1 | ///////////////////////////////////////////////////////////////////////////////// |
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2 | version = "$Id$"; |
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3 | category="Singularities"; |
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4 | |
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5 | info=" |
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6 | LIBRARY: classifyCeq.lib simple hypersurface singularities in characteristic p > 0 |
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7 | AUTHORS: Deeba Afzal deebafzal@gmail.com |
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8 | Faira Kanwal Janjua fairakanwaljanjua@gmail.com |
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9 | |
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10 | OVERVIEW: |
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11 | A library for classifying the simple singularities with respect to contact equivalence in charateristic p > 0 . |
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12 | Simple hypersurface singularities in charateristic p > 0 were classified by Greuel and |
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13 | Kroening [1] with respect to contact equivalence. The classifier we use has been proposed in [2]. |
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14 | |
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15 | REFERENCES: |
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16 | [1] Greuel, G.-M.; Kroening, H.: Simple singularities in positive characteristic. |
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17 | Math.Z. 203, 339-354 (1990). |
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18 | [2] Afzal,D.;Binyamin,M.A.;Janjua,F.K.: On the classification of simple singularities in positive characteristic. |
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19 | |
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20 | PROCEDURES: |
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21 | classifyCeq(f); simple hypersurface singularities in charateristic p > 0 |
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22 | "; |
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23 | LIB "sing.lib"; |
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24 | LIB "classify.lib"; |
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25 | LIB "primdec.lib"; |
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26 | LIB "ring.lib"; |
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27 | ///////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
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28 | proc classifyCeq(poly f) |
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29 | "USAGE: classifyCeq(f); f poly |
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30 | RETURN: string including the Tjurina number of f and its type in the classification of Greuel and Kroening |
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31 | EXAMPLE: example classifyCeq; shows an example |
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32 | " |
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33 | { |
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34 | def R=basering; |
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35 | def S=changeord(list(list("ds",1:nvars(basering)))); |
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36 | setring S; |
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37 | poly f=imap(R,f); |
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38 | string re; |
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39 | if(char(basering)==0) |
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40 | { |
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41 | re=(string(classify(f))); |
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42 | } |
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43 | if(char(basering)!=2) |
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44 | { |
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45 | re=classifyCeq1(f); |
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46 | } |
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47 | if(char(basering)==2) |
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48 | { |
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49 | re=classifyCeq2(f); |
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50 | } |
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51 | setring R; |
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52 | return(re); |
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53 | } |
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54 | example |
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55 | { |
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56 | "EXAMPLE:"; echo=2; |
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57 | ring R=3,(x,y,z,u,v,w),ds; |
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58 | classifyCeq(-x2+xy+y2+xz-yz-z2+w2+u3+v4); |
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59 | } |
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60 | /////////////////////////////////////////////////////////// char p > 2 ////////////////////////////////////////// |
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61 | static proc blowupone(poly f) |
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62 | { |
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63 | //=== input f smooth or isolated simple singularity at zero |
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64 | //=== output var(1) or poly with isolated singularity at zero |
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65 | def R=basering; |
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66 | def S=changeord(list(list("ds",1:nvars(basering)))); |
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67 | setring S; |
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68 | int n=nvars(basering); |
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69 | int i; |
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70 | poly f=imap(R,f); |
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71 | if(deg(lead(f))<=1){setring R;return(var(1));} |
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72 | def T=changeord(list(list("lp",1:nvars(basering)))); |
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73 | setring T; |
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74 | map phi; |
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75 | ideal mphi, sing; |
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76 | poly p,q; |
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77 | //=========== blow up ======================================================= |
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78 | for(i=1;i<=n;i++) |
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79 | { |
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80 | mphi=var(i)*maxideal(1); |
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81 | mphi[i]=var(i); |
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82 | phi=S,mphi; |
