:"; echo = 2; |
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206 | ring r = 0,(x(1..3)),dp; |
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207 | ideal J=x(1)^4*x(2)^2, x(1)^2-x(3)^3; |
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208 | data(J,2,3); |
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209 | } |
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210 | ////////////////////////////////////////////////////// |
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211 | |
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212 | proc Edatalist(list Coef,list Exp,int k,int n,list flaglist) |
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213 | "USAGE: Edatalist(Coef,Exp,k,n,flaglist); |
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214 | Coef,Exp,flaglist lists, k,n, integers |
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215 | Exp is a list of lists of exponents, k=size(Exp) |
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216 | COMPUTE: computes a list with the E-order of each term |
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217 | RETURN: a list with the E-order of each term |
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218 | EXAMPLE: example Edatalist; shows an example |
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219 | " |
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220 | {int i,j,lon,mm; |
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221 | list dd,ss,sums; |
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222 | number aux,aux1,aux2; |
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223 | |
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224 | for (i=1;i<=k;i++){lon=size(Coef[i]); |
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225 | if (lon==1) { for (j=1;j<=n;j++){if (flaglist[j]==0){aux=aux+Exp[i][1][j];}} |
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226 | ss=aux; aux=0;} // monomial |
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227 | else { for (j=1;j<=n;j++){if (flaglist[j]==0){ aux1=aux1+Exp[i][1][j]; |
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228 | aux2=aux2+Exp[i][2][j];}} |
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229 | ss=aux1,aux2; aux1=0; aux2=0; } // binomial |
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230 | sums[i]=ss;} |
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231 | return(sums); |
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232 | } |
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233 | example |
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234 | {"EXAMPLE:"; echo = 2; |
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235 | ring r = 0,(x(1),y(2),x(3),y(4),x(5..7),y(8)),dp; |
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236 | list flag=identifyvar(); |
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237 | ideal J=x(1)^3*x(3)-y(2)*y(4)^2,x(5)*y(2)-x(7)*y(4)^2,x(6)^2*(1-y(4)*y(8)^5); |
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238 | list L=data(J,3,8); |
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239 | list EL=Edatalist(L[1],L[2],3,8,flag); |
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240 | EL; // E-order of each term |
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241 | |
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242 | |
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243 | ring r = 2,(x(1),y(2),x(3),y(4),x(5..7),y(8)),dp; |
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244 | list flag=identifyvar(); |
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245 | ideal J=x(1)^3*x(3)-y(2)*y(4)^2,x(5)*y(2)-x(7)*y(4)^2,x(6)^2*(1-y(4)*y(8)^5); |
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246 | list L=data(J,3,8); |
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247 | list EL=Edatalist(L[1],L[2],3,8,flag); |
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248 | EL; // E-order of each term IN CHAR 2, COMPUTATIONS NEED TO BE DONE IN CHAR 0 |
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249 | |
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250 | |
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251 | ring r = 0,(x(1..3)),dp; |
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252 | list flag=identifyvar(); |
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253 | ideal J=x(1)^4*x(2)^2, x(1)^2-x(3)^3; |
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254 | list L=data(J,2,3); |
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255 | list EL=Edatalist(L[1],L[2],2,3,flag); |
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256 | EL; // E-order of each term |
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257 | } |
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258 | /////////////////////////////////////////////////////////////////////////////////// |
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259 | |
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260 | proc EOrdlist(list Coef,list Exp,int k,int n,list flaglist) |
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261 | "USAGE: EOrdlist(Coef,Exp,k,n,flaglist); |
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262 | Coef,Exp,flaglist lists, k,n, integers |
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263 | Exp is a list of lists of exponents, k=size(Exp) |
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264 | COMPUTE: computes de E-order of an ideal given by a list (Coef,Exp) and extra information |
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265 | RETURN: maximal E-order, and its position=number of generator and term |
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266 | EXAMPLE: example EOrdlist; shows an example |
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267 | " |
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268 | {int i,can,canpost,lon; |
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269 | number canmin; |
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270 | list sums; |
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271 | |
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272 | sums=Edatalist(Coef,Exp,k,n,flaglist); |
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273 | |
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274 | canmin=sums[1][1]; // inicializating, works also with a monomial |
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275 | for (i=1;i<=k; i++){lon=size(sums[i]); // this is 2 for binomial and 1 for monomial generators |
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276 | if (sums[i][1]<=canmin and Coef[i][1]!=0){canmin=sums[i][1]; |
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277 | can=i; canpost=1;} |
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278 | |
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279 | // if the generator is a binomial we check the second term |
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280 | |
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281 | if (lon==2) {if (sums[i][2]<canmin and Coef[i][2]!=0){canmin=sums[i][2]; |
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282 | can=i; canpost=2;}} |
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283 | } |
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284 | return(canmin,can,canpost); |
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285 | } |
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286 | example |
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287 | {"EXAMPLE:"; echo = 2; |
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288 | ring r = 0,(x(1),y(2),x(3),y(4),x(5..7),y(8)),dp; |
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289 | list flag=identifyvar(); |
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290 | ideal J=x(1)^3*x(3)-y(2)*y(4)^2,x(5)*y(2)-x(7)*y(4)^2,x(6)^2*(1-y(4)*y(8)^5),x(7)^4*y(8)^2; |
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291 | list L=data(J,4,8); |
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292 | list Eo=EOrdlist(L[1],L[2],4,8,flag); |
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293 | Eo[1]; // E-order |
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294 | Eo[2]; // generator giving the E-order |
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295 | Eo[3]; // term giving the E-order |
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296 | } |
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297 | |
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298 | ////////////////////////////////////////////////////// |
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299 | |
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300 | proc maxEord(list Coef,list Exp,int k,int n,list flaglist) |
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301 | "USAGE: maxEord(Coef,Exp,k,n,flaglist); |
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302 | Coef,Exp,flaglist lists, k,n, integers |
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303 | Exp is a list of lists of exponents, k=size(Exp) |
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304 | RETURN: computes de maximal E-order of an ideal given by Coef,Exp |
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305 | EXAMPLE: example maxEord; shows an example |
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306 | " |
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307 | { |
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308 | int i,lon; |
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309 | number canmin; // THE ASSIGNMENT IS NOT OK BECAUSE IT IS OF TYPE NUMBER |
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310 | list sums; |
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311 | |
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312 | sums=Edatalist(Coef,Exp,k,n,flaglist); |
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313 | |
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314 | canmin=sums[1][1]; // inicializating, works also with a monomial |
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315 | for (i=1;i<=k; i++){lon=size(sums[i]); // this is 2 for binomial and 1 for monomial generators |
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316 | if (sums[i][1]<=canmin and Coef[i][1]!=0){canmin=sums[i][1];} |
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317 | |
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318 | // if the generator is a binomial we check the second term |
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319 | |
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320 | if (lon==2) {if (sums[i][2]<canmin and Coef[i][2]!=0){canmin=sums[i][2];}} |
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321 | } |
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322 | return(canmin,sums); |
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323 | } |
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324 | example |
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325 | {"EXAMPLE:"; echo = 2; |
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326 | ring r = 0,(x(1),y(2),x(3),y(4),x(5..7),y(8)),dp; |
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327 | list flag=identifyvar(); |
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328 | ideal J=x(1)^3*x(3)-y(2)*y(4)^2*x(3),x(5)*y(2)-x(7)*y(4)^2,x(6)^2*(1-y(4)*y(8)^5),x(7)^4*y(8)^2; |
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329 | list L=data(J,4,8); |
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330 | list M=maxEord(L[1],L[2],4,8,flag); |
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331 | M[1]; // E-order |
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332 | } |
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333 | ////////////////////////////////////////////////////// |
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334 | |
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335 | proc elimrep(list maxvar) |
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336 | "USAGE: elimrep(L); L is a list |
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337 | COMPUTE: Eliminate repeated terms from a list |
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338 | RETURN: the same list without repeated terms |
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339 | EXAMPLE: example elimrep; shows an example |
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340 | " |
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341 | { |
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342 | int i,j; |
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343 | list aux2; |
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344 | |
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345 | aux2=maxvar; |
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346 | for (i=1;i<=size(aux2); i++) |
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347 | { for (j=i+1;j<=size(aux2); j++){if (aux2[i]==aux2[j] and i!=j){aux2=delete(aux2,j);}} |
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348 | } |
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349 | maxvar=aux2; |
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350 | return(maxvar); |
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351 | } |
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352 | example |
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353 | {"EXAMPLE:"; echo = 2; |
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354 | ring r = 0,(x(1..3)),dp; |
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355 | list L=4,5,2,5,7,8,6,3,2; |
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356 | elimrep(L); |
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357 | } |
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358 | ////////////////////////////////////////////////////// |
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359 | |
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360 | proc Emaxcont(list Coef,list Exp,int k,int n,list flag) |
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361 | "USAGE: Emaxcont(Coef,Exp,k,n,flag); |
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362 | Coef,Exp,flag lists, k,n, integers |
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363 | Exp is a list of lists of exponents, k=size(Exp) |
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364 | COMPUTE: Identify ALL the variables of E-maximal contact |
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365 | RETURN: a list with the indexes of the variables of E-maximal contact |
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366 | EXAMPLE: example Emaxcont; shows an example |
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367 | " |
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368 | { |
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369 | int i,j,lon; |
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370 | number maxEo; |
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371 | list L,sums,bx,maxvar; |
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372 | |
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373 | L=maxEord(Coef,Exp,k,n,flag); |
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374 | |
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375 | maxEo=L[1]; |
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376 | sums=L[2]; |
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377 | |
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378 | if (maxEo>0){ |
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379 | |
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380 | for (i=1;i<=k; i++){lon=size(sums[i]); |
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381 | if (lon==2){if (sums[i][1]==maxEo) // variables of the first term |
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382 | {for (j=1;j<=n; j++){if(Exp[i][1][j]!=0 and flag[j]==0){bx=j; maxvar=maxvar + bx;}}} |
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383 | |
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384 | if (sums[i][2]==maxEo) // variables of the second term |
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385 | {for (j=1;j<=n; j++){if(Exp[i][2][j]!=0 and flag[j]==0){bx=j; maxvar=maxvar + bx;}}}} |
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386 | else {if (sums[i][1]==maxEo) |
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387 | {for (j=1;j<=n; j++){if(Exp[i][1][j]!=0 and flag[j]==0){bx=j; maxvar=maxvar + bx;}}}} |
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388 | |
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389 | }} |
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390 | else {maxvar=list();} |
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391 | |
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392 | // eliminating repeated terms |
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393 | maxvar=elimrep(maxvar); |
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394 | |
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395 | // It is necessary to check if flag[j]==0 in order to avoid the selection of y variables |
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396 | |
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397 | return(maxEo,maxvar); |
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398 | } |
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399 | example |
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400 | {"EXAMPLE:"; echo = 2; |
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401 | ring r = 0,(x(1),y(2),x(3),y(4),x(5..7),y(8)),dp; |
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402 | list flag=identifyvar(); |
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403 | ideal J=x(1)^3*x(3)-y(2)*y(4)^2,x(5)*y(2)-x(7)*y(4)^2,x(6)^2*(1-y(4)*y(8)^5),x(7)^4*y(8)^2; |
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404 | list L=data(J,4,8); |
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405 | list hyp=Emaxcont(L[1],L[2],4,8,flag); |
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406 | hyp[1]; // max E-order=0 |
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407 | hyp[2]; // There are no hypersurfaces of E-maximal contact |
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408 | |
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409 | ring r = 0,(x(1),y(2),x(3),y(4),x(5..7),y(8)),dp; |
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410 | list flag=identifyvar(); |
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411 | ideal J=x(1)^3*x(3)-y(2)*y(4)^2*x(3),x(5)*y(2)-x(7)*y(4)^2,x(6)^2*(1-y(4)*y(8)^5),x(7)^4*y(8)^2; |
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412 | list L=data(J,4,8); |
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413 | list hyp=Emaxcont(L[1],L[2],4,8,flag); |
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414 | hyp[1]; // the E-order is 1 |
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415 | hyp[2]; // {x(3)=0},{x(5)=0},{x(7)=0} are hypersurfaces of E-maximal contact |
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416 | |
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417 | } |
---|
418 | /////////////////////////////////////////////////////// |
---|
419 | |
---|
420 | proc cleanunit(list mon,int n,list flaglist) |
---|
421 | "USAGE: cleanunit(mon,n,flaglist); |
---|
422 | mon, flaglist lists, n integer |
---|
423 | COMPUTE: We clean (or forget) the units in a monomial, given by "y" variables |
---|
424 | RETURN: The list defining the monomial ideal already cleaned |
---|
425 | EXAMPLE: example cleanunit; shows an example |
---|
426 | " |
---|
427 | { |
---|
428 | int i; |
---|
429 | |
---|
430 | for (i=1;i<=n;i++){if (flaglist[i]==1){mon[i]=0;}} |
---|
431 | |
---|
432 | // coef[1]=coef[1]*y(i)^mon[i]; IS NOT ALLOWED because mon[i] can be a number |
---|
433 | // therefore, the coefficients remain constant |
---|
434 | |
---|
435 | return(mon); |
---|
436 | } |
---|
437 | example |
---|
438 | {"EXAMPLE:"; echo = 2; |
---|
439 | ring r = 0,(x(1),y(2),x(3),y(4)),dp; |
---|
440 | list flag=identifyvar(); |
---|
441 | ideal J=x(1)^3*y(2)*x(3)^5*y(4)^8; |
---|
442 | list L=data(J,1,4); |
---|
443 | L[2][1][1]; // list of exponents of the monomial J |
---|
444 | list M=cleanunit(L[2][1][1],4,flag); |
---|
445 | M; // new list without units |
---|
446 | } |
---|
447 | ////////////////////////////////////////////////////// |
---|
448 | // Classification of the ideal E-Coeff_V(P): |
---|
449 | // ccase=1, E-Coeff_V(P)=0 |
---|
450 | // 2,3 Bold regular case |
---|
451 | // 4 P=1 monomial case (detected before) |
---|
452 | // 0 Otherwise |
---|
453 | |
---|
454 | proc ECoef(list Coef,list expP,int sP,int V,number auxc,int n,list flaglist) |
---|
455 | "USAGE: ECoef(Coef,expP,sP,V,auxc,n,flaglist); |
---|
456 | Coef, expP, flaglist lists, sP, V, n integers, auxc number |
---|
457 | COMPUTE: The ideal E-Coeff_V(P), where V is a permissible hypersurface which belongs to the center |
---|
458 | RETURN: list of exponents, list of coefficients and classification of the ideal E-Coeff_V(P) |
---|
459 | EXAMPLE: example ECoef; shows an example |
---|
460 | " |
---|
461 | { |
---|
462 | int i,j,k,l,numg,ccase,cont2,cont3,val; |
---|
463 | number aa; |
---|
464 | list Eco,newcoef,auxexp,newL,rs,rs2,aux,aux2,aux3,aux4,L; |
---|
465 | |
---|
466 | auxexp=expP; |
---|
467 | |
---|
468 | l=1; |
---|
469 | for (i=1;i<=sP;i++) |
---|
470 | {rs[i]=size(Coef[i]); |
---|
471 | if (rs[i]==2){ // binomials |
---|
472 | if (auxexp[i][1][V]!=auxexp[i][2][V]) // no common factors for the variable in V |
---|
473 | |
---|
474 | {for (j=1;j<=2;j++){if (auxexp[i][j][V]<auxc){aa=auxc/(auxc-auxexp[i][j][V]); |
---|
475 | auxexp[i][j][V]=0; |
---|
476 | aux4[1]=multiplylist(auxexp[i][j],aa); |
---|
477 | Eco[l]=aux4; |
---|
478 | // newcoef[l]=Coef[i][j]^aa; IT IS NO ALLOWED!!! |
---|
479 | newcoef[l]=Coef[i][j]; // we leave it constant |
---|
480 | l=l+1;}}} |
---|
481 | |
---|
482 | else // common factors for the variable in V, of zero in both terms |
---|
483 | |
---|
484 | {if (auxexp[i][1][V]<auxc){aa=auxc/(auxc-auxexp[i][1][V]); |
---|
485 | auxexp[i][1][V]=0; auxexp[i][2][V]=0; |
---|
486 | |
---|
487 | // this generator is a power of a binomial |
---|
488 | // one possibility is Eco[l]=auxexp[i]; we leave it constant and add some extra number aa, or |
---|
489 | // define a binomial again. The E-order coincides!!! |
---|
490 | |
---|
491 | aux=multiplylist(auxexp[i][1],aa); |
---|
492 | aux2=multiplylist(auxexp[i][2],aa); |
---|
493 | aux3[1]=aux; |
---|
494 | aux3[2]=aux2; |
---|
495 | Eco[l]=aux3; |
---|
496 | newcoef[l]=Coef[i]; |
---|
497 | l=l+1;}} |
---|
498 | } |
---|
499 | |
---|
500 | else // monomials |
---|
501 | {if (auxexp[i][1][V]<auxc){aa=auxc/(auxc-auxexp[i][1][V]); |
---|
502 | auxexp[i][1][V]=0; |
---|
503 | aux4=list(); |
---|
504 | aux4[1]=multiplylist(auxexp[i][1],aa); |
---|
505 | Eco[l]=aux4; |
---|
506 | newcoef[l]=Coef[i]; |
---|
507 | l=l+1;}} |
---|
508 | } |
---|
509 | |
---|
510 | // cleaning units from the monomial generators of Eco |
---|
511 | // If there are hyperbolic equations in Eco, such that Eco=1, we detect it later, computing the E-order |
---|
512 | |
---|
513 | numg=size(Eco); |
---|
514 | for (k=1;k<=numg;k++){ if (size(newcoef[k])==1){Eco[k][1]=cleanunit(Eco[k][1],n,flaglist);}} |
---|
515 | |
---|
516 | // checking Eco |
---|
517 | |
---|
518 | ccase=0; |
---|
519 | cont2=0; |
---|
520 | cont3=0; |
---|
521 | val=0; |
---|
522 | |
---|
523 | // CASE Eco=0: If Eco=empty list then as ideal Eco=0 |
---|
524 | |
---|
525 | if (numg==0){ccase=1;} |
---|
526 | else |
---|
527 | { |
---|
528 | for (i=1;i<=numg;i++) {rs2[i]=size(newcoef[i]); |
---|
529 | if (rs2[i]==1){val=val+n; // monomials |
---|
530 | for (l=1;l<=n; l++) {if (Eco[i][1][l]==0) {cont2=cont2+1;}} |
---|
531 | } |
---|
532 | else{val=val+(2*n); // binomials |
---|
533 | for (l=1;l<=n; l++) {if (Eco[i][1][l]==0) {cont2=cont2+1;} |
---|
534 | if (Eco[i][2][l]==0) {cont2=cont2+1;}} |
---|
535 | } |
---|
536 | } |
---|
537 | |
---|
538 | // If cont2=val then all the entries of Eco are zero!! As ideal Eco=1 |
---|
539 | |
---|
540 | for (i=1;i<=sP;i++){if (rs[i]==2){ // binomials |
---|
541 | for (l=1;l<=n;l++) {if (expP[i][1][l]!=0) {cont3=cont3+1;} |
---|
542 | if (expP[i][2][l]!=0) {cont3=cont3+1;}} |
---|
543 | } |
---|
544 | else{ // monomials |
---|
545 | for (l=1;l<=n;l++) {if (expP[i][1][l]!=0) {cont3=cont3+1;}} |
---|
546 | } |
---|
547 | } |
---|
548 | |
---|
549 | // If cont3=0 all the entries of expP are zero!! As ideal P=1 this is detected before |
---|
550 | // If cont3=1 then P is bold regular |
---|
551 | |
---|
552 | |
---|
553 | // CASE Eco=1 |
---|
554 | |
---|
555 | if (cont2==val and cont3==1){ccase=2;} // BOLD REGULAR CASE |
---|
556 | if (cont2==val and cont3>1){ccase=3;} // CASE P=x^{\alpha},x^{\beta}, IN FACT, BOLD REGULAR |
---|
557 | if (cont2==val and cont3==0){ccase=4;} // P=1, then I=1 monomial case |
---|
558 | |
---|
559 | // Case BOLD REGULAR P=x^{\alpha}*(1-\mu y^{\delta}) |
---|
560 | // IT IS NON NECESSARY TO CHECK IT, Eco=empty list, already done! |
---|
561 | |
---|
562 | L=maxEord(newcoef,Eco,numg,n,flaglist); // L[1] is the E-order of Eco |
---|
563 | if (L[1]==0){ccase=2; print("E-order zero!");} // BOLD REGULAR CASE |
---|
564 | |
---|
565 | // we leave it to check the computations |
---|
566 | |
---|
567 | } // close else |
---|
568 | |
---|
569 | return(Eco,newcoef,ccase); |
---|
570 | } |
---|
571 | example |
---|
572 | {"EXAMPLE:"; echo = 2; |
---|
573 | ring r = 0,(x(1),y(2),x(3),y(4),x(5..7)),dp; |
---|
574 | list flag=identifyvar(); |
---|
575 | ideal P=x(1)^2*x(3)^5-x(5)^7*y(4),x(6)^3*y(2)^5-x(7)^5,x(5)^3*x(6)-y(4)^3*x(1)^5; |
---|
576 | list L=data(P,3,7); |
---|
577 | list L2=ECoef(L[1],L[2],3,1,3,7,flag); |
---|
578 | L2[1]; // exponents of the E-Coefficient ideal respect to x(1) |
---|
579 | L2[2]; // its coefficients |
---|
580 | L2[3]; // classify the type of ideal obtained |
---|
581 | |
---|
582 | ring r = 0,(x(1),y(2),x(3),y(4)),dp; |
---|
583 | list flag=identifyvar(); |
---|
584 | ideal J=x(1)^3*(1-2*y(2)*y(4)^2); // Bold regular case |
---|
585 | list L=data(J,1,4); |
---|
586 | list L2=ECoef(L[1],L[2],1,1,3,4,flag); |
---|
587 | L2; |
---|
588 | |
---|
589 | ring r = 0,(x(1),y(2),x(3),y(4),x(5..7)),dp; |
---|
590 | list flag=identifyvar(); |
---|
591 | ideal J=x(1)^3-x(3)^2*y(4)^2,x(1)*x(7)*y(2)-x(6)^3*x(5)*y(4)^3,x(5)^3-x(5)^3*y(2)^2; |
---|
592 | list L=data(J,3,7); |
---|
593 | list L2=ECoef(L[1],L[2],3,1,2,7,flag); |
---|
594 | L2; |
---|
595 | |
---|
596 | ring r = 3,(x(1),y(2),x(3),y(4),x(5..7)),dp; |
---|
597 | list flag=identifyvar(); |
---|
598 | ideal J=x(1)^3-x(3)^2*y(4)^2,x(1)*x(7)*y(2)-x(6)^3*x(5)*y(4)^3,x(5)^3-x(5)^3*y(2)^2; |
---|
599 | list L=data(J,3,7); |
---|
600 | list L2=ECoef(L[1],L[2],3,1,2,7,flag); |
---|
601 | L2; // THE COMPUTATIONS ARE NOT CORRECT IN CHARACTERISTIC p>0 |
---|
602 | // because numbers are treated as 0 in assignments |
---|
603 | |
---|
604 | } |
---|
605 | //////////////////////////////////////////////////////////////////////////// |
---|
606 | // The intvec a indicates the previous center |
---|
607 | // Hhist = intvec of exceptional divisors of the parent chart |
---|
608 | |
---|
609 | proc determinecenter(list Coef,list expJ,number c,int n,int Y,intvec a,list listmb,list flag,int control3,intvec Hhist) |
---|
610 | "USAGE: determinecenter(Coef,expJ,c,n,Y,a,listmb,flag,control3,Hhist); |
---|
611 | Coef, expJ, listmb, flag lists, c number, n, Y, control3 integers, a, Hhist intvec |
---|
612 | COMPUTE: next center of blowing up and related information, see example |
---|
613 | RETURN: several lists defining the center and related information |
---|
614 | EXAMPLE: example determinecenter; shows an example |
---|
615 | " |
---|
616 | {int i,j,rstep,l,mm,cont,cont1,cont2,cont3,a4,sI,sP,V,V2,ccase,b,Mindx,tip,mval; |
---|
617 | number auxc,a1,a2,ex,maxEo,aux; |
---|
618 | |
---|
619 | list D,H,auxJ; // lists of D_n,D_n-1,...,D_1; H_n,H_n-1,...,H_1; J_n,J_n-1,...,J_1 |
---|
620 | |
---|
621 | list oldOlist,oldC,oldt,oldD,oldH,allH; // information of the previous step |
---|
622 | |
---|
623 | list Olist,C,t,Dstar,center,expI,expP,newJ,maxset; |
---|
624 | |
---|
625 | list maxvar,auxlist,aux3,auxD,auxolist,auxdiv,auxaux,L,rs,auxgamma,auxg2,aux1; // auxiliary lists |
---|
626 | list auxinvlist,newcoef,EL,Ecoaux,Hplus,transH,Hsum,auxset,sumnewH; // auxiliary lists |
---|
627 | list auxcoefI,auxcent,center2; |
---|
628 | |
---|
629 | intvec oldinfobo7,infobo7; |
---|
630 | int infaux,leh,leh2,leh3; |