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83 | p=phi(f); |
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84 | q=p/var(i); |
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85 | while(size(p)==size(q)) |
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86 | { |
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87 | p=q; |
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88 | q=q/var(i); |
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89 | } |
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90 | //=============== p is the strict transform var(i) exceptional divisor ==== |
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91 | //=============== analysis of singularities ================================ |
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92 | sing=jacob(p),p,var(i); |
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93 | sing=radical(sing); |
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94 | option(redSB); |
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95 | sing=std(sing); |
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96 | sing=simplify(sing,1); |
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97 | if(size(sing)>1) |
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98 | { |
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99 | if(size(sing)!=n){ERROR("not simple");} |
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100 | ideal mpsi=std(maxideal(1)); |
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101 | for(i=1;i<=n;i++){mpsi[i]=var(i)-sing[n-i+1][2];} |
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102 | map psi=T,mpsi; |
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103 | p=psi(p); |
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104 | setring R; |
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105 | poly p=imap(T,p); |
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106 | return(p); |
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107 | } |
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108 | } |
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109 | setring R; |
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110 | return(var(1)); |
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111 | } |
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112 | //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
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113 | static proc classifyCeq1(poly f) |
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114 | //====Classify Simple hypersurface singularities when charateristic p!=2 |
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115 | { |
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116 | // char!=2 |
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117 | //====input poly f |
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118 | //====output The function defines ......not an isolated singularity or not a simple singularity and tjurina |
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119 | // of the singularity or a simple singularity ,tjurina of the singularity and type of the singularity. |
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120 | def R=basering; |
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121 | int d; |
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122 | int m=tjurina(f); |
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123 | |
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124 | if(m==-1) |
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125 | { |
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126 | return("The given function defines not an isolated singularity" ); |
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127 | |
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128 | } |
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129 | poly g; |
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130 | int c; |
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131 | c=corank(f); |
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132 | if(c<=1) |
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133 | { |
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134 | |
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135 | if(c==0) |
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136 | { |
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137 | return("The given function defines a simple singularity. |
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138 | The tjurina number is "+string(m)+". |
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139 | A[1]:"+string(var(1))+"^2+"+string(var(2))+"^2"); |
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140 | } |
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141 | |
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142 | if(c==1) |
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143 | |
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144 | { |
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145 | poly w=blowupone(f); |
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146 | int n=tjurina(w); |
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147 | |
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148 | int b=m-n; |
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149 | if(b<=2) |
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150 | { |
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151 | return( "The given function defines a simple singularity. |
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152 | The tjurina number is "+string(m)+". |
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153 | A["+string(m)+"]:"+string(var(1))+"^2+"+string(var(2))+"^"+string(m+1)+""); |
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154 | } |
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155 | else |
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156 | { |
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157 | |
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158 | return("The given function defines a simple singularity. |
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159 | The tjurina number is "+string(m)+". |
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160 | A["+string(m-1)+"]:"+string(var(1))+"^2+"+string(var(2))+"^"+string(m)+""); |
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161 | } |
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162 | |
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163 | } |
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164 | } |
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165 | if(c==2) |
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166 | { |
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167 | g=jet(f,3); |
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168 | if(g==0) |
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169 | { |
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170 | return( "The given function defines not a simple singularity. |
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171 | The tjurina number is "+string(m)+"."); |