---|
631 | |
---|
632 | tip=listmb[1]; // It is not used in this procedure, it is used to compute the lcm of the denominators |
---|
633 | oldOlist=listmb[2]; |
---|
634 | oldC=listmb[3]; |
---|
635 | oldt=listmb[4]; // t= resolution function |
---|
636 | oldD=listmb[5]; |
---|
637 | |
---|
638 | oldH=listmb[6]; |
---|
639 | allH=listmb[7]; |
---|
640 | |
---|
641 | oldinfobo7=listmb[8]; // auxiliary intvec, it is used to define BO[7] |
---|
642 | |
---|
643 | // inicializating lists |
---|
644 | Olist=list(); |
---|
645 | C=list(); |
---|
646 | auxinvlist=list(); |
---|
647 | |
---|
648 | auxJ[1]=expJ; |
---|
649 | rstep=n; // we are in dimension rstep |
---|
650 | auxc=c; |
---|
651 | cont=1; |
---|
652 | |
---|
653 | if (Y==0) {D=iniD(n); H=iniD(n); infobo7=-1;} // first center, inicializate previous information |
---|
654 | |
---|
655 | if (Y!=0 and rstep==n) // In dimension n, D'_n is always of this form |
---|
656 | { auxdiv=list0(n); |
---|
657 | Dstar[1]=oldD[1]; |
---|
658 | |
---|
659 | b=size(a); |
---|
660 | for (i=1;i<=n;i++) {for (j=1;j<=b;j++) {if (a[j]==i) {aux=aux+oldD[1][i];}}} |
---|
661 | Dstar[1][Y]=aux; |
---|
662 | aux=0; |
---|
663 | |
---|
664 | auxdiv[Y]=oldOlist[1]-oldC[1]; |
---|
665 | D[1]=sumlist(Dstar[1],auxdiv);} // list defining D_n |
---|
666 | |
---|
667 | // computing strict transforms of the exceptional divisors H |
---|
668 | |
---|
669 | if (Y!=0){transH=iniD(n); |
---|
670 | for (i=1;i<=size(oldH);i++){transH[i]=oldH[i]; transH[i][Y]=0;} // Note: size(oldH)<=n |
---|
671 | allH[Y]=1;} // transform of |H|=H_nU...UH_1 |
---|
672 | |
---|
673 | // We put here size(oldH) instead of n because maybe we have not |
---|
674 | // calculated all the dimensions in the previous step |
---|
675 | |
---|
676 | // STARTING THE LOOP |
---|
677 | |
---|
678 | while (rstep>=1) |
---|
679 | { |
---|
680 | if (Y!=0 and rstep!=n) // transformation law of D_i for i<n |
---|
681 | { |
---|
682 | if (cont!=0) // the resolution function did not drop in higher dimensions |
---|
683 | { |
---|
684 | if (oldt[n-rstep]==a1/a2 and c==oldC[1] and control3==0) |
---|
685 | {auxD=list0(n); |
---|
686 | auxD[Y]=oldOlist[n-rstep+1]-oldC[n-rstep+1]; |
---|
687 | Dstar[n-rstep+1]=oldD[n-rstep+1]; |
---|
688 | |
---|
689 | for (i=1;i<=n;i++) {for (j=1;j<=b;j++) {if (a[j]==i) {aux=aux+oldD[n-rstep+1][i];}}} |
---|
690 | Dstar[n-rstep+1][Y]=aux; |
---|
691 | aux=0; |
---|
692 | |
---|
693 | D[n-rstep+1]=sumlist(Dstar[n-rstep+1],auxD); |
---|
694 | |
---|
695 | } |
---|
696 | else |
---|
697 | {cont=0; |
---|
698 | for (j=n-rstep+1;j<=n; j++){D[j]=list0(n);} |
---|
699 | } |
---|
700 | } |
---|
701 | } |
---|
702 | |
---|
703 | // Factorizing J=M*I |
---|
704 | |
---|
705 | cont1=0; |
---|
706 | for (i=1;i<=n;i++) {if (D[n-rstep+1][i]==0) {cont1=cont1+1;}} // if it fails write: listO(n)[i] |
---|
707 | |
---|
708 | if (cont1==n) {expI=expJ;} // D[n-rstep+1]=0 (is a list of zeros) |
---|
709 | else { |
---|
710 | for (i=1;i<=size(expJ);i++) |
---|
711 | {rs[i]=size(Coef[i]); |
---|
712 | if (rs[i]==2){ aux1=list(); |
---|
713 | aux1[1]=reslist(expJ[i][1],D[n-rstep+1]); |
---|
714 | aux1[2]=reslist(expJ[i][2],D[n-rstep+1]); |
---|
715 | expI[i]=aux1;} // binomial |
---|
716 | else {aux1=list(); |
---|
717 | aux1[1]=reslist(expJ[i][1],D[n-rstep+1]); |
---|
718 | expI[i]=aux1;}} // monomial |
---|
719 | } |
---|
720 | |
---|
721 | // NOTE: coeficients of I = coeficients of J, because I and J differ in a monomial |
---|
722 | |
---|
723 | // Detecting errors, negative exponents in expI |
---|
724 | |
---|
725 | sI=size(expI); |
---|
726 | |
---|
727 | for (i=1;i<=sI;i++) |
---|
728 | {rs[i]=size(Coef[i]); |
---|
729 | if (rs[i]==2){for (j=1;j<=2;j++){for (l=1;l<=n; l++) |
---|
730 | {if (expI[i][j][l]<0) {print("ERROR, the BINOMIAL ideal I has negative components"); |
---|
731 | // print("M ideal"); print(D[n-rstep+1]); print(expI); print("dimension"); print(rstep); |
---|
732 | // print("previous chart"); print(size(finalchart)); ~; |
---|
733 | }}}} |
---|
734 | else {for (l=1;l<=n; l++) |
---|
735 | {if (expI[i][1][l]<0) {print("ERROR, the MONOMIAL ideal I has negative components"); |
---|
736 | // print("M ideal"); print(D[n-rstep+1]); print(expI); print("dimension"); print(rstep); |
---|
737 | // print("previous chart"); print(size(finalchart)); ~; |
---|
738 | }}} |
---|
739 | } |
---|
740 | |
---|
741 | // Compute the maximal E-order of I |
---|
742 | |
---|
743 | L=maxEord(Coef,expI,sI,n,flag); |
---|
744 | maxEo=L[1]; // E-order of I |
---|
745 | |
---|
746 | // Inicializating information |
---|
747 | |
---|
748 | auxolist=maxEo; |
---|
749 | a1=maxEo; |
---|
750 | a2=auxc; |
---|
751 | Olist=Olist+auxolist; // list of new maximal E-orders o_n,o_{n-1},...o_1 |
---|
752 | aux3=auxc; |
---|
753 | C=C+aux3; // list of new critical values c=c_{n+1},c_{n},...c_2 |
---|
754 | |
---|
755 | // It is necessary to check if the first coordinate of the invariant has dropped or not |
---|
756 | // NOTE: By construction, the first coordinate is always 1 !! |
---|
757 | // It has dropped is equivalent to: CURRENT C<c of the previous step |
---|
758 | |
---|
759 | // Calculate new H, this is done for every dimension |
---|
760 | |
---|
761 | if (Y!=0){a4=size(oldt); |
---|
762 | if (n-rstep+1>a4){cont=0; oldt[n-rstep+1]=0; } // VERIFICAR!!!! |
---|
763 | |
---|
764 | if (cont!=0 and oldt[n-rstep+1]==a1/a2 and c==oldC[1] and control3==0){H[n-rstep+1]=transH[n-rstep+1]; |
---|
765 | |
---|
766 | // we fill now the value for BO[7] |
---|
767 | if (oldinfobo7[n-rstep+1]==-1){leh=size(Hhist); |
---|
768 | infobo7[n-rstep+1]=Hhist[leh];} // suitable index !!! |
---|
769 | else{ infaux=oldinfobo7[n-rstep+1]; |
---|
770 | infobo7[n-rstep+1]=infaux;} // the same as the previous step |
---|
771 | |
---|
772 | } |
---|
773 | else { |
---|
774 | if (rstep<n) {sumnewH=list0(n); |
---|
775 | for (i=1;i<n-rstep+1;i++){sumnewH=sumlist(sumnewH,H[i]);} |
---|
776 | H[n-rstep+1]=reslist(allH,sumnewH);} |
---|
777 | else {H[n-rstep+1]=allH;} |
---|
778 | |
---|
779 | // we fill the value for BO[7] too, we complete it at the end if necessary |
---|
780 | infobo7[n-rstep+1]=-1; |
---|
781 | } |
---|
782 | } |
---|
783 | |
---|
784 | // It is necessary to detect the monomial case AFTER inicializate the information |
---|
785 | // OTHERWISE WE WILL HAVE EMPTY COMPONENTS IN THE RESOLUTION FUNCTION |
---|
786 | |
---|
787 | // If maxEo=0 but maxo!=0 MONOMIAL CASE (because E-Sing(J,c) still !=emptyset) |
---|
788 | // If maxEo=0 and maxo=0 then I=1, (real) monomial case, the same case for us |
---|
789 | // NOTE THAT IT DOESN'T MATTER IF THERE IS A p-TH POWER OF A HYPERBOLIC EQ, THE E-ORDER IS ZERO ANYWAY |
---|
790 | |
---|
791 | if (maxEo==0){auxgamma=Gamma(D[n-rstep+1],auxc,n); // Gamma gives (maxlist,gamma,center) |
---|
792 | auxg2=auxgamma[3]; |
---|
793 | center=center+auxg2; |
---|
794 | center=elimrep(center); |
---|
795 | auxinvlist=auxgamma[2]; |
---|
796 | |
---|
797 | // print("gamma"); print(auxg2); |
---|
798 | |
---|
799 | break;} |
---|
800 | |
---|
801 | // Calculate P // P=I+M^{o/(c-o)} with weight o |
---|
802 | |
---|
803 | if (maxEo>=auxc) {expP=expI; Mindx=0;} // The coefficients also remain constant |
---|
804 | else {ex=maxEo/(auxc-maxEo); |
---|
805 | auxlist=list(); |
---|
806 | Mindx=1; |
---|
807 | auxlist[1]=multiplylist(D[n-rstep+1],ex); // weighted monomial part: D[n-rstep+1]^ex; |
---|
808 | expP=insert(expI,auxlist); // P=I+D[n-rstep+1]^ex; |
---|
809 | auxcoefI=Coef; |
---|
810 | Coef=insert(Coef,list(1));} // Adding the coefficient for M |
---|
811 | |
---|
812 | // NOTE: IT IS NECESSARY TO ADD COEFFICIENT 1 TO THE MONOMIAL PART M |
---|
813 | // E-ord(P_i)=E-ord(I_i) so to compute the E-order of P_i we can compute E-ord(I_i) |
---|
814 | |
---|
815 | // Calculate variables of E-maximal contact, ALWAYS WITH RESPECT TO THE IDEAL I !! |
---|
816 | |
---|
817 | sP=size(expP); // Can be different from size(expI) |
---|
818 | |
---|
819 | if (Mindx==1){ maxvar=Emaxcont(auxcoefI,expI,sI,n,flag);} |
---|
820 | else{ maxvar=Emaxcont(Coef,expP,sP,n,flag);} |
---|
821 | |
---|
822 | auxc=maxvar[1]; // E-order of P, critical value for the next step, ALSO VALID auxc=maxEo; |
---|
823 | if (auxc!=maxEo){print("ERROR, the E-order is not well computed");} |
---|
824 | |
---|
825 | maxset=maxvar[2]; |
---|
826 | |
---|
827 | // center=center + maxset; // HACER DESPUES Y A?ADIR SOLO V!!!!!! |
---|
828 | // Cleaning the center: eliminating repeated variables |
---|
829 | // center=elimrep(center); |
---|
830 | |
---|
831 | // if (rstep==1) {break;} // Induction finished, is not necessary to compute the rest |
---|
832 | |
---|
833 | // Calculate Hplus=set of non permissible hypersurfaces |
---|
834 | // RESET Hplus if c has dropped or we have eliminated hyperbolic generators |
---|
835 | |
---|
836 | // ES NECESARIO PONER CONDICION DE SI INVARIANTE BAJA O NO??? SI BAJA HPLUS NO SE USA... |
---|
837 | |
---|
838 | if (Y==0 or c<oldC[1] or control3==1) {Hplus=list0(n);} |
---|
839 | else {Hsum=list0(n); |
---|
840 | Hplus=allH; |
---|
841 | for (i=1;i<=n-rstep+1;i++){Hsum=sumlist(Hsum,H[i]);} |
---|
842 | Hplus=reslist(Hplus,Hsum); // CHEQUEAR QUE NO SALEN -1'S |
---|
843 | } |
---|
844 | |
---|
845 | // Taking into account variables of maxset outside of Hplus (so inside Hminus) |
---|
846 | |
---|
847 | if (Y==0 or c<oldC[1] or control3==1){V=maxset[1]; // Hplus=0 so any variable is permissible |
---|
848 | maxset=delete(maxset,1);} // eliminating this variable V from maxset |
---|
849 | else{ |
---|
850 | // If the invariant remains constant V comes from the previous step |
---|
851 | |
---|
852 | if (cont!=0 and oldt[n-rstep+1]==a1/a2 and c==oldC[1]){ |
---|
853 | if (Mindx==1){ |
---|
854 | //----------------------------USING HPLUS---------------------------------------- |
---|
855 | // REMIND THAT IN THIS CASE maxset=HYPERSURFACES OF E-MAXIMAL CONTACT FOR I, INSTEAD OF P |
---|
856 | |
---|
857 | V2=a[n-rstep+1]; // V can be different from the variable coming from the previous step |
---|
858 | // check that V2 belongs to maxset |
---|
859 | |
---|
860 | for (i=1;i<=size(maxset);i++){ |
---|
861 | if (V2==maxset[i]){mval=1; break;} |
---|
862 | else{mval=0;} |
---|
863 | } |
---|
864 | |
---|
865 | if (Hplus[V2]==0 and mval==1){V=V2;} // V2 is permissible |
---|
866 | else{ |
---|
867 | cont2=1; |
---|
868 | cont3=1; |
---|
869 | auxset=maxset; |
---|
870 | while (cont2!=0){mm=auxset[1]; |
---|
871 | if (Hplus[mm]!=0) {auxset=delete(auxset,1); cont3=cont3+1;} |
---|
872 | // eliminating non permissible variables from maxset |
---|
873 | else {cont2=0;}} |
---|
874 | V=maxset[cont3]; // first permissible variable |
---|
875 | maxset=delete(maxset,cont3); |
---|
876 | } |
---|
877 | } |
---|
878 | |
---|
879 | //------------------------------------------------------------------------------- |
---|
880 | else{ V=a[n-rstep+1];} |
---|
881 | } |
---|
882 | else {V=maxset[1]; // Hplus=0 so any variable is permissible |
---|
883 | maxset=delete(maxset,1); |
---|
884 | } |
---|
885 | |
---|
886 | } |
---|
887 | |
---|
888 | |
---|
889 | // if (V!=V2 and V2!=0){print(a); print(rstep); print(V); print(V2); print("num cartas"); print(size(finalchart)); ~;} |
---|
890 | |
---|
891 | V2=0; |
---|
892 | |
---|
893 | // Adding the new hypersurface of E-maximal contact to the center |
---|
894 | |
---|
895 | auxcent[1]=V; |
---|
896 | |
---|
897 | center=center + auxcent; // print("num cartas"); print(size(finalchart)); print(center); if (size(finalchart)==2){~~;} |
---|
898 | |
---|
899 | auxcent=list(); |
---|
900 | |
---|
901 | // Cleaning the center: eliminating repeated variables CREO QUE NO HACE FALTA |
---|
902 | |
---|
903 | center2=elimrep(center); // print(center2); print("-----------"); |
---|
904 | |
---|
905 | // if (size(center2)!=size(center)){print("MAL");} |
---|
906 | |
---|
907 | // for (i=1;i<=size(center);i++){if (center2[i]!=center[i]){print("cambia");}} |
---|
908 | |
---|
909 | |
---|
910 | if (rstep==1) {break;} // Induction finished, is not necessary to compute the rest |
---|
911 | |
---|
912 | |
---|
913 | // Calculate Eco=E-Coeff_V(P) where V is a permissible hypersurface which belongs to the center |
---|
914 | // Eco can have rational exponents |
---|
915 | |
---|
916 | Ecoaux=ECoef(Coef,expP,sP,V,auxc,n,flag); |
---|
917 | |
---|
918 | // SPECIAL CASES: BOLD REGULAR CASE |
---|
919 | //-------------------------------------------------------------------- |
---|
920 | |
---|
921 | if (Ecoaux[3]==1){ // Eco=EMPTY LIST, Eco=0 AS IDEAL |
---|
922 | aux1[1]=list0(n); |
---|
923 | newJ[1]=aux1; // monomial with zero entries, newJ=1 as ideal |
---|
924 | newcoef[1]=list(1); // the new coefficient is only 1 |
---|
925 | auxaux=list(); |
---|
926 | auxaux[1]=newJ; |
---|
927 | auxJ=auxJ+auxaux; // auxJ list of ideals J_i |
---|
928 | auxinvlist=1; |
---|
929 | break;} |
---|
930 | |
---|
931 | //----------------------------------------------------------- |
---|
932 | // THIS CASE IS NOT GOING TO APPEAR, BUT WE LEAVE IT TO CHECK COMPUTATIONS |
---|
933 | |
---|
934 | if (Ecoaux[3]==2 or Ecoaux[3]==3){ // Eco=0 LIST, Eco=1 AS IDEAL |
---|
935 | aux1[1]=list0(n); |
---|
936 | newJ[1]=aux1; |
---|
937 | newcoef[1]=list(1); // print("Strange case happens"); ~; |
---|
938 | auxaux=list(); |
---|
939 | auxaux[1]=newJ; |
---|
940 | auxJ=auxJ + auxaux; // auxJ list of ideals J_i |
---|
941 | auxinvlist=1; |
---|
942 | break;} |
---|
943 | //----------------------------------------------------------- |
---|
944 | // THIS CASE IS NOT GOING TO APPEAR, BUT WE LEAVE IT TO CHECK COMPUTATIONS |
---|
945 | |
---|
946 | // P=1 THIS CANNOT HAPPEN SINCE P=1 IFF I=1 (or I is equivalent to 1) |
---|
947 | // and this is the monomial case, already checked |
---|
948 | |
---|
949 | if (Ecoaux[3]==4){print("ERROR in ECoef"); break;} |
---|
950 | //----------------------------------------------------------- |
---|
951 | |
---|
952 | // If we are here Ecoaux[3]=0, then continue |
---|
953 | |
---|
954 | // Filling the list of "ideals", auxJ=J_n,J_{n-1},... |
---|
955 | |
---|
956 | newJ=Ecoaux[1]; |
---|
957 | newcoef=Ecoaux[2]; |
---|
958 | |
---|
959 | auxJ=insert(auxJ,newJ,n-rstep+1); // newJ is inserted after n-rstep+1 position, so in position n-rstep+2 |
---|
960 | |
---|
961 | // New input for the loop, if we are here newJ is different from 0 |
---|
962 | |
---|
963 | expJ=newJ; |
---|
964 | Coef=newcoef; |
---|
965 | |
---|
966 | newJ=list(); |
---|
967 | expI=list(); |
---|
968 | expP=list(); |
---|
969 | rstep=rstep-1; // print(size(auxJ)); |
---|
970 | } |
---|
971 | |
---|
972 | // EXIT LOOP "while" |
---|
973 | // we do NOT construct the center as an ideal because WE USE LISTS |
---|
974 | |
---|
975 | t=dividelist(Olist,C); // resolution function t |
---|
976 | |
---|
977 | // Complete the intvec infobo7 if necessary |
---|
978 | |
---|
979 | if (control3==1){infobo7=-1;} // We reset the value after clean hyperbolic equations |
---|
980 | leh2=size(Olist); |
---|
981 | leh3=size(infobo7); |
---|
982 | if (leh3<leh2){for (j=leh3+1;j<=leh2; j++){infobo7[j]=-1;}} |
---|
983 | |
---|
984 | // Auxiliary list to complete the resolution function in special cases |
---|
985 | if (size(auxinvlist)==0) {auxinvlist[1]=0;} |
---|
986 | |
---|
987 | return(center,auxJ,Olist,C,t,D,H,allH,auxinvlist,infobo7); |
---|
988 | } |
---|
989 | example |
---|
990 | {"EXAMPLE:"; echo = 2; |
---|
991 | ring r = 0,(x(1..4)),dp; |
---|
992 | list flag=identifyvar(); |
---|
993 | ideal J=x(1)^2-x(2)^2*x(3)^5, x(1)*x(3)^3+x(4)^6; |
---|
994 | list Lmb=1,list(0,0,0,0),list(0,0,0,0),list(0,0,0,0),iniD(4),iniD(4),list(0,0,0,0),-1; |
---|
995 | list L=data(J,2,4); |
---|
996 | list LL=determinecenter(L[1],L[2],2,4,0,0,Lmb,flag,0,-1); // Compute the first center |
---|
997 | LL[1]; // index of variables in the center |
---|
998 | LL[2]; // exponents of ideals J_4,J_3,J_2,J_1 |
---|
999 | LL[3]; // list of orders of J_4,J_3,J_2,J_1 |
---|
1000 | LL[4]; // list of critical values |
---|
1001 | LL[5]; // components of the resolution function t |
---|
1002 | LL[6]; // list of D_4,D_3,D_2,D_1 |
---|
1003 | LL[7]; // list of H_4,H_3,H_2,H_1 (exceptional divisors) |
---|
1004 | LL[8]; // list of all exceptional divisors acumulated |
---|
1005 | LL[9]; // auxiliary invariant |
---|
1006 | LL[10]; // intvec pointing out the last step where the function t has dropped |
---|
1007 | |
---|
1008 | ring r= 0,(x(1..4)),dp; |
---|
1009 | list flag=identifyvar(); |
---|
1010 | ideal J=x(1)^3-x(2)^2*x(3)^5, x(1)*x(3)^3+x(4)^5; |
---|
1011 | list Lmb=2,list(0,0,0,0),list(0,0,0,0),list(0,0,0,0),iniD(4),iniD(4),list(0,0,0,0),-1; |
---|
1012 | list L2=data(J,2,4); |
---|
1013 | list L3=determinecenter(L2[1],L2[2],2,4,0,0,Lmb,flag,0,-1); // Example with rational exponents in E-Coeff |
---|
1014 | L3[1]; // index of variables in the center |
---|
1015 | L3[2]; // exponents of ideals J_4,J_3,J_2,J_1 |
---|
1016 | L3[3]; // list of orders of J_4,J_3,J_2,J_1 |
---|
1017 | L3[4]; // list of critical values |
---|
1018 | L3[5]; // components of the resolution function |
---|
1019 | } |
---|
1020 | //////////////////////////////////////////////////////// |
---|
1021 | // idchart= identity number of the current chart |
---|
1022 | // infochart=chart[idchart] information related to the chart to blow up |
---|
1023 | // infochart= int parent,int Y,intvec a,list expJ,list Coef, list flag, // NEEDED FOR THE RESOLUTION |
---|
1024 | // intvec Hhist, list blwhist, module path, list hipercoef, list hiperexp // NEEDED FOR THE OUTPUT |
---|
1025 | |
---|
1026 | // NOTE: IT IS NOT NECESSARY TAKE INTO ACCOUNT "y" VARIABLES BECAUSE THE CENTER IS ALREADY GIVEN |
---|
1027 | |
---|
1028 | proc Blowupcenter(list center,int idchart,int nchart,list infochart,number c,int n,int currentstep) |
---|
1029 | "USAGE: Blowupcenter(center,id,nchart,infochart,c,n,cstep); |
---|
1030 | center, infochart lists, id, nchart, n, cstep integers, c number |
---|
1031 | COMPUTE: The blowing up at the chart IDCHART along the given center |
---|
1032 | RETURN: new affine charts and related information, see example |
---|
1033 | EXAMPLE: example Blowupcenter; shows an example |
---|
1034 | " |
---|
1035 | {int num,i,j,k,l,parent,Y,lon,m,m2; |
---|
1036 | intvec a,Hhist,auxHhist; |
---|
1037 | number auxsum, auxsum2; |
---|
1038 | list sons,aux,expJ,blexpJ,blD; |
---|
1039 | list auxstep,Coef; |
---|
1040 | list auxchart,auxchart1,info,flaglist; |
---|
1041 | list auxblwhist,blwhist,hipercoef,hiperexp; |
---|
1042 | module auxpath,auxp2; |
---|
1043 | |
---|
1044 | parent=idchart; |
---|
1045 | num=size(center); |
---|
1046 | |
---|
1047 | // Transform to intvec the list of variables defining the center |
---|
1048 | a=center[1]; |
---|
1049 | for (i=2;i<=num;i++){a=a,center[i];} |
---|
1050 | |
---|
1051 | expJ=infochart[4]; |
---|
1052 | Coef=infochart[5]; |
---|
1053 | flaglist=infochart[6]; |
---|
1054 | Hhist=infochart[7]; |
---|
1055 | blwhist=infochart[8]; |
---|
1056 | auxpath=infochart[9]; |
---|
1057 | hipercoef=infochart[10]; |
---|
1058 | hiperexp=infochart[11]; |
---|
1059 | |
---|
1060 | l=size(expJ); |
---|
1061 | |
---|
1062 | // input for the loop |
---|
1063 | blexpJ=expJ; |
---|
1064 | |
---|
1065 | // making the blowing up in the i-th chart |
---|
1066 | for (i=1;i<=num;i++) |
---|
1067 | { |
---|
1068 | // we assign the current number of charts +1 to the i-th chart |
---|
1069 | idchart=nchart+1; |
---|
1070 | nchart=nchart+1; |
---|
1071 | aux=idchart; |
---|
1072 | sons=sons+aux; |
---|
1073 | |
---|
1074 | auxstep[i]=currentstep+1; |
---|
1075 | |
---|
1076 | Y=center[i]; |
---|
1077 | |
---|
1078 | // The blowing up |
---|
1079 | |
---|
1080 | for (j=1;j<=l;j++){lon=size(Coef[j]); |
---|
1081 | if (lon==1){for (m=1;m<=n;m++){for (m2=1;m2<=num;m2++){ |
---|
1082 | if (m==center[m2]){auxsum=auxsum+ expJ[j][1][m];}}} |
---|
1083 | blexpJ[j][1][Y]=auxsum-c; |
---|
1084 | auxsum=0;} // monomial |
---|
1085 | else {for (m=1;m<=n;m++){for (m2=1;m2<=num;m2++){ |
---|
1086 | if (m==center[m2]){auxsum=auxsum+expJ[j][1][m]; |
---|
1087 | auxsum2=auxsum2+expJ[j][2][m];}}} |
---|
1088 | blexpJ[j][1][Y]=auxsum-c; |
---|
1089 | blexpJ[j][2][Y]=auxsum2-c; |
---|
1090 | auxsum=0; auxsum2=0;} // binomial |
---|
1091 | } |
---|
1092 | |
---|
1093 | |
---|
1094 | auxHhist=Hhist,Y; // history of the exceptional divisors in this chart |
---|
1095 | auxblwhist=tradblwup(blwhist,n,Y,a,num); // history of the blow ups in this chart |
---|
1096 | |
---|
1097 | auxp2=auxpath,[parent,i]; |
---|
1098 | |
---|
1099 | auxchart1=parent,Y,a,blexpJ,Coef,flaglist,auxHhist,auxblwhist,auxp2,hipercoef,hiperexp; |
---|
1100 | |
---|
1101 | // Coef, flaglist are not modified after the blowing-up, the hyperbolic information is the same as in the parent chart |
---|
1102 | |
---|
1103 | auxchart[i]=auxchart1; |
---|
1104 | |
---|
1105 | // Inicializating the exponents of J for the next chart |
---|
1106 | |
---|
1107 | blexpJ=expJ; |
---|
1108 | } |
---|
1109 | // end of the loop |
---|
1110 | |
---|
1111 | // we add its sons to the current chart |
---|
1112 | infochart=infochart+sons; |
---|
1113 | info[1]=infochart; |
---|
1114 | |
---|
1115 | return(info,auxchart,nchart,auxstep,num); |
---|
1116 | } |
---|
1117 | example |
---|
1118 | {"EXAMPLE:"; echo = 2; |
---|
1119 | ring r = 0,(x(1),y(2),x(3),y(4),x(5..7)),dp; |
---|
1120 | list flag=identifyvar(); |
---|
1121 | ideal J=x(1)^3-x(3)^2*y(4)^2,x(1)*x(7)*y(2)-x(6)^3*x(5)*y(4)^3,x(5)^3-x(5)^3*y(2)^2; |
---|
1122 | list Lmb=2,list(0,0,0,0,0,0,0),list(0,0,0,0,0,0,0),list(0,0,0,0,0,0,0),iniD(7),iniD(7),list(0,0,0,0,0,0,0),-1; |
---|
1123 | list L=data(J,3,7); |
---|
1124 | list L2=determinecenter(L[1],L[2],2,7,0,0,Lmb,flag,0,-1); // Computing the center |
---|
1125 | module auxpath=[0,-1]; |
---|
1126 | list infochart=0,0,0,L[2],L[1],flag,0,list(0,0,0,0,0,0,0),auxpath,list(),list(); |
---|
1127 | list L3=Blowupcenter(L2[1],1,1,infochart,2,7,0); |
---|
1128 | L3[1]; // current chart (parent,Y,center,expJ,Coef,flag,Hhist,blwhist,path,hipercoef,hiperexp) with sons: [12],...,[16] |
---|
1129 | L3[2][1]; // information of its first son, write L3[2][2],...,L3[2][5] to see the other sons |
---|
1130 | L3[3]; // current number of charts |
---|
1131 | L3[4]; // step/level associated to each son |
---|
1132 | L3[5]; // number of variables in the center |
---|
1133 | } |
---|
1134 | ////////////////////////////////////////////////////////////// |
---|
1135 | |
---|
1136 | proc tradblwup(list blwhist,int n,int Y,intvec a,int num) |
---|
1137 | "Internal procedure - no help and no example available |
---|
1138 | " |
---|
1139 | { |
---|
1140 | int i,j,blwnew; |
---|
1141 | intvec aux,aux2; |
---|
1142 | |
---|
1143 | for (j=1;j<=n;j++){ |
---|
1144 | for (i=1;i<=num;i++){ |
---|
1145 | if (j==a[i] and a[i]!=Y){blwnew=Y; break;} |
---|
1146 | else {blwnew=0;} |
---|
1147 | } |
---|
1148 | aux=blwhist[j]; |
---|
1149 | aux2=aux,blwnew; |
---|
1150 | blwhist[j]=aux2; |
---|
1151 | } |
---|
1152 | return(blwhist); |
---|
1153 | } |
---|
1154 | ////////////////////////////////////////////////////////////// |
---|
1155 | // It is called only when Eord(J)=0, and J!=1 it is checked inside |
---|
1156 | // SO IT IS CALLED AFTER: maxEord(Coef,expJ,sJ,n,flaglist); --> gives (max E-order,sums) |
---|
1157 | |
---|
1158 | proc Nonhyp(list Coef,list expJ,int sJ,int n,list flaglist,list sums) |
---|
1159 | "USAGE: Nonhyp(Coef,expJ,sJ,n,flaglist,sums); |
---|
1160 | Coef, expJ, flaglist, sums lists, sJ, n integers |
---|
1161 | COMPUTE: The "ideal" generated by the non hyperbolic generators of J |
---|
1162 | RETURN: lists with the following information |
---|
1163 | newcoef,newJ: coefficients and exponents of the non hyperbolic generators |
---|
1164 | totalhyp,totalgen: coefficients and exponents of the hyperbolic generators |
---|
1165 | flaglist: new list saying status of variables |
---|
1166 | NOTE: the basering r is supposed to be a polynomial ring K[x,y], |
---|
1167 | in fact, we work in a localization of K[x,y], of type K[x,y]_y with y invertible variables. |
---|
1168 | EXAMPLE: example Nonhyp; shows an example |
---|
1169 | " |
---|
1170 | { |
---|
1171 | int i,j,k,h,lon,lon2,cont; |
---|
1172 | number eordcontrol; |
---|
1173 | list genhyp,listgen,listid,posnumJ,newJ,newcoef,hypcoef,hyp,aux1,aux2,aux3,aux,midlist; |
---|
1174 | list totalhyp,totalgen; |
---|
1175 | |
---|
1176 | eordcontrol=0; |
---|
1177 | |
---|
1178 | while (eordcontrol==0 and sJ!=0) |
---|
1179 | { |
---|
1180 | |
---|
1181 | // Give a positional number/flag to each generator of expJ |
---|
1182 | |
---|
1183 | for (i=1;i<=sJ; i++){listgen=expJ[i]; listid=i; posnumJ[i]=listgen+listid; } |
---|
1184 | |
---|
1185 | // Select the non hyperbolic and hyperbolic generators |
---|
1186 | |
---|
1187 | for (j=1;j<=sJ; j++){lon=size(Coef[j]); |
---|
1188 | if (lon==1){ |
---|
1189 | |
---|
1190 | // IS NOT NECESSARY TO CHECK IF THERE EXIST A MONOMIAL WITH ONLY UNITS, ALREADY DONE!! |
---|
1191 | |
---|
1192 | aux1=aux1+posnumJ[j]; |
---|
1193 | aux3=list(); |
---|
1194 | aux3[1]=expJ[j]; |
---|
1195 | newJ=newJ+aux3; |
---|
1196 | aux3[1]=Coef[j]; |
---|
1197 | newcoef=newcoef+aux3; |
---|
1198 | } |
---|
1199 | |
---|
1200 | else{ // CHECKING BINOMIALS, ONE TERM WITH E-ORDER ZERO GIVES HYPERBOLIC EQ |
---|
1201 | |
---|
1202 | if (sums[j][1]==0 or sums[j][2]==0){aux2=aux2+posnumJ[j]; |
---|
1203 | aux3=list(); |
---|
1204 | aux3[1]=expJ[j]; |
---|
1205 | genhyp=genhyp+aux3; |
---|
1206 | aux3[1]=Coef[j]; |
---|
1207 | hypcoef=hypcoef+aux3; |
---|
1208 | if (sums[j][1]==0){aux3[1]=expJ[j][2]; hyp=hyp+aux3;} |
---|
1209 | if (sums[j][2]==0){aux3[1]=expJ[j][1]; hyp=hyp+aux3;} |
---|
1210 | } |
---|
1211 | else {aux1=aux1+posnumJ[j]; |
---|
1212 | aux3=list(); |
---|
1213 | aux3[1]=expJ[j]; |
---|
1214 | newJ=newJ+aux3; |
---|
1215 | aux3[1]=Coef[j]; |
---|
1216 | newcoef=newcoef+aux3;} |
---|
1217 | |
---|
1218 | } |
---|
1219 | } |
---|
1220 | |
---|
1221 | // NOTE: aux1 and aux2 are no needed right now! |
---|
1222 | |
---|
1223 | // Identify new y variables, that is, x variables in the monomials contained in hyp |
---|
1224 | |
---|
1225 | h=size(hyp); |
---|
1226 | |
---|
1227 | for (k=1;k<=h; k++){ for(i=1;i<=n; i++){ if (hyp[k][i]!=0 and flaglist[i]==0) {flaglist[i]=1;}}} |
---|
1228 | |
---|
1229 | // To replace x by y IT IS NECESSARY TO CHANGE THE BASERING!!! We change only the list flaglist |
---|
1230 | |
---|
1231 | // CHECK IF THE IDEAL IS ALREADY GENERATED BY MONOMIALS, in this case |
---|
1232 | // WE HAVE FINISHED THE E-RESOLUTION PART, J GENERATED BY MONOMIALS AND HYPERBOLIC EQS |
---|
1233 | |
---|
1234 | cont=0; |
---|
1235 | lon2=size(newJ); |
---|
1236 | for (j=1;j<=lon2; j++){if (size(newJ[j])==1){cont=cont+1;}} |
---|
1237 | |
---|
1238 | if (cont==lon2){newcoef=list(); |
---|
1239 | newJ=list(); |
---|
1240 | totalgen=totalgen+genhyp; |
---|
1241 | totalhyp=totalhyp+hypcoef; |
---|
1242 | break;} |
---|
1243 | |
---|
1244 | // CHECK IF THERE ARE MORE HYPERBOLIC EQUATIONS AFTER UPDATE THE FLAG LIST |
---|
1245 | // CHECK THE MAXIMAL E-ORDER AGAIN |
---|
1246 | |
---|
1247 | if (lon2==0){ // we are in the previous case, newJ=empty list, save values and exit |
---|
1248 | |
---|
1249 | totalgen=totalgen+genhyp; |
---|
1250 | totalhyp=totalhyp+hypcoef; |
---|
1251 | break; |
---|
1252 | } |
---|
1253 | |
---|
1254 | midlist=maxEord(newcoef,newJ,lon2,n,flaglist); |
---|
1255 | |
---|
1256 | eordcontrol=midlist[1]; |
---|
1257 | |
---|
1258 | if (eordcontrol==0){ // new input for the loop |
---|
1259 | Coef=newcoef; |
---|
1260 | expJ=newJ; |
---|
1261 | sJ=lon2; |
---|
1262 | sums=midlist[2]; // flaglist is already updated |
---|
1263 | |
---|
1264 | totalgen=totalgen+genhyp; |
---|
1265 | totalhyp=totalhyp+hypcoef; |
---|
1266 | |
---|
1267 | hypcoef=list(); |
---|
1268 | genhyp=list(); |
---|
1269 | |
---|
1270 | newJ=list(); |
---|
1271 | newcoef=list(); |
---|
1272 | } |
---|
1273 | else{ // If the process is already finished we save the values and exit |
---|
1274 | |
---|
1275 | totalgen=totalgen+genhyp; |
---|
1276 | totalhyp=totalhyp+hypcoef; |
---|
1277 | } |
---|
1278 | |
---|
1279 | } // closing while |
---|
1280 | |
---|
1281 | return(newcoef,newJ,totalhyp,totalgen,flaglist); |
---|
1282 | } |
---|
1283 | example |
---|
1284 | {"EXAMPLE:"; echo = 2; |
---|
1285 | ring r = 0,(x(1),y(2),x(3),y(4),x(5..7)),dp; |
---|
1286 | list flag=identifyvar(); // List giving flag=1 to invertible variables: y(2),y(4) |
---|
1287 | ideal J=x(1)^3-x(3)^2*y(4)^2,x(1)*x(7)*y(2)-x(6)^3*x(5)*y(4)^3,1-x(5)^2*y(2)^2; |
---|
1288 | list L=data(J,3,7); |
---|
1289 | list L2=maxEord(L[1],L[2],3,7,flag); |
---|
1290 | L2[1]; // Maximum E-order |
---|
1291 | list New=Nonhyp(L[1],L[2],3,7,flag,L2[2]); |
---|
1292 | New[1]; // Coefficients of the non hyperbolic part |