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172 | } |
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173 | if(g!=0) |
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174 | { |
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175 | |
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176 | def S=GDsplitting(f); |
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177 | setring S; |
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178 | |
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179 | |
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180 | poly g=jet(f,3); |
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181 | if(g==0) |
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182 | { |
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183 | setring(R); |
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184 | return("The given function defines not a simple singularity. |
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185 | The tjurina number is "+string(m)+"."); |
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186 | } |
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187 | list L=factorize(g); |
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188 | if(size(L[1])==2) |
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189 | { |
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190 | ideal M=var(1),var(2)^2; |
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191 | ideal N=std(M^3); |
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192 | poly h=reduce(f,N); |
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193 | if(h==0) |
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194 | { |
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195 | return(" The given function defines not a simple singularity |
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196 | The tjurina number is "+string(m)+"."); |
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197 | } |
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198 | if(h!=0) |
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199 | { |
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200 | if((char(R)!=3)&&(char(R)!=5)) |
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201 | { |
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202 | setring R; |
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203 | if(m==6) |
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204 | { |
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205 | return("The given function defines a simple singularity. |
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206 | The tjurina number is "+string(m)+". |
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207 | E^0[6]:"+string(var(1))+"^3+"+string(var(2))+"^4"); |
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208 | } |
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209 | if(m==7) |
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210 | { |
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211 | return("The given function defines a simple singularity. |
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212 | The tjurina number is "+string(m)+". |
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213 | E^0[7]:"+string(var(1))+"^3+"+string(var(1))+"*"+string(var(2))+"^3"); |
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214 | } |
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215 | if(m==8) |
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216 | { |
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217 | return("The given function defines a simple singularity. |
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218 | The tjurina number is "+string(m)+". |
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219 | E^0[8]:"+string(var(1))+"^3+"+string(var(2))+"^5"); |
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220 | } |
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221 | } |
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222 | if(char(R)==5) |
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223 | { |
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224 | setring R; |
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225 | if(m==6) |
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226 | { |
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227 | return("The given function defines a simple singularity. |
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228 | The tjurina number is "+string(m)+". |
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229 | E^0[6]:"+string(var(1))+"^3+"+string(var(2))+"^4"); |
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230 | } |
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231 | if(m==7) |
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232 | { |
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233 | return("The given function defines a simple singularity. |
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234 | The tjurina number is "+string(m)+". |
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235 | E^0[7]:"+string(var(1))+"^3+"+string(var(1))+"*"+string(var(2))+"^3"); |
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236 | } |
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237 | if(m==10) |
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238 | { |
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239 | return("The given function defines a simple singularity. |
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240 | The tjurina number is "+string(m)+". |
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241 | E^0[8]:"+string(var(1))+"^3+"+string(var(2))+"^5"); |
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242 | } |
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243 | if(m==8) |
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244 | { |
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245 | return("The given function defines a simple singularity. |
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246 | The tjurina number is "+string(m)+". |
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247 | E^1[8]:"+string(var(1))+"^3+"+string(var(2))+"^5+"+string(var(1))+"*"+string(var(2))+"^4"); |
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248 | } |
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249 | } |
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250 | if(char(R)==3) |
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251 | { |
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252 | poly p=blowupone(f); |
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253 | int e=(std(jacob(p)+ideal(p))==1); |
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254 | setring R; |
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255 | if((m==7)&&e) |
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256 | { |
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257 | return("The given function defines a simple singularity. |
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258 | The tjurina number is "+string(m)+". |
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259 | E^1[6]:"+string(var(1))+"^3+"+string(var(2))+"^4+"+string(var(1))+"^2*"+string(var(2))+"^2"); |
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260 | } |
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261 | if((m==7)&&!e) |
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262 | { |
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263 | return("The given function defines a simple singularity. |
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264 | The tjurina number is "+string(m)+". |
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265 | E^1[7]:"+string(var(1))+"^3+"+string(var(1))+"*"+string(var(2))+"^3+"+string(var(1))+"^2*"+string(var(2))+"^2"); |
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266 | } |
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267 | if(m==8) |
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268 | { |
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269 | |
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270 | return("The given function defines a simple singularity. |
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271 | The tjurina number is "+string(m)+". |
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272 | E^2[8]:"+string(var(1))+"^3+"+string(var(2))+"^5+"+string(var(1))+"^2*"+string(var(2))+"^2"); |
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273 | } |
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274 | if(m==10) |
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275 | { |
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276 | return("The given function defines a simple singularity. |
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277 | The tjurina number is "+string(m)+". |
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278 | E^1[8]:"+string(var(1))+"^3+"+string(var(2))+"^5+"+string(var(1))+"^2*"+string(var(2))+"^3"); |
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279 | } |
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280 | if((m==9)&&e) |
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281 | { |
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282 | |
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283 | return("The given function defines a simple singularity. |
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284 | The tjurina number is "+string(m)+". |
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285 | E^0[6]:"+string(var(1))+"^3+"+string(var(2))+"^4"); |
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286 | } |
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287 | if((m==9)&&!e) |
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288 | { |
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289 | |
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290 | return("The given function defines a simple singularity. |
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291 | The tjurina number is "+string(m)+". |
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292 | E^0[7]:"+string(var(1))+"^3+"+string(var(1))+"*"+string(var(2))+"^3"); |
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293 | } |
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294 | if(m==12) |
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295 | { |
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296 | |
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297 | return("The given function defines a simple singularity. |
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298 | The tjurina number is "+string(m)+". |
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299 | E^0[8]:"+string(var(1))+"^3+"+string(var(2))+"^5"); |
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300 | } |
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301 | |
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302 | } |
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303 | else |
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304 | { |
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305 | return( "The given function defines not a simple singularity"); |
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306 | } |
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307 | |
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308 | } |
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309 | |
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310 | } |
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311 | |
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312 | if(size(L[1])==3) |
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313 | { |
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314 | setring R; |
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315 | return("The given function defines a simple singularity. |
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316 | The tjurina number is "+string(m)+". |
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317 | D["+string(m)+"]:"+string(var(1))+"^2*"+string(var(2))+"+"+string(var(2))+"^"+string(m-1)+""); |
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318 | } |
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319 | if(size(L[1])==4) |
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320 | { |
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321 | setring R; |
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322 | return("The given function defines a simple singularity. |
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323 | The tjurina number is "+string(m)+". |
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324 | D[4]:"+string(var(1))+"^2*"+string(var(2))+"+"+string(var(1))+"^3"); |
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325 | } |
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326 | } |
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327 | } |
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328 | if(c>=3) |
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329 | { |
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330 | return( "The given function defines not a simple singularity. |
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331 | The tjurina number is "+string(m)+"."); |
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332 | } |
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333 | |
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334 | } // ends classifyCeq1 |
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335 | example |
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336 | { |
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337 | "EXAMPLE:"; echo=2; |
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338 | ring R=5,(x,y),ds; |
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339 | classifyCeq1(x2y+y22); |
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340 | } |
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341 | ////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
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342 | static proc GDsplitting(poly f) |
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343 | { // the result of the splitting lemma (myMorsesplitting) in a special ring of 2 variables |