---|
1293 | New[2]; // Exponents of the non hyperbolic part |
---|
1294 | New[3]; // Coefficients of the hyperbolic part |
---|
1295 | New[4]; // New hyperbolic equations |
---|
1296 | New[5]; // New list giving flag=1 to invertible variables: y(2),y(4),y(5) |
---|
1297 | |
---|
1298 | ring r = 0,(x(1..4)),dp; |
---|
1299 | list flag=identifyvar(); |
---|
1300 | ideal J=1-x(1)^5*x(2)^2*x(3)^5, x(1)^2*x(3)^3+x(1)^4*x(4)^6; |
---|
1301 | list L=data(J,2,4); |
---|
1302 | list L2=maxEord(L[1],L[2],2,4,flag); |
---|
1303 | L2[1]; // Maximum E-order |
---|
1304 | list New=Nonhyp(L[1],L[2],2,4,flag,L2[2]); |
---|
1305 | New; |
---|
1306 | |
---|
1307 | } |
---|
1308 | ////////////////////////////////////////////////////////////// |
---|
1309 | |
---|
1310 | proc calculateI(list Coef,list expJ,number c,int n,int Y,intvec a,number oldordI,list oldD) |
---|
1311 | "USAGE: calculateI(Coef,expJ,c,n,Y,a,b,D); |
---|
1312 | Coef, expJ, D lists, c, b numbers, n,Y integers, a intvec |
---|
1313 | RETURN: ideal I, non monomial part of J |
---|
1314 | EXAMPLE: example calculateI; shows an example |
---|
1315 | " |
---|
1316 | { |
---|
1317 | int i,cont1,b,j; |
---|
1318 | number EordI,aux; |
---|
1319 | list D,L,expI; |
---|
1320 | list auxdiv,Dstar,aux1,rs; |
---|
1321 | |
---|
1322 | // WE NEED THE MONOMIAL PART, BUT ONLY IN DIMENSION n |
---|
1323 | |
---|
1324 | auxdiv=list0(n); |
---|
1325 | auxdiv[Y]=oldordI-c; |
---|
1326 | Dstar[1]=oldD[1]; |
---|
1327 | |
---|
1328 | b=size(a); |
---|
1329 | for (i=1;i<=n;i++) {for (j=1;j<=b;j++) {if (a[j]==i) {aux=aux+oldD[1][i];}}} |
---|
1330 | Dstar[1][Y]=aux; |
---|
1331 | aux=0; |
---|
1332 | |
---|
1333 | D[1]=sumlist(Dstar[1],auxdiv); |
---|
1334 | |
---|
1335 | cont1=0; |
---|
1336 | for (i=1;i<=n;i++) {if (D[1][i]==0) {cont1=cont1+1;}} // if it fails write listO(n)[i] |
---|
1337 | |
---|
1338 | if (cont1==n) {expI=expJ;} |
---|
1339 | else { |
---|
1340 | for (i=1;i<=size(expJ);i++) |
---|
1341 | {rs[i]=size(Coef[i]); |
---|
1342 | if (rs[i]==2){ aux1=list(); |
---|
1343 | aux1[1]=reslist(expJ[i][1],D[1]); |
---|
1344 | aux1[2]=reslist(expJ[i][2],D[1]); |
---|
1345 | expI[i]=aux1;} // binomial |
---|
1346 | else {aux1=list(); |
---|
1347 | aux1[1]=reslist(expJ[i][1],D[1]); |
---|
1348 | expI[i]=aux1;}} // monomial |
---|
1349 | } |
---|
1350 | |
---|
1351 | return(expI); |
---|
1352 | } |
---|
1353 | example |
---|
1354 | {"EXAMPLE:"; echo = 2; |
---|
1355 | ring r = 0,(x(1..3)),dp; |
---|
1356 | list flag=identifyvar(); |
---|
1357 | ideal J=x(1)^4*x(2)^2, x(3)^3; |
---|
1358 | list Lmb=1,list(0,0,0),list(0,0,0),list(3),iniD(3),iniD(3),list(0,0,0),-1; |
---|
1359 | list L=data(J,2,3); |
---|
1360 | list LL=determinecenter(L[1],L[2],3,3,0,0,Lmb,flag,0,-1); // Calculate the center |
---|
1361 | module auxpath=[0,-1]; |
---|
1362 | list infochart=0,0,0,L[2],L[1],flag,0,list(0,0,0),auxpath,list(),list(); |
---|
1363 | list L3=Blowupcenter(LL[1],1,1,infochart,3,3,0); // blowing-up and looking to the x(3) chart |
---|
1364 | calculateI(L3[2][1][5],L3[2][1][4],3,3,3,L3[2][1][3],3,iniD(3)); // (I_3) |
---|
1365 | // looking to the x(1) chart |
---|
1366 | calculateI(L3[2][2][5],L3[2][2][4],3,3,1,L3[2][2][3],3,iniD(3)); // (I_3) |
---|
1367 | } |
---|
1368 | ////////////////////////////////////////////////////////////////////////////////////// |
---|
1369 | // // |
---|
1370 | // E-RESOLUTION: Eresol(J) subroutine computing the E-resolution of J, char 0 // |
---|
1371 | // // |
---|
1372 | ////////////////////////////////////////////////////////////////////////////////////// |
---|
1373 | |
---|
1374 | proc Eresol(ideal J) |
---|
1375 | "USAGE: Eresol(J); J ideal |
---|
1376 | RETURN: The E-resolution of singularities of J in terms of the affine charts, see example |
---|
1377 | EXAMPLE: example Eresol; shows an example |
---|
1378 | " |
---|
1379 | {int i,n,k,idchart,nchart,parent,Y,oldid,tnum,s,cont,control,control2,control3,cont2,val,rs2,l,cont3,tip; |
---|
1380 | intvec a,Hhist; |
---|
1381 | number c,EordJ,EordI,oldordI; |
---|
1382 | list L,LL,oldD,t,auxL,finalchart,chart,auxchart,newL,auxp,auxfchart,L2; |
---|
1383 | list Coef,expJ,expI,sons,oldOlist,oldC,oldt,oldH,allH,auxordJ,auxordI,auxmb,mobile,invariant; |
---|
1384 | list step,nsons,auxinv,extraL,totalinv,auxsum; |
---|
1385 | string empstring; |
---|
1386 | module auxpath; |
---|
1387 | |
---|
1388 | // ADDED LATER |
---|
1389 | |
---|
1390 | list flag,newflag,blwhist,hipercoef,hiperexp,hipercoefson,hiperexpson; |
---|
1391 | intvec infobo7; |
---|
1392 | |
---|
1393 | export finalchart; |
---|
1394 | // export nsons; |
---|
1395 | // export tnum; |
---|
1396 | // export nchart; |
---|
1397 | // export step; |
---|
1398 | export invariant; |
---|
1399 | export auxinv; |
---|
1400 | export mobile; |
---|
1401 | |
---|
1402 | n=nvars(basering); |
---|
1403 | flag=identifyvar(); |
---|
1404 | |
---|
1405 | k=size(J); |
---|
1406 | // Checking input data |
---|
1407 | if (inidata(J,k)==0){return("This library only works for binomial ideals.");} |
---|
1408 | |
---|
1409 | idchart=1; |
---|
1410 | nchart=1; |
---|
1411 | parent=0; |
---|
1412 | step=0; |
---|
1413 | control=0; |
---|
1414 | control2=0; |
---|
1415 | control3=0; |
---|
1416 | |
---|
1417 | // Translate the input ideal to a list |
---|
1418 | auxL=data(J,k,n); // data gives (Coef,Exp) |
---|
1419 | |
---|
1420 | // THEREAFTER WE WORK ALL THE TIME WITH LISTS |
---|
1421 | |
---|
1422 | L=maxEord(auxL[1],auxL[2],k,n,flag); // gives (max E-ord, sums) |
---|
1423 | EordJ=L[1]; |
---|
1424 | |
---|
1425 | // before the first blow up I=J |
---|
1426 | EordI=EordJ; |
---|
1427 | |
---|
1428 | // main loop AT EACH CHART WE MUST INICIALIZATE ALL THE VALUES AND |
---|
1429 | // CONSTRUCT THE FIRST CHART chart[1] BEFORE THE LOOP |
---|
1430 | |
---|
1431 | // at the first step, before the blow up, there are no exceptional divisors, Y=0 |
---|
1432 | Y=0; |
---|
1433 | expJ=auxL[2]; |
---|
1434 | Coef=auxL[1]; |
---|
1435 | Hhist=0; |
---|
1436 | blwhist=list0(n); |
---|
1437 | auxpath=[0,-1]; |
---|
1438 | hipercoef=list(); // this is for the first chart |
---|
1439 | hiperexp=list(); |
---|
1440 | auxp=parent,Y,a,expJ,Coef,flag,Hhist,blwhist,auxpath,hipercoef,hiperexp; |
---|
1441 | chart[1]=auxp; // information of the first chart |
---|
1442 | |
---|
1443 | tip=1; |
---|
1444 | oldOlist=list0(n); |
---|
1445 | oldC=list0(n); |
---|
1446 | oldC[1]=EordJ; // non necessary here |
---|
1447 | c=EordJ; // the value c is given by the previous step |
---|
1448 | oldt=list0(n); |
---|
1449 | oldD=iniD(n); |
---|
1450 | oldH=iniD(n); |
---|
1451 | allH=list0(n); |
---|
1452 | |
---|
1453 | for (i=1;i<=n;i++){infobo7[i]=-1;} |
---|
1454 | |
---|
1455 | auxmb=tip,oldOlist,oldC,oldt,oldD,oldH,allH,infobo7; |
---|
1456 | mobile[1]=auxmb; // mobile corresponding to the first chart |
---|
1457 | auxinv[1]=list(0); |
---|
1458 | |
---|
1459 | // NOTE: oldC[1] is the value c to classify the chart in one of the next cases |
---|
1460 | |
---|
1461 | // HERE BEGIN THE LOOP |
---|
1462 | |
---|
1463 | while (idchart<=nchart) // WE PROCEED WHILE THERE EXIST UNSOLVED CHARTS |
---|
1464 | { |
---|
1465 | if (idchart!=1) // WE ARE NOT IN THE FIRST CHART, INICIALIZATE ALL THE VALUES |
---|
1466 | { |
---|
1467 | |
---|
1468 | parent=chart[idchart][1]; |
---|
1469 | Y=chart[idchart][2]; |
---|
1470 | a=chart[idchart][3]; |
---|
1471 | expJ=chart[idchart][4]; |
---|
1472 | Coef=chart[idchart][5]; |
---|
1473 | flag=chart[idchart][6]; |
---|
1474 | Hhist=chart[idchart][7]; // it is not necessary for the computations |
---|
1475 | blwhist=chart[idchart][8]; |
---|
1476 | auxpath=chart[idchart][9]; |
---|
1477 | hipercoef=chart[idchart][10]; |
---|
1478 | hiperexp=chart[idchart][11]; |
---|
1479 | |
---|
1480 | k=size(Coef); // IT IS NECESSARY TO COMPUTE IT BECAUSE IT DECREASES IF THERE ARE HYPERBOLIC EQS |
---|
1481 | |
---|
1482 | auxordJ=maxEord(Coef,expJ,k,n,flag); |
---|
1483 | EordJ=auxordJ[1]; |
---|
1484 | |
---|
1485 | if (control==0){c=mobile[parent+1][3][1];} // we keep c from the last step |
---|
1486 | else {c=EordJ; control=0; } // we reset the value of c |
---|
1487 | |
---|
1488 | if (control2==1){c=EordJ; control2=0; control3=1;} // we reset the value of c |
---|
1489 | |
---|
1490 | // NOTE: oldC[1] is the value c to classify the chart in one of the next cases |
---|
1491 | |
---|
1492 | } |
---|
1493 | |
---|
1494 | // The E-order must be computed here |
---|
1495 | |
---|
1496 | oldid=idchart; |
---|
1497 | |
---|
1498 | if (EordJ<0) {print("ERROR in J in chart"); print(idchart); ~; break;} |
---|
1499 | |
---|
1500 | |
---|
1501 | //------------------------------------------------------------- |
---|
1502 | // CASE J=1, if we reset c, can happen Eord=c=0 |
---|
1503 | |
---|
1504 | // or if there are hyperbolic equations at the beginning!!! A?ADIR!!!! |
---|
1505 | |
---|
1506 | // if (EordJ==0){auxfchart[1]=chart[idchart]; // WE HAVE FINISHED |
---|
1507 | // finalchart=finalchart+auxfchart; |
---|
1508 | // empstring="#"; print("reset c and Eord=c=0"); print(idchart); |
---|
1509 | // invariant[idchart]=empstring; |
---|
1510 | // auxinv[idchart]=list(0); |
---|
1511 | // nsons[idchart]=0; |
---|
1512 | // idchart=idchart+1;} |
---|
1513 | |
---|
1514 | |
---|
1515 | //---------------------------------------------------------------------- |
---|
1516 | if (EordJ>=c and EordJ!=0) // subroutine: E-RESOLUTION OF PAIRS |
---|
1517 | { |
---|
1518 | if (parent>0) |
---|
1519 | { LL=determinecenter(Coef,expJ,c,n,Y,a,mobile[parent+1],flag,control3,chart[parent][7]); } |
---|
1520 | else { LL=determinecenter(Coef,expJ,c,n,Y,a,mobile[parent+1],flag,control3,Hhist); } |
---|
1521 | |
---|
1522 | // determinecenter gives (center,auxJ,Olist,C,t,D,H,allH,auxinvlist,infobo7) |
---|
1523 | |
---|
1524 | // save current values, before the blow up |
---|
1525 | oldOlist=LL[3]; |
---|
1526 | tip=computemcm(oldOlist); |
---|
1527 | oldC=LL[4]; |
---|
1528 | oldt=LL[5]; |
---|
1529 | oldD=LL[6]; |
---|
1530 | oldH=LL[7]; |
---|
1531 | allH=LL[8]; |
---|
1532 | auxinv[idchart]=LL[9]; |
---|
1533 | infobo7=LL[10]; |
---|
1534 | |
---|
1535 | auxmb=tip,oldOlist,oldC,oldt,oldD,oldH,allH,infobo7; |
---|
1536 | mobile[idchart+1]=auxmb; |
---|
1537 | invariant[idchart]=oldt; |
---|
1538 | |
---|
1539 | newL=Blowupcenter(LL[1],idchart,nchart,chart[idchart],c,n,step[idchart]); |
---|
1540 | |
---|
1541 | // Blowupcenter gives (info,auxchart,nchart,auxstep,num) |
---|
1542 | |
---|
1543 | // IMPORTANT: ADD THE NEW CHARTS AFTER EACH BLOW UP, IN ORDER TO KEEP THEM CORRECTLY |
---|
1544 | |
---|
1545 | step=step+newL[4]; |
---|
1546 | nsons[idchart]=newL[5]; |
---|
1547 | |
---|
1548 | chart=chart+newL[2]; |
---|
1549 | finalchart=finalchart+newL[1]; |
---|
1550 | |
---|
1551 | // new input for the loop |
---|
1552 | |
---|
1553 | idchart=idchart+1; |
---|
1554 | nchart=newL[3]; |
---|
1555 | |
---|
1556 | control3=0; |
---|
1557 | |
---|
1558 | } // END OF CASE EordJ>=c |
---|
1559 | //--------------------------------------------------------------------- |
---|
1560 | |
---|
1561 | else{ |
---|
1562 | |
---|
1563 | // compute EordI=max E-order(I) |
---|
1564 | |
---|
1565 | expI=calculateI(Coef,expJ,c,n,Y,a,mobile[parent+1][2][1],mobile[parent+1][5]); |
---|
1566 | k=size(expJ); // probably non necessary |
---|
1567 | auxordI=maxEord(Coef,expI,k,n,flag); |
---|
1568 | EordI=auxordI[1]; |
---|
1569 | auxsum=auxordI[2]; |
---|
1570 | |
---|
1571 | // CASE EordI>0 DROP c AND CONTINUE |
---|
1572 | |
---|
1573 | if (EordI>0){idchart=idchart; // keep the chart and back to the main loop while, dropping the value of c |
---|
1574 | control=1;} |
---|
1575 | else{ // EordI=0, so check if I=1 or not |
---|
1576 | |
---|
1577 | cont2=0; // If cont2=val then all the entries of expI are zero!! |
---|
1578 | val=0; |
---|
1579 | |
---|
1580 | for (i=1;i<=k;i++) {rs2=size(Coef[i]); |
---|
1581 | if (rs2==1){if (auxsum[i][1]==0){cont2=val; break;} // THERE EXIST A MONOMIAL WITH ONLY UNITS |
---|
1582 | |
---|
1583 | val=val+n; // monomials |
---|
1584 | for (l=1;l<=n; l++) {if (expI[i][1][l]==0) {cont2=cont2+1;}} |
---|
1585 | } |
---|
1586 | else{val=val+(2*n); // binomials |
---|
1587 | for (l=1;l<=n; l++) {if (expI[i][1][l]==0) {cont2=cont2+1;} |
---|
1588 | if (expI[i][2][l]==0) {cont2=cont2+1;}} |
---|
1589 | } |
---|
1590 | } |
---|
1591 | |
---|
1592 | |
---|
1593 | // CASE EordI==0 AND I=1 THIS CHART IS DONE, FINISH |
---|
1594 | |
---|
1595 | // NOTE: THIS CASE IS NOT MONOMIAL BECAUSE E-Sing(J,c) is empty |
---|
1596 | |
---|
1597 | if (cont2==val){auxfchart[1]=chart[idchart]; |
---|
1598 | finalchart=finalchart+auxfchart; |
---|
1599 | empstring="#"; |
---|
1600 | invariant[idchart]=empstring; |
---|
1601 | auxinv[idchart]=list(0); |
---|
1602 | nsons[idchart]=0; |
---|
1603 | |
---|
1604 | // information for the mobile |
---|
1605 | tip=1; |
---|
1606 | oldOlist=list(0); |
---|
1607 | oldC=list(0); |
---|
1608 | oldt=list(0); |
---|
1609 | oldD=list(0); |
---|
1610 | oldH=list(0); |
---|
1611 | allH=list(0); // the value of the parent + the new one |
---|
1612 | infobo7=-1; |
---|
1613 | |
---|
1614 | auxmb=tip,oldOlist,oldC,oldt,oldD,oldH,allH,infobo7; |
---|
1615 | mobile[idchart+1]=auxmb; |
---|
1616 | |
---|
1617 | idchart=idchart+1;} |
---|
1618 | |
---|
1619 | else{ // CASE EordI==0 AND I!=1 --> HYPERBOLIC EQUATIONS |
---|
1620 | |
---|
1621 | // COMPUTE THE IDEAL OF NON HYPERBOLIC ELEMENTS |
---|
1622 | |
---|
1623 | extraL=Nonhyp(Coef,expI,k,n,flag,auxordI[2]); // gives (newcoef,newI,hypcoef,genhyp,flaglist) |
---|
1624 | |
---|
1625 | // CHECK IF ALL THE VARIABLES ARE ALREADY INVERTIBLE |
---|
1626 | |
---|
1627 | newflag=extraL[5]; |
---|
1628 | chart[idchart][6]=extraL[5]; // update the status of variables |
---|
1629 | |
---|
1630 | cont3=0; |
---|
1631 | for (i=1;i<=n;i++){if (newflag[i]==1){cont3=cont3+1;}} |
---|
1632 | |
---|
1633 | if (cont3==n){ // ALL THE VARIABLES ARE INVERTIBLE |
---|
1634 | auxfchart[1]=chart[idchart]; |
---|
1635 | finalchart=finalchart+auxfchart; |
---|
1636 | empstring="@"; |
---|
1637 | invariant[idchart]=empstring; |
---|
1638 | auxinv[idchart]=list(0); |
---|
1639 | nsons[idchart]=0; |
---|
1640 | |
---|
1641 | // information for the mobile |
---|
1642 | tip=1; |
---|
1643 | oldOlist=list(0); |
---|
1644 | oldC=list(0); |
---|
1645 | oldt=list(0); |
---|
1646 | oldD=list(0); |