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344 | // input a poly f of order >=2 |
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345 | // output ring S=char(basering),(x,y),ds; and a poly f in <x,y>^3 in S with the following properties: |
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346 | // f + a sum of squares of further variables is contact equivalentt to the input polynomial |
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347 | def R=basering; |
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348 | intvec t; |
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349 | int b,i; |
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350 | b=tjurina(f); |
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351 | f=myMorsesplitting(f,b); |
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352 | t=forv(findVar(f)); |
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353 | ring S=char(basering),(x,y),ds; |
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354 | ideal M; |
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355 | M[t[1]]=var(1); |
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356 | M[t[2]]=var(2); |
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357 | i=1; |
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358 | if((i!=t[1])&&(i!=t[2])) |
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359 | { |
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360 | M[i]=0; |
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361 | } |
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362 | map phi=R,M; |
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363 | poly f=phi(f); |
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364 | export(f); |
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365 | setring(R); |
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366 | return(S); |
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367 | } |
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368 | //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
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369 | static proc myMorsesplitting(poly f,int b) |
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370 | { |
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371 | // splitting lemma |
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372 | // input poly f=f(x(1),...,x(n)) jet(f,2)!=0 |
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373 | // output a polynomial in 2 variables of order >=3 such that a sum of squares of the remaining variables plus this |
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374 | // polynomial is contact equivalent to the input polynomial |
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375 | intvec w; |
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376 | f=simplify(jet(f,b),1); |
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377 | while(jet(f,2)!=0) |
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378 | { |
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379 | w=findlead(f); |
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380 | if(w[1]==w[2]) |
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381 | { |
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382 | f=splitting_one(f,w[1],b); |
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383 | } |
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384 | else |
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385 | { |
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386 | f=splitting_two(f,w[1],w[2],b); |
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387 | } |
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388 | } |
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389 | return(f); |
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390 | } |
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391 | example |
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392 | { |
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393 | "EXAMPLE:"; echo=2; |
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394 | ring R=3,(x,y,z,u,v),ds; |
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395 | poly f=x2+x3z+u2+v2+z3+y11; |
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396 | myMorsesplitting(f,30); |
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397 | } |
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398 | //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
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399 | static proc findlead(poly f) |
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400 | |
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401 | { |
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402 | // input poly f=x(i)^2+h or poly f=x(i)*x(j)+h , h of order >=2 |
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403 | // output intvec w w[1]=i,w[2]=i or w[1]=i ,w[2]=j |
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404 | intvec v=leadexp(f); |
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405 | int i,k,n; |
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406 | intvec w; |
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407 | n=nvars(basering); |
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408 | for(i=1;i<=n;i++) |
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409 | { |
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410 | if(v[i]==2) |
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411 | { |
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412 | w[1]=i; |
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413 | w[2]=i; |
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414 | break; |
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415 | } |
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416 | if(v[i]==1) |
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417 | { |
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418 | k++; |
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419 | w[k]=i; |
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420 | } |
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421 | } |
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422 | return(w); |
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423 | } |
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424 | example |
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425 | { |
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426 | "EXAMPLE:"; echo=2; |
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427 | ring R=3,(x,y,z,u,v),ds; |
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428 | poly f=x2+x3z+u2+v2+z3+y11; |
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429 | findlead(f); |
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430 | } |
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431 | //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
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432 | static proc splitting_one(poly p, int i, int b) |
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433 | { |
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434 | // assumes that p=x_i^2+h, no x_i^2 in h, h of order >=2 |
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435 | // returns q(x_1,??x_i-1,x_i+1,...,x_n) such that x_i^2 +q is right equivalent to p mod |
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436 | // <x_1,??,x_n>^b |
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437 | |
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438 | if(b<2){b=2;} |
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439 | def R=basering; |
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440 | ideal M=maxideal(1); |
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441 | map phi; |
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442 | poly q=jet((p-var(i)^2)/var(i),b); |
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443 | while(q!=0) |
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444 | { |
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445 | p=quickSubst(p,var(i)-1/2*q,i,b); |
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446 | q=jet((p-var(i)^2)/var(i),b); |
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447 | } |
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448 | return(simplify(jet(p,b)-var(i)^2,1)); //make the leading coefficient 1 |
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449 | } |
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450 | example |
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451 | { |
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452 | "EXAMPLE:"; echo=2; |
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453 | ring R=3,(x,y,z,u,v),ds; |
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454 | poly f=x2+y3+z4+xy3+u2+v2; |
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455 | splitting_one(f,1,18); |
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456 | } |
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457 | //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
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458 | static proc splitting_two(poly p, int i, int j, int b) |
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459 | { |
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460 | // assumes that p=x_i*x_j+h, no x_i^2 in h, h of order >=2 |
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461 | // returns q(x_1,???,x_i-1,x_i+1,...,x_j-1,x_j+1,...,x_n) such that x_i*x_j +q is right equivalent to p mod |
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462 | // <x_1,??,x_n>^b |
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463 | |
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464 | if(b<2){b=2;} |
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465 | def R=basering; |
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466 | ideal M=maxideal(1); |
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467 | map phi; |
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468 | poly q=jet((p-var(i)*var(j))/var(i),b); |
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469 | while(q!=0) |
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470 | { |
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471 | p=quickSubst(p,var(j)-q,j,b); |
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472 | q=jet((p-var(i)*var(j))/var(i),b); |
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473 | } |
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474 | return(simplify(substitute(jet(p,b),var(j),0),1)); //make the leading coefficient 1 |
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475 | } |
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476 | example |
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477 | { |
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478 | "EXAMPLE:"; echo=2; |
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479 | ring R=5,(u,v,w,s,t),ds; |
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480 | poly f=uv+t2+u4s+u11+v7+s9+w8; |
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481 | splitting_two(f,1,2,128); |
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482 | } |
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483 | //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
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484 | static proc quickSubst(poly h, poly r, int i, int b) |
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485 | { |
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486 | //=== assume h, r is in Q[x_1,...,x_n], computes jet(h(x_i=r),b) |
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487 | h=jet(h,b); |
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488 | r=jet(r,b); |
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489 | matrix M=coef(h,var(i)); |
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490 | poly q = 1; |
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491 | int j,k,d; |
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492 | intvec v; |
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493 | d=deg(M[1,1]); |
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494 | v[d+1]=1; |
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495 | for(k = 2; k <= ncols(M); k++) |
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496 | { |
---|
497 | v[deg(M[1,k])+1]=1; |
---|
498 | } |
---|
499 | h=0; |
---|
500 | for(k=1;k<=d+1;k++) |
---|
501 | { |
---|
502 | if(v[k]==1) |
---|
503 | { |
---|
504 | h=h+jet(q*M[2,ncols(M)-j],b); |
---|
505 | j++; |
---|
506 | } |
---|
507 | q=jet(q*r,b); |
---|
508 | } |
---|
509 | return(h); |
---|
510 | } |
---|
511 | //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
512 | static proc findVar(poly p) |
---|
513 | { |
---|
514 | // input poly f |
---|
515 | // output intvec v v[i]=0 if f is not depending on var(i) and v[j]=1 if f is depending on var(j) |
---|
516 | intvec v; |
---|
517 | int i,n; |
---|
518 | n=nvars(basering); |
---|
519 | for(i=1;i<=n;i++) |
---|
520 | { |
---|
521 | if(subst(p,var(i),0)==p) |
---|
522 | { |
---|
523 | v[i]=0; |
---|
524 | } |
---|
525 | else |
---|
526 | { |
---|
527 | v[i]=1; |
---|
528 | } |
---|
529 | } |
---|
530 | return(v); |
---|
531 | } |
---|
532 | example |
---|
533 | { |
---|
534 | "EXAMPLE:"; echo=2; |
---|
535 | ring R=3,(x,y,z,u,v),ds; |
---|
536 | poly f=y3+u5; |
---|
537 | findlead(f); |
---|
538 | } |
---|
539 | //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
540 | static proc forv(intvec v) |
---|
541 | { |
---|
542 | //returns the places of v which are different from zero |
---|
543 | intvec w; |
---|
544 | int i,j; |
---|
545 | j=1; |
---|
546 | for(i=1;i<=nvars(basering);i++) |
---|
547 | { |
---|
548 | if(v[i]!=0) |
---|
549 | { |
---|
550 | w[j]=i; |
---|
551 | j++; |
---|
552 | } |
---|
553 | } |
---|
554 | return(w); |
---|
555 | } |
---|
556 | |
---|
557 | /////////////////////////////////////////// char p=2 //////////////////////////////////////////////////////////////////////////// |
---|
558 | static proc classifyCeq2(poly p) |
---|
559 | //====Classification of hypersurface singularities in Characteristic p=2. |
---|
560 | { |
---|
561 | // input poly p |
---|
562 | // output The normal form to which f is contact equvalent or the function is not simple. |
---|
563 | def R=basering; |