---|
1647 | oldH=list(0); |
---|
1648 | allH=list(0); |
---|
1649 | infobo7=-1; |
---|
1650 | |
---|
1651 | auxmb=tip,oldOlist,oldC,oldt,oldD,oldH,allH,infobo7; |
---|
1652 | mobile[idchart+1]=auxmb; |
---|
1653 | |
---|
1654 | idchart=idchart+1;} |
---|
1655 | else{ // OTHERWISE, CONTINUE CHEKING IF newI=0 or not |
---|
1656 | |
---|
1657 | Coef=extraL[1]; |
---|
1658 | expI=extraL[2]; |
---|
1659 | |
---|
1660 | hipercoefson=extraL[3]; // Information about hyperbolic generators |
---|
1661 | hiperexpson=extraL[4]; |
---|
1662 | |
---|
1663 | k=size(expI); |
---|
1664 | |
---|
1665 | if (k==0){auxfchart[1]=chart[idchart]; // WE HAVE FINISHED |
---|
1666 | finalchart=finalchart+auxfchart; |
---|
1667 | empstring="#"; // no more non-hyperbolic generators in this chart |
---|
1668 | invariant[idchart]=empstring; |
---|
1669 | auxinv[idchart]=list(0); |
---|
1670 | nsons[idchart]=0; |
---|
1671 | |
---|
1672 | // information for the mobile |
---|
1673 | tip=1; |
---|
1674 | oldOlist=list(0); |
---|
1675 | oldC=list(0); |
---|
1676 | oldt=list(0); |
---|
1677 | oldD=list(0); |
---|
1678 | oldH=list(0); |
---|
1679 | allH=list(0); |
---|
1680 | infobo7=-1; |
---|
1681 | |
---|
1682 | auxmb=tip,oldOlist,oldC,oldt,oldD,oldH,allH,infobo7; |
---|
1683 | mobile[idchart+1]=auxmb; |
---|
1684 | |
---|
1685 | idchart=idchart+1;} |
---|
1686 | |
---|
1687 | else{ // CONTINUE WITH THE IDEAL OF NON HYPERBOLIC EQS |
---|
1688 | |
---|
1689 | chart[idchart][4]=expI; // new input ideal and coefficients |
---|
1690 | chart[idchart][5]=Coef; |
---|
1691 | chart[idchart][10]=hipercoef+hipercoefson; |
---|
1692 | chart[idchart][11]=hiperexp+hiperexpson; |
---|
1693 | |
---|
1694 | idchart=idchart; |
---|
1695 | control2=1; // it is necessary to reset the value of c |
---|
1696 | control3=1; // and the previous exceptional divisors |
---|
1697 | } |
---|
1698 | |
---|
1699 | // PROBABLY IT IS NEC MORE INFORMATION !!! |
---|
1700 | |
---|
1701 | } // closing else otherwise |
---|
1702 | |
---|
1703 | } // closing else case I!=1 |
---|
1704 | |
---|
1705 | } // closing else for EordI=0 |
---|
1706 | |
---|
1707 | if (EordI<0) {print("ERROR in chart"); print(idchart); ~; break;} |
---|
1708 | |
---|
1709 | //----------------------- guardar de momento-------- |
---|
1710 | // if (EordI==0) {auxfchart[1]=chart[idchart]; |
---|
1711 | // finalchart=finalchart+auxfchart; |
---|
1712 | // L2=Gamma(expJ,c,n); // HAY QUE APLICARLO AL M NO AL J |
---|
1713 | // invariant[idchart]=L2[2]; |
---|
1714 | // auxinv[idchart]=list(0); |
---|
1715 | // nsons[idchart]=0; |
---|
1716 | // idchart=idchart+1;} |
---|
1717 | //------------------------------------------------ |
---|
1718 | |
---|
1719 | |
---|
1720 | } // END ELSE |
---|
1721 | //--------------------------------------------------- |
---|
1722 | |
---|
1723 | } // END LOOP WHILE |
---|
1724 | |
---|
1725 | tnum=step[nchart]; |
---|
1726 | |
---|
1727 | totalinv=resfunction(invariant,auxinv,nchart,n); |
---|
1728 | |
---|
1729 | return(chart,finalchart,invariant,nchart,step,nsons,auxinv,mobile,totalinv); |
---|
1730 | } |
---|
1731 | example |
---|
1732 | {"EXAMPLE:"; echo = 2; |
---|
1733 | ring r = 0,(x(1..2)),dp; |
---|
1734 | ideal J=x(1)^2-x(2)^3; |
---|
1735 | list L=Eresol(J); |
---|
1736 | "Please press return after each break point to see the next element of the output list"; |
---|
1737 | L[1][1]; // information of the first chart, L[1] list of charts |
---|
1738 | ~; |
---|
1739 | L[2]; // list of charts with information about sons |
---|
1740 | ~; |
---|
1741 | L[3]; // invariant, "#" means solved chart |
---|
1742 | ~; |
---|
1743 | L[4]; // number of charts, 7 in this example |
---|
1744 | ~; |
---|
1745 | L[5]; // height corresponding to each chart |
---|
1746 | ~; |
---|
1747 | L[6]; // number of sons |
---|
1748 | ~; |
---|
1749 | L[7]; // auxiliary invariant |
---|
1750 | ~; |
---|
1751 | L[8]; // H exceptional divisors and more information |
---|
1752 | ~; |
---|
1753 | L[9]; // complete resolution function |
---|
1754 | |
---|
1755 | "Second example, write L[i] to see the i-th component of the list"; |
---|
1756 | ring r = 0,(x(1..3)),dp; |
---|
1757 | ideal J=x(1)^2*x(2),x(3)^3; // SOLVED! |
---|
1758 | list L=Eresol(J); |
---|
1759 | L[4]; // 16 charts |
---|
1760 | L[9]; // complete resolution function |
---|
1761 | ~; |
---|
1762 | |
---|
1763 | "Third example, write L[i] to see the i-th component of the list"; |
---|
1764 | ring r = 0,(x(1..2)),dp; |
---|
1765 | ideal J=x(1)^3-x(1)*x(2)^3; |
---|
1766 | list L=Eresol(J); |
---|
1767 | L[4]; // 8 charts, rational exponents |
---|
1768 | L[9]; // complete resolution function |
---|
1769 | ~; |
---|
1770 | } |
---|
1771 | |
---|
1772 | ////////////////////////////////////////////////////////////////////////////////////// |
---|
1773 | |
---|
1774 | proc resfunction(list invariant, list auxinv, int nchart,int n) |
---|
1775 | "USAGE: resfunction(invariant,auxinv,nchart,n); |
---|
1776 | invariant, auxinv lists, nchart, n integers |
---|
1777 | COMPUTE: Patch the resolution function |
---|
1778 | RETURN: The complete resolution function |
---|
1779 | EXAMPLE: example resfunction; shows an example |
---|
1780 | " |
---|
1781 | { |
---|
1782 | int i,j,l,k; |
---|
1783 | list patchfun,aux; |
---|
1784 | |
---|
1785 | for (i=1;i<=nchart;i++){patchfun[i]=invariant[i];} |
---|
1786 | |
---|
1787 | for (i=1;i<=nchart;i++){if (auxinv[i][1]!=0 and size(auxinv[i])==3){l=size(invariant[i]); |
---|
1788 | for (j=1;j<=l;j++){ |
---|
1789 | if (invariant[i][j]==0){aux=auxinv[i]; |
---|
1790 | patchfun[i][j]=aux; |
---|
1791 | if (l<n){for (k=j+1;k<=n;k++){patchfun[i][k]="*";}}}} |
---|
1792 | |
---|
1793 | } |
---|
1794 | else{ |
---|
1795 | if (auxinv[i][1]==1 and size(auxinv[i])==1){l=size(invariant[i]); |
---|
1796 | if (l<n){for (k=l+1;k<=n;k++){patchfun[i][k]="*";}} |
---|
1797 | } |
---|
1798 | } |
---|
1799 | } |
---|
1800 | |
---|
1801 | return(patchfun); |
---|
1802 | } |
---|
1803 | example |
---|
1804 | {"EXAMPLE:"; echo = 2; |
---|
1805 | ring r = 0,(x(1..2)),dp; |
---|
1806 | ideal J=x(1)^2-x(2)^3; |
---|
1807 | list L=Eresol(J); |
---|
1808 | L[3]; // incomplete resolution function |
---|
1809 | resfunction(L[3],L[7],7,2); // complete resolution function |
---|
1810 | } |
---|
1811 | ////////////////////////////////////////////////////////////////////////////////////// |
---|
1812 | // // |
---|
1813 | // MAIN PROCEDURE // |
---|
1814 | // // |
---|
1815 | ////////////////////////////////////////////////////////////////////////////////////// |
---|
1816 | |
---|
1817 | proc BINresol(ideal J) |
---|
1818 | "USAGE: BINresol(J); J ideal |
---|
1819 | RETURN: E-resolution of singularities of a binomial ideal J in terms of the affine charts, see example |
---|
1820 | EXAMPLE: example BINresol; shows an example |
---|
1821 | " |
---|
1822 | { |
---|
1823 | |
---|
1824 | int p,n; |
---|
1825 | |
---|
1826 | p=char(basering); |
---|
1827 | n=nvars(basering); // already computed in Eresol, it can be improved |
---|
1828 | |
---|
1829 | def rr=basering; |
---|
1830 | |
---|
1831 | // INTERNAL CHANGE: changing the name of the variables, only if it is necessary |
---|
1832 | |
---|
1833 | list Mout=changeoriginalvar(); |
---|
1834 | |
---|
1835 | if (Mout[2]==1){ |
---|
1836 | def r=Mout[1]; |
---|
1837 | setring r; |
---|
1838 | ideal chy=maxideal(1); |
---|
1839 | map frr=rr,chy; |
---|
1840 | ideal J=frr(J); |
---|
1841 | } |
---|
1842 | // else{def r=basering;} // CHECK THAT IS NECESSARY !!! |
---|
1843 | |
---|
1844 | // IF WE ARE IN POSTIVE CHAR |
---|
1845 | |
---|
1846 | if (p>0){list Lring=ringlist(basering); |
---|
1847 | Lring[1]=0; |
---|
1848 | // def r=basering; |
---|
1849 | def Rnew=ring(Lring); |
---|
1850 | setring Rnew; |
---|
1851 | ideal chy=maxideal(1); |
---|
1852 | map fRnew=r,chy; |
---|
1853 | ideal J=fRnew(J); |
---|
1854 | |
---|
1855 | // E-RESOLUTION, Computations in char 0 |
---|
1856 | |
---|
1857 | list L=Eresol(J); |
---|
1858 | |
---|
1859 | // STEP 2: WRITE THE LOCALLY MONOMIAL IDEAL AS A MONOMIAL IDEAL |
---|
1860 | |
---|
1861 | // not implemented yet, CHAR p !!!! |
---|
1862 | |
---|
1863 | // STEP 3: DO THE E-RESOLUTION AGAIN (char 0 again) |
---|
1864 | |
---|
1865 | |
---|
1866 | // generating output in char p |
---|
1867 | |
---|
1868 | int q=lcmofall(L[4],L[8]); // lcm of the denominators |
---|
1869 | |
---|
1870 | list B=genoutput(L[1],L[8],L[4],L[6],n,q,p); // generate output needed for visualization |
---|
1871 | |
---|
1872 | |
---|
1873 | // setring r; // Back to the basering |
---|
1874 | // ideal chy=maxideal(1); |
---|
1875 | // map fr=Rnew,chy; |
---|
1876 | // list L=fr(L); |
---|
1877 | // list B=fr(B); |
---|
1878 | |
---|
1879 | } |
---|
1880 | |
---|
1881 | else{ |
---|
1882 | |
---|
1883 | // E-RESOLUTION |
---|
1884 | |
---|
1885 | list L=Eresol(J); |
---|
1886 | |
---|
1887 | // STEP 2: WRITE THE LOCALLY MONOMIAL IDEAL AS A MONOMIAL IDEAL |
---|
1888 | |
---|
1889 | // not implemented yet |
---|
1890 | |
---|
1891 | // STEP 3: DO THE E-RESOLUTION AGAIN |
---|
1892 | |
---|
1893 | |
---|
1894 | // generating output |
---|
1895 | |
---|
1896 | int q=lcmofall(L[4],L[8]); |
---|
1897 | |
---|
1898 | list B=genoutput(L[1],L[8],L[4],L[6],n,q,p); |
---|
1899 | |
---|
1900 | } |
---|
1901 | |
---|
1902 | return(B); |
---|
1903 | } |
---|
1904 | example |
---|
1905 | {"EXAMPLE:"; echo = 2; |
---|
1906 | ring r = 0,(x(1..2)),dp; |
---|
1907 | ideal J=x(1)^2-x(2)^3; |
---|
1908 | list B=BINresol(J); |
---|
1909 | B[1]; // list of final charts |
---|
1910 | B[2]; // list of all charts |
---|
1911 | |
---|
1912 | ring r = 2,(x(1..3)),dp; |
---|
1913 | ideal J=x(1)^2-x(2)^2*x(3)^2; |
---|
1914 | list B=BINresol(J); |
---|
1915 | B[2]; // list of all charts |
---|
1916 | } |
---|
1917 | /////////////////////////////////////////////////////// |
---|
1918 | |
---|
1919 | proc Maxord(list L,int n) |
---|
1920 | "USAGE: Maxord(L,n); L list, n integer |
---|
1921 | COMPUTE: Find the maximal entry of a list, input is a list defining a monomial |
---|
1922 | RETURN: maximum entry of a list and its position |
---|
1923 | EXAMPLE: example Maxord; shows an example |
---|
1924 | " |
---|
1925 | {int i,can; |
---|
1926 | number canmax; |
---|
1927 | list aux; |
---|
1928 | |
---|
1929 | canmax=1; |
---|
1930 | can=1; |
---|
1931 | for (i=1;i<=n;i++) |
---|
1932 | { if (L[i]>=canmax and i>=can) |
---|
1933 | {canmax=L[i]; can=i;}} |
---|
1934 | |
---|
1935 | return(canmax,can); |
---|
1936 | } |
---|
1937 | example |
---|
1938 | {"EXAMPLE:"; echo = 2; |
---|
1939 | ring r = 0,(x(1..3)),dp; |
---|
1940 | ideal J=x(1)^2*x(2)*x(3)^5; |
---|
1941 | list L=data(J,1,3); |
---|
1942 | L[2]; // list of exponents |
---|
1943 | Maxord(L[2][1][1],3); |
---|
1944 | } |
---|
1945 | /////////////////////////////////////////////////////// |
---|
1946 | |
---|
1947 | proc Gamma(list expM,number c,int n) |
---|
1948 | "USAGE: Gamma(L,c,n); L list, c number, n integer |
---|
1949 | COMPUTE: The Gamma function, resolution function corresponding to the monomial case |
---|
1950 | RETURN: lists of maximum exponents in L, value of Gamma function, center of blow up |
---|
1951 | EXAMPLE: example Gamma; shows an example |
---|
1952 | " |
---|
1953 | {int i,j,k,l,cont,can; |
---|
1954 | intvec upla; |
---|
1955 | number canmax; |
---|
1956 | list expM2,gamma,L,aux,maxlist,center,aux2; |
---|
1957 | |
---|
1958 | i=1; |
---|
1959 | cont=0; |
---|
1960 | expM2=expM; |
---|
1961 | |
---|
1962 | while (cont==0 and i<=n) |
---|
1963 | { |
---|
1964 | |
---|
1965 | L=Maxord(expM2,n); |
---|
1966 | aux=L[1]; |
---|
1967 | maxlist=maxlist + aux; |
---|
1968 | can=L[2]; |
---|
1969 | |
---|
1970 | if (i==1) {upla=can; center=can;} |
---|
1971 | else {upla=upla,can; aux2=can; center=center+aux2;} |
---|
1972 | |
---|
1973 | canmax=sum(maxlist); |
---|
1974 | if (canmax>=c) |
---|
1975 | {gamma[1]=-i; gamma[2]=canmax/c; gamma[3]=upla; cont=1;} |
---|
1976 | else {expM2[can]=0;} |
---|
1977 | i=i+1; |
---|
1978 | } |
---|
1979 | return(maxlist,gamma,center); |
---|
1980 | } |
---|
1981 | example |
---|
1982 | {"EXAMPLE:"; echo = 2; |
---|
1983 | ring r = 0,(x(1..5)),dp; |
---|
1984 | ideal J=x(1)^2*x(2)*x(3)^5*x(4)^2*x(5)^3; |
---|
1985 | list L=data(J,1,5); |
---|
1986 | list G=Gamma(L[2][1][1],9,5); // critical value c=9 |
---|
1987 | G[1]; // maximum exponents in the ideal |
---|
1988 | G[2]; // maximal value of Gamma function |
---|
1989 | G[3]; // center given by Gamma |
---|
1990 | } |
---|
1991 | /////////////////////////////////////////////////////// |
---|
1992 | |
---|
1993 | proc convertdata(list C,list L, int n, list flaglist) |
---|
1994 | "USAGE: convertdata(C,L,n,flaglist); |
---|
1995 | C, L, flaglist lists, n integer |
---|
1996 | COMPUTE: Compute the ideal corresponding to the given lists C,L |
---|
1997 | RETURN: an ideal whose coefficients are given by C, exponents given by L |
---|
1998 | EXAMPLE: example convertdata; shows an example |
---|
1999 | " |
---|
2000 | {int i,j,k,a,b,lon; |
---|
2001 | poly aux,aux1,aux2,aux3,f; |
---|
2002 | ideal J; |
---|
2003 | |
---|
2004 | aux=poly(0); |
---|
2005 | aux1=poly(1); |
---|
2006 | aux2=poly(0); |
---|
2007 | aux3=poly(1); |
---|
2008 | |
---|
2009 | |
---|
2010 | k=size(L); |
---|
2011 | for (i=1;i<=k;i++){lon=size(C[i]); |
---|
2012 | if (lon==1){ // variables in the monomial |
---|
2013 | for (j=1;j<=n;j++){a=int(poly(L[i][1][j])); |
---|
2014 | if (a!=0){ |
---|
2015 | if (flaglist[j]==0){aux=poly(x(j)^a); |
---|
2016 | aux1=aux1*aux;} |
---|
2017 | else {aux=poly(y(j)^a); |
---|
2018 | aux1=aux1*aux;} |
---|
2019 | } |
---|
2020 | } |
---|
2021 | if (C[i][1]!=0){aux1=C[i][1]*aux1;} // we add the coefficient |
---|
2022 | else {aux1=0;} |
---|
2023 | |
---|
2024 | J[i]=aux1; |
---|
2025 | aux1=poly(1); |
---|
2026 | } |
---|
2027 | |
---|
2028 | else{ // variables in the binomial |
---|
2029 | |
---|
2030 | for (j=1;j<=n;j++){a=int(poly(L[i][1][j])); b=int(poly(L[i][2][j])); |
---|
2031 | |
---|
2032 | if (a!=0){ |
---|
2033 | if (flaglist[j]==0){aux=poly(x(j)^a); |
---|
2034 | aux1=aux1*aux;} |
---|
2035 | else {aux=poly(y(j)^a); |
---|
2036 | aux1=aux1*aux;} |
---|
2037 | } |
---|
2038 | |
---|
2039 | if (b!=0){ |
---|
2040 | if (flaglist[j]==0){aux2=poly(x(j)^b); |
---|
2041 | aux3=aux3*aux2;} |
---|
2042 | else {aux2=poly(y(j)^b); |
---|
2043 | aux3=aux3*aux2;} |
---|
2044 | } |
---|
2045 | } |
---|