---|
564 | int t=tjurina(p); |
---|
565 | list T; |
---|
566 | if(t==-1) |
---|
567 | { |
---|
568 | return("The given function defines not an isolated singularity"); |
---|
569 | } |
---|
570 | int b=t+1; |
---|
571 | p=SPILPRO(p,b); |
---|
572 | list L=findVAR(p); |
---|
573 | if(L[2]>=4) |
---|
574 | { |
---|
575 | |
---|
576 | return("The given function defines not a simple singularity. |
---|
577 | The Tjurina Number is "+string(t)+". "); |
---|
578 | } |
---|
579 | if(L[2]==1) |
---|
580 | { |
---|
581 | def S=redvar2(p); |
---|
582 | setring S; |
---|
583 | |
---|
584 | string a="The given function defines an isolated Singularity. |
---|
585 | The Tjurina number is "+string(t)+". |
---|
586 | A_"+string(leadexp(p)[1]-1)+":xy+z"+string(leadexp(p)[1])+"."; |
---|
587 | setring R; |
---|
588 | return(a); |
---|
589 | } |
---|
590 | |
---|
591 | if(jet(p,2)==0) |
---|
592 | { |
---|
593 | if(L[2]==3) |
---|
594 | { |
---|
595 | return("The given function defines not a simple singularity. |
---|
596 | The Tjurina Number is "+string(t)+". "); |
---|
597 | |
---|
598 | } |
---|
599 | def S=redvar2(p); |
---|
600 | setring S; |
---|
601 | string a=curCeq2(p); |
---|
602 | setring R; |
---|
603 | return(a); |
---|
604 | } |
---|
605 | else |
---|
606 | { |
---|
607 | if(L[2]==3) |
---|
608 | { |
---|
609 | int B=t div 2+1; |
---|
610 | p=splitting_SQUA(p,B); |
---|
611 | def S=redvar2(p); |
---|
612 | setring S; |
---|
613 | string a=surCeq2(p); |
---|
614 | setring R; |
---|
615 | return(a); |
---|
616 | } |
---|
617 | p=splitting_SQUA(p,b); |
---|
618 | def S=redvar2(p); |
---|
619 | setring S; |
---|
620 | string a=curCeq2(p); |
---|
621 | setring R; |
---|
622 | return(a); |
---|
623 | } |
---|
624 | } |
---|
625 | example |
---|
626 | { |
---|
627 | "EXAMPLE:"; echo=2; |
---|
628 | ring r=2,(x,y,z,w,t,v),ds; |
---|
629 | poly f=xy+zw+t5+v3; |
---|
630 | classifyCeq2(f); |
---|
631 | } |
---|
632 | ////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
633 | static proc SPILPRO(poly p,int B) |
---|
634 | { |
---|
635 | //Splitting lemma in characteristic 2 |
---|
636 | //input a polynomial f and a bound B |
---|
637 | //output a polynomial q, the result of the splitting lemma applied to f up to the order B |
---|
638 | int i,j; |
---|
639 | def R=basering; |
---|
640 | int n=nvars(R); |
---|
641 | poly q=splittingLchar2(p); |
---|
642 | list T=FindPRO(q); |
---|
643 | |
---|
644 | while(size(T)!=0) |
---|
645 | { |
---|
646 | i=T[1]; j=T[2]; |
---|
647 | q=splitting_two2(q,i,j,B); |
---|
648 | |
---|
649 | T=FindPRO(q); |
---|
650 | } |
---|
651 | return(q); |
---|
652 | } |
---|
653 | //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
654 | static proc surCeq2(poly f) |
---|
655 | //====Classification of Surface Singularaties in case when p=2 |
---|
656 | { |
---|
657 | //===input a funation defined in the ring whose Characteristic is 2. |
---|
658 | //===output is a Normal Form to which Given function is contact equivalent or function is not simple. |
---|
659 | def R=basering; |
---|
660 | int n=nvars(R); |
---|
661 | list L; |
---|
662 | int k=tjurina(f); |
---|
663 | if(k==-1) |
---|
664 | { |
---|
665 | return("The given function defines not an isolated singularity."); |
---|
666 | } |
---|
667 | return(surEsing(f)); |
---|
668 | } |
---|
669 | example |
---|
670 | { |
---|
671 | "EXAMPLE:"; echo=2; |
---|
672 | ring R=2,(x,y,z),Ds; |
---|
673 | surCeq(xy+y2+yz+x2y+xy2+xyz+z21+xz21); |
---|
674 | |
---|
675 | } |
---|
676 | ///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
677 | static proc surEsing(poly f) |
---|
678 | //====Classifcation of Ek singularities. |
---|
679 | { |
---|
680 | //===input: A function which is an isolated singularity. |
---|
681 | //===output: Normal form to which given function is contact equivalent. |
---|
682 | list M; |
---|
683 | def R=basering; |
---|
684 | f=formf(f); |
---|
685 | poly p=jet(f,2); |
---|
686 | int k=tjurina(f); |
---|
687 | |
---|
688 | list L=factorize(p); |
---|
689 | poly g=subst(f,leadmonom(L[1][2]),0); |
---|
690 | poly j=jet(g,3)-jet(g,2); |
---|
691 | poly C=f-g; |
---|
692 | C=C/leadmonom(L[1][2]); |
---|
693 | poly h=subst(C,leadmonom(L[1][2]),0); |
---|
694 | h=jet(h,2); |
---|
695 | poly h1=j+h*leadmonom(L[1][2]); |
---|
696 | |
---|
697 | if(h!=0) |
---|
698 | { |
---|
699 | list P=factorize(h1); |
---|
700 | list N=factorize(j); |
---|
701 | |
---|
702 | if((size(P[1])==2)&&(size(P[2])==2)&&(P[2][2]==1)&&(N[2][2]==3)) |
---|
703 | { |
---|
704 | if(k==6) |
---|
705 | { |
---|
706 | |
---|
707 | return(" The given function defines an isolated Singularity. |
---|
708 | The Tjurina number is "+string(tjurina(f))+". |
---|
709 | E^1[6]:z2+x3+y2z+xyz."); |
---|
710 | |
---|
711 | } |
---|
712 | if(k==8) |
---|
713 | { |
---|
714 | return("The given function defines an isolated Singularity. |
---|
715 | The Tjurina number is "+string(tjurina(f))+". |
---|
716 | E^0[6]:z2+x3+y2z."); |
---|
717 | |
---|
718 | } |
---|
719 | } |
---|
720 | if(((size(P[1])==4)&&(size(P[2])==4))||((size(N[1])==4)&&(size(N[2])==4))) |
---|
721 | { |
---|
722 | if(k==8) |
---|
723 | { |
---|
724 | return("The given function defines an isolated Singularity. |
---|
725 | The Tjurina number is "+string(tjurina(f))+". |
---|
726 | D[4]:z2+x2y+xy2."); |
---|
727 | } |
---|
728 | if(k==6) |
---|
729 | { |
---|
730 | return("The given function defines an isolated Singularity. |
---|
731 | The Tjurina number is "+string(tjurina(f))+". |
---|
732 | D^1[4]:z2+x2y+xy2+xyz."); |
---|
733 | |
---|
734 | } |
---|
735 | } |
---|
736 | if((size(P[1])<=3)&&(size(P[2])<=3)) |
---|
737 | { |
---|
738 | |
---|
739 | if(k==6) |
---|
740 | { |
---|
741 | return("The given function defines an isolated Singularity. |
---|
742 | The Tjurina number is "+string(tjurina(f))+". |
---|
743 | D^1[5]:z2+x2y+y2z+xyz."); |
---|
744 | } |
---|
745 | if(k==8) |
---|
746 | { |
---|
747 | |
---|
748 | if(size(lengthBL(f))==4) |
---|
749 | { |
---|
750 | return("The given function defines an isolated Singularity. |
---|
751 | The Tjurina number is "+string(tjurina(f))+". |
---|
752 | E^3[7]:z2+x3+xy3+xyz."); |
---|
753 | |
---|
754 | } |
---|
755 | else |
---|
756 | { |
---|
757 | if(size(lengthBL(f))==5) |
---|
758 | { |
---|
759 | |
---|
760 | return("The given function defines an isolated Singularity. |
---|
761 | The Tjurina number is "+string(tjurina(f))+". |
---|
762 | E^4[8]:z2+x3+y5+xyz."); |
---|
763 | } |
---|
764 | else |
---|
765 | { |
---|
766 | return("The given function defines an isolated Singularity. |
---|
767 | The Tjurina number is "+string(tjurina(f))+". |
---|
768 | D^0[5]:z2+x2y+y2z."); |
---|
769 | |
---|
770 | } |
---|
771 | } |
---|
772 | |
---|
773 | } |
---|
774 | |
---|
775 | } |
---|
776 | list Q=factorize(j); |
---|
777 | if((size(Q[1])==3)&&(size(Q[2])==3)) |
---|
778 | { |
---|
779 | |
---|
780 | return(surDsing(f)); |
---|
781 | } |
---|
782 | else |
---|
783 | { |
---|
784 | if(k==10) |
---|
785 | { |
---|
786 | if(size(lengthBL(f))==4) |
---|
787 | { |
---|
788 | return("The given function defines an isolated Singularity. |
---|
789 | The Tjurina number is "+string(tjurina(f))+". |
---|
790 | E^2[7]:z2+x3+xy3+y3z."); |
---|
791 | |
---|
792 | } |