2046 | // we add the coefficients |
---|
2047 | if (C[i][1]!=0){aux1=C[i][1]*aux1;} |
---|
2048 | else {aux1=0;} |
---|
2049 | if (C[i][2]!=0){aux3=C[i][2]*aux3;} |
---|
2050 | else {aux3=0;} |
---|
2051 | |
---|
2052 | f=aux1+aux3; |
---|
2053 | J[i]=f; |
---|
2054 | aux1=poly(1); |
---|
2055 | aux3=poly(1); |
---|
2056 | |
---|
2057 | } |
---|
2058 | } |
---|
2059 | return(J); |
---|
2060 | } |
---|
2061 | example |
---|
2062 | {"EXAMPLE:"; echo = 2; |
---|
2063 | ring r = 0,(x(1..4),y(5)),dp; |
---|
2064 | list M=identifyvar(); |
---|
2065 | ideal J=x(1)^2*y(5)^2-x(2)^2*x(3)^2,6*x(4)^2; |
---|
2066 | list L=data(J,2,5); |
---|
2067 | L[1]; // Coefficients |
---|
2068 | L[2]; // Exponents |
---|
2069 | ideal J2=convertdata(L[1],L[2],5,M); |
---|
2070 | J2; |
---|
2071 | } |
---|
2072 | |
---|
2073 | ///////////////////////////////////////////////////////////////////////////// |
---|
2074 | |
---|
2075 | proc lcmofall(int nchart,list mobile) |
---|
2076 | "USAGE: lcmofall(nchart,mobile); |
---|
2077 | nchart integer, mobile list of lists |
---|
2078 | COMPUTE: Compute the lcm of the denominators of the E-orders of all the charts |
---|
2079 | RETURN: an integer given the lcm |
---|
2080 | NOTE: CALL BEFORE salida |
---|
2081 | EXAMPLE: example lcmofall; shows an example |
---|
2082 | " |
---|
2083 | |
---|
2084 | { |
---|
2085 | int i,m,tip,mcmall; |
---|
2086 | intvec numall; |
---|
2087 | |
---|
2088 | for (i=2;i<=nchart+1;i++){ |
---|
2089 | tip=mobile[i][1]; |
---|
2090 | if (tip!=1){numall=numall,tip;} |
---|
2091 | } |
---|
2092 | m=size(numall); |
---|
2093 | |
---|
2094 | if (m==1){mcmall=1;} |
---|
2095 | else{ |
---|
2096 | if (numall[1]==0){numall=numall[2..m];} |
---|
2097 | mcmall=lcm(numall);} |
---|
2098 | |
---|
2099 | return(mcmall); |
---|
2100 | } |
---|
2101 | example |
---|
2102 | {"EXAMPLE:"; echo = 2; |
---|
2103 | ring r = 0,(x(1..2)),dp; |
---|
2104 | ideal J=x(1)^3-x(1)*x(2)^3; |
---|
2105 | list L=Eresol(J); |
---|
2106 | L[4]; // 8 charts, rational exponents |
---|
2107 | L[8][2][2]; // E-orders at the first chart |
---|
2108 | lcmofall(8,L[8]); |
---|
2109 | } |
---|
2110 | ///////////////////////////////////////////////////////////////////////////// |
---|
2111 | |
---|
2112 | proc salida(int idchart,list chart,list mobile,int numson,intvec previousa,int n,int q,int p) |
---|
2113 | "USAGE: salida(idchart,chart,mobile,numson,previousa,n,q,p); |
---|
2114 | idchart, numson, n, q, p integers, chart, mobile, lists, previousa intvec |
---|
2115 | COMPUTE: CONVERT THE OUTPUT OF A CHART IN A RING, WHERE DEFINE A BASIC OBJECT (BO) |
---|
2116 | RETURN: the ring corresponding to the chart |
---|
2117 | EXAMPLE: example salida; shows an example |
---|
2118 | " |
---|
2119 | { |
---|
2120 | int l,i,m,aux,parent,m4,j; |
---|
2121 | intvec Hhist,EOhist,aux7,aux9; |
---|
2122 | list expJ,Coef,BO,blwhist,Eolist,hipercoef,hiperexp; |
---|
2123 | list flag; |
---|
2124 | |
---|
2125 | // chart gives: parent,Y,a,expJ,Coef,flag,Hhist,blwhist,path,hipercoef,hiperexp |
---|
2126 | // mobile gives: tip,oldOlist,oldC,oldt,oldD,oldH,allH,infobo7; NOTE: Eolist=mobile[2]; |
---|
2127 | |
---|
2128 | // we need to define the suitable ring at this chart |
---|
2129 | |
---|
2130 | list Lring=ringlist(basering); |
---|
2131 | def RR2=basering; |
---|
2132 | |
---|
2133 | flag=chart[6]; |
---|
2134 | string newl; |
---|
2135 | |
---|
2136 | for (l=1;l<=n; l++){if (flag[l]==1){newl=string(l); |
---|
2137 | Lring[2][l]="y("+newl+")";} } |
---|
2138 | |
---|
2139 | |
---|
2140 | def RRnew=ring(Lring); |
---|
2141 | setring RRnew; |
---|
2142 | ideal chy=maxideal(1); |
---|
2143 | map fRnew=RR2,chy; |
---|
2144 | |
---|
2145 | list chart=fRnew(chart); |
---|
2146 | |
---|
2147 | list mobile2=fRnew(mobile); |
---|
2148 | |
---|
2149 | |
---|
2150 | flag=chart[6]; |
---|
2151 | |
---|
2152 | // we need to convert expJ and Coef to an ideal |
---|
2153 | |
---|
2154 | expJ=chart[4]; |
---|
2155 | Coef=chart[5]; |
---|
2156 | Hhist=chart[7]; |
---|
2157 | blwhist=chart[8]; |
---|
2158 | |
---|
2159 | // now the ideal will be correctly defined in the ring Rnew |
---|
2160 | |
---|
2161 | ideal J2=convertdata(Coef,expJ,n,flag); // Computations in RRnew |
---|
2162 | |
---|
2163 | //------------------------------------------------------------------------------ |
---|
2164 | // START TO CREATE THE BO corresponding to this chart |
---|
2165 | |
---|
2166 | BO=createBO(J2); |
---|
2167 | |
---|
2168 | // MODIFY BO WITH THE INFORMATION OF THE CHART |
---|
2169 | |
---|
2170 | // BO[1] an ideal, say W_i, defining the ambient space of the i-th chart of the blowing up |
---|
2171 | // If there are hyperbolic equations, we put them here |
---|
2172 | |
---|
2173 | hipercoef=chart[10]; |
---|
2174 | hiperexp=chart[11]; |
---|
2175 | |
---|
2176 | if (size(hipercoef)!=0){ |
---|
2177 | ideal ambJ=convertdata(hipercoef,hiperexp,n,flag); |
---|
2178 | BO[1]=ambJ; |
---|
2179 | } |
---|
2180 | |
---|
2181 | // BO[2] an ideal defining the controlled transform |
---|
2182 | |
---|
2183 | BO[2]=J2; |
---|
2184 | |
---|
2185 | // BO[3] intvec, tupla containing the maximal E-order of BO[2] |
---|
2186 | |
---|
2187 | if (numson==0){BO[3]=1;} // we write 1 if the chart is a final chart |
---|
2188 | else{ |
---|
2189 | Eolist=mobile2[2]; // otherwise, convert the list of E-orders in an intvec |
---|
2190 | m=size(Eolist); |
---|
2191 | aux=int(Eolist[1]*q); |
---|
2192 | EOhist=aux; |
---|
2193 | |
---|
2194 | if (m>1){for (i=2;i<=m;i++){aux=int(Eolist[i]*q); EOhist=EOhist,aux;}} |
---|
2195 | |
---|
2196 | BO[3]=EOhist; |
---|
2197 | } |
---|
2198 | |
---|
2199 | // BO[4] the list of exceptional divisors given by Hhist |
---|
2200 | |
---|
2201 | BO[4]=constructH(Hhist,n,flag); |
---|
2202 | |
---|
2203 | // BO[5] an ideal defining the map K[W] ----> K[Wi] given by blwhist |
---|
2204 | |
---|
2205 | BO[5]=constructblwup(blwhist,n,chy,flag); |
---|
2206 | |
---|
2207 | // BO[6] an intvec, BO[6][j]=1 indicates that <BO[4][j],BO[2]>=1, i.e. the |
---|
2208 | // strict transform does not meet the j-th exceptional divisor |
---|
2209 | |
---|
2210 | m4=size(BO[4]); |
---|
2211 | ideal auxydeal; |
---|
2212 | ideal Jint; |
---|
2213 | |
---|
2214 | for (j=1;j<=m4;j++){ |
---|
2215 | |
---|
2216 | auxydeal=BO[4][j]+J2; |
---|
2217 | Jint=std(auxydeal); |
---|
2218 | |
---|
2219 | if (size(Jint)==1 and Jint[1]==1){BO[6][j]=1;} |
---|
2220 | else{BO[6][j]=0;} |
---|
2221 | } |
---|
2222 | |
---|
2223 | // BO[7] intvec, the index of the first blown-up object in the resolution process |
---|
2224 | // leading to this object for which the value of b was BO[3] |
---|
2225 | // the subsequent ones are the indices for the Coeff-Objects |
---|
2226 | // of BO[2] used when determining the center |
---|
2227 | // index of last element of H^- in H |
---|
2228 | |
---|
2229 | |
---|
2230 | if (numson!=0){BO[7]=mobile2[8];} // it is always -1 at the final charts |
---|
2231 | |
---|
2232 | // BO[8] a matrix indicating that BO[4][i] meets BO[4][j] by BO[8][i,j]=1 for i < j |
---|
2233 | |
---|
2234 | if (m4>0){ |
---|
2235 | matrix aux8[m4][m4]; |
---|
2236 | |
---|
2237 | BO[8]=aux8; |
---|
2238 | |
---|
2239 | ideal auxydeal2; |
---|
2240 | ideal Jint2; |
---|
2241 | |
---|
2242 | for (i=1;i<=m4;i++){ |
---|
2243 | for (j=i+1;j<=m4;j++){ |
---|
2244 | auxydeal2=BO[4][i]+BO[4][j]; |
---|
2245 | Jint2=std(auxydeal2); |
---|
2246 | |
---|
2247 | if (size(Jint2)==1 and Jint2[1]==1){BO[8][i,j]=0;} |
---|
2248 | else{ for (l=1;l<j;l++){BO[8][l,j]=1;} } |
---|
2249 | } |
---|
2250 | |
---|
2251 | } |
---|
2252 | } |
---|
2253 | else{ matrix aux8[1][1]; |
---|
2254 | BO[8]=aux8;} |
---|
2255 | |
---|
2256 | |
---|
2257 | // BO[9] INTERNAL DATA, second component of Villamayor resolution function, |
---|
2258 | // only needed to use the visualization procedures |
---|
2259 | |
---|
2260 | int m3=size(BO[3]); |
---|
2261 | |
---|
2262 | if (m3==1){aux9=intvec(0);} |
---|
2263 | else{ aux9[1]=0; |
---|
2264 | for (i=2;i<=m3;i++){aux9=aux9,0;} |
---|
2265 | } |
---|
2266 | |
---|
2267 | BO[9]=aux9; |
---|
2268 | |
---|
2269 | //------------------------------------------------------------------------------ |
---|
2270 | |
---|
2271 | // START TO CREATE THE extra information corresponding to this chart |
---|
2272 | |
---|
2273 | /////////////// Short description of data in a chart /////////////////// |
---|
2274 | // All chart data is stored in an object of type ring, the following |
---|
2275 | // variables are always present in such a ring: |
---|
2276 | |
---|
2277 | // BO: already created |
---|
2278 | |
---|
2279 | // cent: ideal, describing the upcoming center determined by the algorithm |
---|
2280 | |
---|
2281 | ideal cent=tradtoideal(previousa,J2,flag); |
---|
2282 | |
---|
2283 | |
---|
2284 | // path= module (autoconverted to matrix) |
---|
2285 | // path[1][idchart]=parent[idchart] index of the parent-chart in resolution history of this chart |
---|
2286 | // path[2][idchart]=index of this chart in relation with its brother-charts |
---|
2287 | |
---|
2288 | module path=chart[9]; |
---|
2289 | |
---|
2290 | |
---|
2291 | // lastMap: ideal, describing the preceding blow up leading to this chart |
---|
2292 | |
---|
2293 | ideal lastMap=constructlastblwup(blwhist,n,chy,flag); |
---|
2294 | |
---|
2295 | |
---|
2296 | //------------------------------------------------------------------------------ |
---|
2297 | |
---|
2298 | // EXTRA INFORMATION NEEDED |
---|
2299 | |
---|
2300 | list invSat=ideal(0),aux9; |
---|
2301 | |
---|
2302 | |
---|
2303 | // BACK TO THE CHAR OF THE ORIGINAL RING, IF IT HAD p>0 |
---|
2304 | |
---|
2305 | if (p>0){ |
---|
2306 | |
---|
2307 | list Lring; |
---|
2308 | Lring=ringlist(RRnew); |
---|
2309 | Lring[1]=p; |
---|
2310 | def auxRnew=ring(Lring); |
---|
2311 | |
---|
2312 | kill Lring; |
---|
2313 | setring auxRnew; |
---|
2314 | ideal chy=maxideal(1); |
---|
2315 | map frnew=RRnew,chy; |
---|
2316 | def BO=frnew(BO); |
---|
2317 | |
---|
2318 | // def chart=frr(chart); |
---|
2319 | def invSat=frnew(invSat); |
---|
2320 | def lastMap=frnew(lastMap); |
---|
2321 | def cent=frnew(cent); |
---|
2322 | def path=frnew(path); |
---|
2323 | |
---|
2324 | } |
---|
2325 | |
---|
2326 | // export everything needed |
---|
2327 | |
---|
2328 | export BO; |
---|
2329 | export(invSat); |
---|
2330 | export lastMap; |
---|
2331 | export path; |
---|
2332 | export cent; |
---|
2333 | |
---|
2334 | if (p==0){return(RRnew);} |
---|
2335 | else{ |
---|
2336 | return(auxRnew);} |
---|
2337 | } |
---|
2338 | example |
---|
2339 | {"EXAMPLE:"; echo = 2; |
---|
2340 | ring r = 0,(x(1..2)),dp; |
---|
2341 | ideal J=x(1)^2-x(2)^3; |
---|
2342 | list L=Eresol(J); |
---|
2343 | list B=salida(5,L[1][5],L[8][6],2,L[1][3][3],2,1,0); // chart 5 |
---|
2344 | def RR=B[1]; |
---|
2345 | setring RR; |
---|
2346 | BO; |
---|
2347 | "press return to see next example"; ~; |
---|
2348 | |
---|
2349 | ring r = 0,(x(1..2)),dp; |
---|
2350 | ideal J=x(1)^2-x(2)^3; |
---|
2351 | list L=Eresol(J); |
---|
2352 | list B=salida(7,L[1][7],L[8][8],0,L[1][5][3],2,1,0); // chart 7 |
---|
2353 | def RR=B[1]; |
---|
2354 | setring RR; |
---|
2355 | BO; |
---|
2356 | showBO(BO); |
---|
2357 | "press return to see next example"; ~; |
---|
2358 | |
---|
2359 | ring r = 0,(x(1..2)),dp; |
---|
2360 | ideal J=x(1)^3-x(1)*x(2)^3; |
---|
2361 | list L=Eresol(J); // 8 charts, rational exponents |
---|
2362 | list B=salida(1,L[1][1],L[8][2],2,0,2,2,0); // CHART 1 |
---|
2363 | def RR=B[1]; |
---|
2364 | setring RR; |
---|
2365 | BO; |
---|
2366 | |
---|
2367 | } |
---|
2368 | |
---|
2369 | ///////////////////////////////////////////////////////////////////////////// |
---|
2370 | // CONVERT THE OUTPUT OF Eresol IN A LIST OF RINGS, WHERE A BASIC OBJECT (BO) IS DEFINED |
---|
2371 | // IN ORDER TO INTEGRATE THIS LIBRARY INSIDE THE LIBRARY resolve.lib |
---|
2372 | |
---|
2373 | proc genoutput(list chart,list mobile,int nchart,list nsons,int n,int q, int p) |
---|
2374 | "USAGE: genoutput(chart,mobile,nchart,nsons,n,q,p); |
---|
2375 | chart, mobile, nsons lists, nchart, n,q, p integers |
---|
2376 | RETURN: two lists, the first one gives the rings corresponding to the final charts, |
---|
2377 | the second one is the list of all rings corresponding to the affine charts of the resolution process |
---|
2378 | EXAMPLE: example genoutput; shows an example |
---|
2379 | " |
---|
2380 | { |
---|
2381 | int idchart,parent; |
---|
2382 | list auxlist,solvedrings,totalringlist,previousa; |
---|
2383 | list auxlistenp,solvedringsenp,totalringenp; |
---|
2384 | |
---|
2385 | // chart gives: parent,Y,a,expJ,Coef,flag,Hhist,blwhist,path,hipercoef,hiperexp |
---|
2386 | // mobile gives: tip,oldOlist,oldC,oldt,oldD,oldH,allH,infobo7; NOTE: Eolist=mobile[2]; |
---|
2387 | |
---|
2388 | idchart=1; |
---|
2389 | |
---|
2390 | // first loop, construct list previousa |
---|
2391 | |
---|
2392 | while (idchart<=nchart) |
---|
2393 | { |
---|
2394 | if (idchart==1){previousa[1]=chart[2][3];} |
---|
2395 | else |
---|
2396 | { |
---|
2397 | // if there are no sons, the next center is nothing |
---|
2398 | if (nsons[idchart]==0){previousa[idchart]=0;} |
---|
2399 | // always fill the parent |
---|
2400 | parent=chart[idchart][1]; |
---|
2401 | previousa[parent]=chart[idchart][3]; |
---|
2402 | } |
---|
2403 | idchart=idchart+1; |
---|
2404 | } |
---|
2405 | // HERE BEGIN THE LOOP |
---|
2406 | idchart=1; |
---|
2407 | while (idchart<=nchart) |
---|
2408 | { |
---|
2409 | def auxexit=salida(idchart,chart[idchart],mobile[idchart+1],nsons[idchart],previousa[idchart],n,q,p); |
---|
2410 | if (p>0) |
---|
2411 | { // we need the computations in char 0 too |
---|
2412 | def auxexitenp=salida(idchart,chart[idchart],mobile[idchart+1],nsons[idchart],previousa[idchart],n,q,0); |
---|
2413 | } |
---|
2414 | else{def auxexitenp=auxexit;} |
---|
2415 | // we add the ring to the list of all rings |
---|
2416 | auxlist[1]=auxexit; |
---|
2417 | totalringlist=totalringlist+auxlist; |
---|
2418 | auxlistenp[1]=auxexitenp; |
---|
2419 | totalringenp=totalringenp+auxlistenp; |
---|
2420 | // if the chart has no sons, add it to the list of final charts |
---|
2421 | if (nsons[idchart]==0) |
---|
2422 | { |
---|
2423 | solvedrings=solvedrings+auxlist; |
---|
2424 | solvedringsenp=solvedringsenp+auxlistenp; |
---|
2425 | } |
---|
2426 | auxlist=list(); |
---|
2427 | auxlistenp=list(); |
---|
2428 | kill auxexit; |
---|
2429 | kill auxexitenp; |
---|
2430 | idchart=idchart+1; |
---|
2431 | } // EXIT WHILE |
---|
2432 | return(solvedrings,totalringlist,solvedringsenp,totalringenp); |
---|
2433 | } |
---|
2434 | example |
---|
2435 | {"EXAMPLE:"; echo = 2; |
---|
2436 | ring r = 0,(x(1..2)),dp; |
---|
2437 | ideal J=x(1)^3-x(1)*x(2)^3; |
---|
2438 | list L=Eresol(J); // 8 charts, rational exponents |
---|
2439 | list B=genoutput(L[1],L[8],L[4],L[6],2,2,0); // generates the output |
---|
2440 | presentTree(B); |
---|
2441 | list iden0=collectDiv(B); |
---|
2442 | ResTree(B,iden0[1]); // generates the resolution tree |
---|
2443 | |
---|
2444 | // Use presentTree(B); to see the final charts |
---|
2445 | // To see the tree type in another shell |
---|
2446 | // dot -Tjpg ResTree.dot -o ResTree.jpg |
---|
2447 | // /usr/bin/X11/xv ResTree.jpg |
---|
2448 | |
---|
2449 | } |
---|
2450 | ///////////////////////////////////////////////////////////////////// |
---|
2451 | |
---|
2452 | proc computemcm(list Eolist) |
---|
2453 | "USAGE: computemcm(Eolist); Eolist list |
---|
2454 | RETURN: an integer, the least common multiple of the denominators of the E-orders |
---|
2455 | NOTE: Make the same as lcmofall but for one chart. NECESSARY BECAUSE THE E-ORDERS ARE OF TYPE NUMBER!! |
---|
2456 | EXAMPLE: example computemcm; shows an example |
---|
2457 | " |
---|
2458 | { |
---|
2459 | int m,i,aux,mcmchart; |
---|
2460 | intvec num; |
---|
2461 | m=size(Eolist); |
---|
2462 | if (m==1){mcmchart=int(denominator(Eolist[1])); return(mcmchart);} |
---|
2463 | if (m>1) |
---|
2464 | { |
---|
2465 | num=int(denominator(Eolist[1])); |
---|
2466 | for (i=2;i<=m;i++) |
---|
2467 | {aux=int(denominator(Eolist[i])); num=num,aux; } |
---|
2468 | } |
---|
2469 | mcmchart=lcm(num); |
---|
2470 | return(mcmchart); |
---|
2471 | } |
---|
2472 | example |
---|
2473 | {"EXAMPLE:"; echo = 2; |
---|
2474 | ring r = 0,(x(1..2)),dp; |
---|
2475 | ideal J=x(1)^3-x(1)*x(2)^3; |
---|
2476 | list L=Eresol(J); // 8 charts, rational exponents |
---|
2477 | L[8][2][2]; // maximal E-order at the first chart |
---|
2478 | computemcm(L[8][2][2]); |
---|
2479 | |
---|
2480 | } |
---|
2481 | ///////////////////////////////////////////////////////////////////// |
---|
2482 | |
---|
2483 | proc constructH(intvec Hhist,int n,list flag) |
---|
2484 | "USAGE: constructH(Hhist,n,flag); |
---|
2485 | Hhist intvec, n integer, flag list |
---|
2486 | RETURN: the list of exceptional divisors accumulated at this chart |
---|
2487 | EXAMPLE: example constructH; shows an example |
---|
2488 | " |
---|
2489 | { |
---|
2490 | int i,j,m,l; |
---|
2491 | list exceplist; |
---|
2492 | ideal aux; |
---|
2493 | m=size(Hhist); |
---|
2494 | if (Hhist[1]==0 and m>1) |
---|
2495 | { |
---|
2496 | Hhist=Hhist[2..m]; m=m-1; |
---|
2497 | for (i=1;i<=m;i++) |
---|
2498 | { |
---|
2499 | l=Hhist[i]; |
---|
2500 | if (flag[l]==0){aux=ideal(poly(x(l))); } |
---|
2501 | else {aux=ideal(poly(y(l))); } |
---|
2502 | exceplist[i]=aux; |
---|
2503 | } |
---|
2504 | // eliminate repeated variables |
---|
2505 | for (i=1;i<=m;i++) |
---|
2506 | { |
---|
2507 | for (j=1;j<=m;j++) |
---|
2508 | { |
---|
2509 | if (Hhist[i]==Hhist[j] and i!=j) |
---|
2510 | { |
---|
2511 | if (i<j){exceplist[i]=ideal(1);} |
---|
2512 | if (i>j){exceplist[j]=ideal(1);} |
---|
2513 | } |
---|
2514 | } |
---|
2515 | } |
---|
2516 | } |
---|
2517 | else {exceplist=list();} |
---|
2518 | // else {exceplist=list(ideal(0));} // IF IT FAILS USE THIS |
---|
2519 | return(exceplist); |
---|
2520 | } |
---|
2521 | example |
---|
2522 | {"EXAMPLE:"; echo = 2; |
---|
2523 | ring r = 0,(x(1..3)),dp; |
---|
2524 | list flag=identifyvar(); |
---|
2525 | ideal J=x(1)^4*x(2)^2, x(1)^2+x(3)^3; |
---|
2526 | list L=Eresol(J); // 7 charts |
---|
2527 | // history of the exceptional divisors at the 7-th chart |
---|
2528 | L[1][7][7]; // blow ups at x(3)-th, x(1)-th and x(1)-th charts |
---|
2529 | constructH(L[1][7][7],3,flag); |
---|
2530 | } |
---|
2531 | ///////////////////////////////////////////////////////////////////// |
---|
2532 | |
---|
2533 | proc constructblwup(list blwhist,int n,ideal chy,list flag) |
---|
2534 | "USAGE: constructblwup(blwhist,n,chy,flag); |
---|
2535 | blwhist, flag lists, n integer, chy ideal |
---|
2536 | RETURN: the ideal defining the map K[W] --> K[Wi], |
---|
2537 | which gives the composition map of all the blowing up leading to this chart |
---|
2538 | NOTE: NECESSARY START WITH COLUMNS |
---|
2539 | EXAMPLE: example constructblwup; shows an example |
---|
2540 | " |
---|
2541 | { |
---|
2542 | int i,j,m,m2; |
---|
2543 | poly aux2; |
---|
2544 | |
---|
2545 | m=size(blwhist[1]); |
---|
2546 | |
---|
2547 | for (j=1;j<=m;j++) |
---|
2548 | { |
---|
2549 | for (i=1;i<=n;i++) |
---|
2550 | { |
---|
2551 | m2=blwhist[i][j]; |
---|
2552 | // If m2!=0 this variable changes. First decide if the variable to multiply is invertible or not |
---|
2553 | if (m2!=0) |
---|
2554 | { |
---|
2555 | if (flag[m2]==0){aux2=poly(x(m2));} |
---|
2556 | else {aux2=poly(y(m2));} |
---|
2557 | // And then substitute this variable for the corresponding product in the whole ideal |
---|
2558 | if (flag[i]==0){chy=subst(chy,x(i),x(i)*aux2);} |
---|
2559 | else {chy=subst(chy,y(i),y(i)*aux2);} |
---|
2560 | } |
---|
2561 | } |
---|
2562 | } |
---|
2563 | return(chy); |
---|
2564 | } |
---|
2565 | example |
---|
2566 | {"EXAMPLE:"; echo = 2; |
---|
2567 | ring r = 0,(x(1..3)),dp; |
---|
2568 | list flag=identifyvar(); |
---|
2569 | ideal chy=maxideal(1); |
---|
2570 | ideal J=x(1)^4*x(2)^2, x(1)^2+x(3)^3; |
---|
2571 | list L=Eresol(J); // 7 charts |
---|
2572 | // history of the blow ups at the 7-th chart, center {x(1)=x(3)=0} every time |
---|
2573 | L[1][7][8]; // blow ups at x(3)-th, x(1)-th and x(1)-th charts |
---|
2574 | constructblwup(L[1][7][8],3,chy,flag); |
---|
2575 | } |
---|
2576 | ///////////////////////////////////////////////////////////////////// |
---|
2577 | |
---|
2578 | proc constructlastblwup(list blwhist,int n,ideal chy,list flag) |
---|
2579 | "USAGE: constructlastblwup(blwhist,n,chy,flag); |
---|
2580 | blwhist, flag lists, n integer, chy ideal |
---|
2581 | RETURN: the ideal defining the last blow up |
---|
2582 | NOTE: NECESSARY START WITH COLUMNS |
---|
2583 | EXAMPLE: example constructlastblwup; shows an example |
---|
2584 | " |
---|
2585 | { |
---|
2586 | int i,j,m,m2; |
---|
2587 | poly aux2; |
---|
2588 | m=size(blwhist[1]); |
---|
2589 | |
---|
2590 | if (m>0) |
---|
2591 | { |
---|
2592 | for (i=1;i<=n;i++){ m2=blwhist[i][m]; |
---|
2593 | |
---|
2594 | // If m2!=0 this variable changes. First decide if the variable to multiply is invertible or not |
---|
2595 | |
---|
2596 | if (m2!=0) |
---|
2597 | { |
---|
2598 | if (flag[m2]==0){aux2=poly(x(m2));} |
---|
2599 | else {aux2=poly(y(m2));} |
---|
2600 | |
---|
2601 | // And then substitute this variable for the corresponding product in the whole ideal |
---|
2602 | |
---|
2603 | if (flag[i]==0){chy=subst(chy,x(i),x(i)*aux2);} |
---|
2604 | else {chy=subst(chy,y(i),y(i)*aux2);} |
---|
2605 | } |
---|
2606 | } |
---|
2607 | } |
---|
2608 | |
---|
2609 | return(chy); |
---|
2610 | } |
---|
2611 | example |
---|
2612 | {"EXAMPLE:"; echo = 2; |
---|
2613 | ring r = 0,(x(1..3)),dp; |
---|
2614 | list flag=identifyvar(); |
---|
2615 | ideal chy=maxideal(1); |
---|
2616 | ideal J=x(1)^4*x(2)^2, x(1)^2+x(3)^3; |
---|
2617 | list L=Eresol(J); // 7 charts |
---|
2618 | // history of the blow ups at the 7-th chart, center {x(1)=x(3)=0} every time |
---|
2619 | L[1][7][8]; // blow ups at x(3)-th, x(1)-th and x(1)-th charts |
---|
2620 | constructlastblwup(L[1][7][8],3,chy,flag); |
---|
2621 | } |
---|
2622 | ///////////////////////////////////////////////////////////////////// |
---|
2623 | |
---|
2624 | proc tradtoideal(intvec a,ideal J2,list flag) |
---|
2625 | "USAGE: tradtoideal(a,J2,flag); |
---|
2626 | a intvec, J2 ideal, flag list |
---|
2627 | COMPUTE: traslate to an ideal the intvec defining the center |
---|
2628 | RETURN: the ideal of the center, given by the intvec a, or J2 if a=0 |
---|
2629 | EXAMPLE: example tradtoideal; shows an example |
---|
2630 | " |
---|
2631 | { |
---|
2632 | int i,m; |
---|
2633 | ideal acenter,aux2; |
---|
2634 | |
---|
2635 | if (a==0) |
---|
2636 | {acenter=J2;} |
---|
2637 | else |
---|
2638 | { |
---|
2639 | m=size(a); |
---|
2640 | for (i=1;i<=m;i++) |
---|
2641 | { |
---|
2642 | if (flag[a[i]]==0){aux2=poly(x(a[i]));} |
---|
2643 | else {aux2=poly(y(a[i]));} |
---|
2644 | |
---|
2645 | acenter=acenter+aux2; |
---|
2646 | } |
---|
2647 | } |
---|
2648 | return(acenter); |
---|
2649 | } |
---|
2650 | example |
---|
2651 | {"EXAMPLE:"; echo = 2; |
---|
2652 | ring r = 0,(x(1..3)),dp; |
---|
2653 | list flag=identifyvar(); |
---|
2654 | ideal J=x(1)^4*x(2)^2, x(1)^2+x(3)^3; |
---|
2655 | intvec a=1,3; // first center of blowing up |
---|
2656 | tradtoideal(a,J,flag); |
---|
2657 | } |
---|
2658 | ////////////////////////////////////////////////////////////////////////////////////// |
---|
2659 | // OPERATIONS WITH LISTS |
---|
2660 | ////////////////////////////////////////////////////////////////////////////////////// |
---|
2661 | |
---|
2662 | proc iniD(int n) |
---|
2663 | "USAGE: iniD(n); n integer |
---|
2664 | RETURN: list of lists of zeros of size n |
---|
2665 | EXAMPLE: example iniD; shows an example |
---|
2666 | " |
---|
2667 | {int i,j; |
---|
2668 | list D,auxD; |
---|
2669 | for (j=1;j<=n; j++) {auxD[j]=0;} |
---|
2670 | for (i=1;i<=n; i++) {D[i]=auxD;} |
---|
2671 | return(D); |
---|
2672 | } |
---|
2673 | example |
---|
2674 | {"EXAMPLE:"; echo = 2; |
---|
2675 | iniD(3); |
---|
2676 | } |
---|
2677 | ///////////////////////////////////////////////////////// |
---|
2678 | |
---|
2679 | proc sumlist(list L1,list L2) |
---|
2680 | "USAGE: sumlist(L1,L2); L1,L2 lists, (size(L1)==size(L2)) |
---|
2681 | RETURN: a list, sum of L1 and L2 |
---|
2682 | EXAMPLE: example sumlist; shows an example |
---|
2683 | " |
---|
2684 | { |
---|
2685 | int i,k; |
---|
2686 | list sumL; |
---|
2687 | k=size(L1); |
---|
2688 | if (size(L2)!=k) {return("ERROR en sumlist, lists must have the same size");} |
---|
2689 | for (i=1;i<=k;i++) {sumL[i]=L1[i]+L2[i];} |
---|
2690 | return(sumL); |
---|
2691 | } |
---|
2692 | example |
---|
2693 | {"EXAMPLE:"; echo = 2; |
---|
2694 | list L1=1,2,3; |
---|
2695 | list L2=5,9,7; |
---|
2696 | sumlist(L1,L2); |
---|
2697 | } |
---|
2698 | /////////////////////////////////////////////////////// |
---|
2699 | |
---|
2700 | proc reslist(list L1,list L2) |
---|
2701 | "USAGE: reslist(L1,L2); L1,L2 lists, (size(L1)==size(L2)) |
---|
2702 | RETURN: a list, subtraction of L1 and L2 |
---|
2703 | EXAMPLE: example reslist; shows an example |
---|
2704 | " |
---|
2705 | { |
---|
2706 | int i,k; |
---|
2707 | list resL; |
---|
2708 | k=size(L1); |
---|
2709 | if (size(L2)!=k) {return("ERROR en reslist, lists must have the same size");} |
---|
2710 | for (i=1;i<=k;i++) {resL[i]=L1[i]-L2[i];} |
---|
2711 | return(resL); |
---|
2712 | } |
---|
2713 | example |
---|
2714 | {"EXAMPLE:"; echo = 2; |
---|
2715 | list L1=1,2,3; |
---|
2716 | list L2=5,9,7; |
---|
2717 | reslist(L1,L2); |
---|
2718 | } |
---|
2719 | ////////////////////////////////////////////////////// |
---|
2720 | |
---|
2721 | proc multiplylist(list L,number a) |
---|
2722 | "USAGE: multiplylist(L,a); L list, a number |
---|
2723 | RETURN: list of elements of type number, multiplication of L times a |
---|
2724 | EXAMPLE: example multiplylist; shows an example |
---|
2725 | " |
---|
2726 | {int i,k; |
---|
2727 | list newL,bb; |
---|
2728 | number b; |
---|
2729 | k=size(L); |
---|
2730 | for (i=1;i<=k;i++) {b=L[i]*a; bb=b; newL=newL+bb;} |
---|
2731 | return(newL); |
---|
2732 | } |
---|
2733 | example |
---|
2734 | {"EXAMPLE:"; echo = 2; |
---|
2735 | ring r = 0,(x(1..3)),dp; |
---|
2736 | list L=1,2,3; |
---|
2737 | multiplylist(L,1/5); |
---|
2738 | } |
---|
2739 | /////////////////////////////////////////////////////// |
---|
2740 | |
---|
2741 | proc dividelist(list L1,list L2) |
---|
2742 | "USAGE: dividelist(L1,L2); L1,L2 lists |
---|
2743 | RETURN: list of elements of type number, division of L1 by L2 |
---|
2744 | EXAMPLE: example dividelist; shows an example |
---|
2745 | " |
---|
2746 | {int i,k,k1,k2; |
---|
2747 | list LL,bb; |
---|
2748 | number a1,a2,b; |
---|
2749 | k1=size(L1); |
---|
2750 | k2=size(L2); |
---|
2751 | if (k2!=k1) {print("ERROR en dividelist, lists must have the same size");} |
---|
2752 | if (k1<=k2) {k=k1;} |
---|
2753 | else {k=k2;} |
---|
2754 | for (i=1;i<=k;i++) |
---|
2755 | {a1=L1[i]; a2=L2[i]; b=a1/a2; bb=b; LL=LL+bb;} |
---|
2756 | return(LL); |
---|
2757 | } |
---|
2758 | example |
---|
2759 | {"EXAMPLE:"; echo = 2; |
---|
2760 | ring r = 0,(x(1..3)),dp; |
---|
2761 | list L1=1,2,3; |
---|
2762 | list L2=5,9,7; |
---|
2763 | dividelist(L1,L2); |
---|
2764 | } |
---|
2765 | /////////////////////////////////////////////////////// |
---|
2766 | |
---|
2767 | proc createlist(list L1,list L2) |
---|
2768 | "USAGE: createlist(L1,L2); L1,L2 lists, (size(L1)==size(L2)) |
---|
2769 | RETURN: list of lists of two elements, the first one of L1 and the second of L2 |
---|
2770 | EXAMPLE: example createlist; shows an example |
---|
2771 | " |
---|
2772 | {int i,k; |
---|
2773 | list L,aux; |
---|
2774 | k=size(L1); |
---|
2775 | if (size(L2)!=k) {return("ERROR en createlist, lists must have the same size");} |
---|
2776 | L=list0(k); |
---|
2777 | for (i=1;i<=k;i++) {if (L1[i]!=0) {aux=L1[i],L2[i]; L[i]=aux;} |
---|
2778 | else {L=delete(L,i);}} |
---|
2779 | return(L); |
---|
2780 | } |
---|
2781 | example |
---|
2782 | {"EXAMPLE:"; echo = 2; |
---|
2783 | list L1=1,2,3; |
---|
2784 | list L2=5,9,7; |
---|
2785 | createlist(L1,L2); |
---|
2786 | } |
---|
2787 | /////////////////////////////////////////////////////// |
---|
2788 | static proc list0(int n) |
---|
2789 | "USAGE: list0(n); n integer |
---|
2790 | RETURN: list of n zeros |
---|
2791 | EXAMPLE: example list0; shows an example |
---|
2792 | " |
---|
2793 | {int i; |
---|
2794 | list L0; |
---|
2795 | for (i=1;i<=n;i++) {L0[i]=0;} |
---|
2796 | return(L0); |
---|
2797 | } |
---|
2798 | example |
---|
2799 | {"EXAMPLE:"; echo = 2; |
---|
2800 | list0(4); |
---|
2801 | } |
---|
2802 | //////////////////////////////////////////////////////////// |
---|