---|
793 | if(size(lengthBL(f))==5) |
---|
794 | { |
---|
795 | return("The given function defines an isolated Singularity. |
---|
796 | The Tjurina number is "+string(tjurina(f))+". |
---|
797 | E^3[8]:z2+x3+y5+y3z."); |
---|
798 | |
---|
799 | } |
---|
800 | } |
---|
801 | if(k==12) |
---|
802 | { |
---|
803 | |
---|
804 | if(size(lengthBL(f))==4) |
---|
805 | { |
---|
806 | return("The given function defines an isolated Singularity. |
---|
807 | The Tjurina number is "+string(tjurina(f))+". |
---|
808 | E^1[7]:z2+x3+xy3+xy3z."); |
---|
809 | |
---|
810 | } |
---|
811 | if(size(lengthBL(f))==5) |
---|
812 | { |
---|
813 | return("The given function defines an isolated Singularity. |
---|
814 | The Tjurina number is "+string(tjurina(f))+". |
---|
815 | E^2[8]:z2+x3+y5+xy2z."); |
---|
816 | |
---|
817 | } |
---|
818 | } |
---|
819 | if(k==14) |
---|
820 | { |
---|
821 | |
---|
822 | if(size(lengthBL(f))==4) |
---|
823 | { |
---|
824 | retrun("The given function defines an isolated Singularity. |
---|
825 | The Tjurina number is "+string(tjurina(f))+". |
---|
826 | E^0[7]:z2+x3+xy3."); |
---|
827 | |
---|
828 | } |
---|
829 | if(size(lengthBL(f))==5) |
---|
830 | { |
---|
831 | return("The given function defines an isolated Singularity. |
---|
832 | The Tjurina number is "+string(tjurina(f))+". |
---|
833 | E^1[8]:z2+x3+y5+xy3z."); |
---|
834 | } |
---|
835 | } |
---|
836 | if(k==16) |
---|
837 | { |
---|
838 | return("The given function defines an isolated Singularity. |
---|
839 | The Tjurina number is "+string(tjurina(f))+". |
---|
840 | E^0[8]:z2+x3+y5."); |
---|
841 | } |
---|
842 | } |
---|
843 | } |
---|
844 | if(h==0) |
---|
845 | { |
---|
846 | list P=factorize(j); |
---|
847 | if((size(P[1])==2)&&(size(P[2])==2)&&((P[2][2])==1)) |
---|
848 | { |
---|
849 | if(k==6) |
---|
850 | { |
---|
851 | return("The given function defines an isolated Singularity. |
---|
852 | The Tjurina number is "+string(tjurina(f))+". |
---|
853 | E^1[6]:z2+x3+y2z+xyz."); |
---|
854 | } |
---|
855 | if(k==8) |
---|
856 | { |
---|
857 | return("The given function defines an isolated Singularity. |
---|
858 | The Tjurina number is "+string(tjurina(f))+". |
---|
859 | E^0[6]:z2+x3+y2z."); |
---|
860 | |
---|
861 | } |
---|
862 | } |
---|
863 | if((size(P[1])==4)&&(size(P[2])==4)) |
---|
864 | { |
---|
865 | if(k==8) |
---|
866 | { |
---|
867 | return("The given function defines an isolated Singularity. |
---|
868 | The Tjurina number is "+string(tjurina(f))+". |
---|
869 | D^0[4]:z2+x2y+xy2."); |
---|
870 | |
---|
871 | } |
---|
872 | if(k==6) |
---|
873 | { |
---|
874 | return("The given function defines an isolated Singularity. |
---|
875 | The Tjurina number is "+string(tjurina(f))+". |
---|
876 | D^1[4]:z2+x2y+xy2+xyz."); |
---|
877 | |
---|
878 | } |
---|
879 | } |
---|
880 | |
---|
881 | if((size(P[1])==3)&&(size(P[2])==3)) |
---|
882 | { |
---|
883 | if(((P[2][2])==1)&&((P[2][3])==1)) |
---|
884 | { |
---|
885 | if(k==6) |
---|
886 | { |
---|
887 | return("The given function defines an isolated Singularity. |
---|
888 | The Tjurina number is "+string(tjurina(f))+". |
---|
889 | D^1[5]:z2+x2y+y2z+xyz."); |
---|
890 | } |
---|
891 | if(k==8) |
---|
892 | { |
---|
893 | if(size(lengthBL(f))==4) |
---|
894 | { |
---|
895 | return("The given function defines an isolated Singularity. |
---|
896 | The Tjurina number is "+string(tjurina(f))+". |
---|
897 | E^3[7]:z2+x3+xy3+xyz."); |
---|
898 | } |
---|
899 | else |
---|
900 | { |
---|
901 | if(size(lengthBL(f))==5) |
---|
902 | { |
---|
903 | return("The given function defines an isolated Singularity. |
---|
904 | The Tjurina number is "+string(tjurina(f))+". |
---|
905 | E^4[8]:z2+x3+y5+xyz."); |
---|
906 | |
---|
907 | } |
---|
908 | else |
---|
909 | { |
---|
910 | return("The given function defines an isolated Singularity. |
---|
911 | The Tjurina number is "+string(tjurina(f))+". |
---|
912 | D^0[5]:z2+x2y+y2z."); |
---|
913 | |
---|
914 | } |
---|
915 | } |
---|
916 | } |
---|
917 | } |
---|
918 | |
---|
919 | } |
---|
920 | |
---|
921 | if((size(P[1])==2)&&((P[2][2])==3)) |
---|
922 | { |
---|
923 | if(k==10) |
---|
924 | { |
---|
925 | if(size(lengthBL(f))==4) |
---|
926 | { |
---|
927 | return("The given function defines an isolated Singularity. |
---|
928 | The Tjurina number is "+string(tjurina(f))+". |
---|
929 | E^2[7]:z2+x3+xy3+y3z."); |
---|
930 | |
---|
931 | } |
---|
932 | if(size(lengthBL(f))==5) |
---|
933 | { |
---|
934 | return("The given function defines an isolated Singularity. |
---|
935 | The Tjurina number is "+string(tjurina(f))+". |
---|
936 | E^3[8]:z2+x3+y5+y3z."); |
---|
937 | |
---|
938 | } |
---|
939 | } |
---|
940 | if(k==12) |
---|
941 | { |
---|
942 | |
---|
943 | if(size(lengthBL(f))==4) |
---|
944 | { |
---|
945 | return("The given function defines an isolated Singularity. |
---|
946 | The Tjurina number is "+string(tjurina(f))+". |
---|
947 | E^1[7]:z2+x3+xy3+xy3z."); |
---|
948 | |
---|
949 | } |
---|
950 | if(size(lengthBL(f))==5) |
---|
951 | { |
---|
952 | return("The given function defines an isolated Singularity. |
---|
953 | The Tjurina number is "+string(tjurina(f))+". |
---|
954 | E^2[8]:z2+x3+y5+xy2z."); |
---|
955 | |
---|
956 | } |
---|
957 | } |
---|
958 | if(k==14) |
---|
959 | { |
---|
960 | |
---|
961 | if(size(lengthBL(f))==4) |
---|
962 | { |
---|
963 | return("The given function defines an isolated Singularity. |
---|
964 | The Tjurina number is "+string(tjurina(f))+". |
---|
965 | E^0[7]:z2+x3+xy3."); |
---|
966 | |
---|
967 | } |
---|
968 | if(size(lengthBL(f))==5) |
---|
969 | { |
---|
970 | return("The given function defines an isolated Singularity. |
---|
971 | The Tjurina number is "+string(tjurina(f))+". |
---|
972 | E^1[8]:z2+x3+y5+xy3z."); |
---|
973 | |
---|
974 | } |
---|
975 | } |
---|
976 | if(k==16) |
---|
977 | { |
---|
978 | return("The given function defines an isolated Singularity. |
---|
979 | The Tjurina number is "+string(tjurina(f))+". |
---|
980 | E^0[8]:z2+x3+y5."); |
---|
981 | |
---|
982 | } |
---|
983 | } |
---|
984 | return(surDsing(f)); |
---|
985 | } |
---|
986 | } |
---|
987 | |
---|
988 | /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
989 | static proc surDsing(poly f) |
---|
990 | //====Classification of Dk singularaties======================== |
---|
991 | |
---|
992 | { |
---|
993 | //===input: A function which is an isolated singularity. |
---|
994 | //===output: Normal form to which given function is contact equivalent. |
---|
995 | |
---|
996 | def R=basering; |
---|
997 | int n=nvars(R); |
---|
998 | f=formf(f); |
---|
999 | int k=tjurina(f); |
---|
1000 | |
---|
1001 | list M=factorize(jet(f,2)); |
---|
1002 | |
---|
1003 | poly g=subst(f,leadmonom(M[1][2]),0); |
---|
1004 | poly j=jet(g,3); |
---|
1005 | if(j==0) |
---|
1006 | { |
---|
1007 | |
---|
1008 | return("The given function defines not a simple singularity. |
---|
1009 | The Tjurina Number is "+string(k)+"."); |
---|
1010 | |
---|
1011 | } |
---|
1012 | list P=factorize(j); |
---|
1013 | if((size(P[1])==4)&&(size(P[2])==4)) |
---|
1014 | { |
---|
1015 | if(k==8) |
---|
1016 | { |
---|
1017 | |
---|
1018 | return("The given function defines an isolated Singularity. |
---|
1019 | The Tjurina number is "+string(tjurina(f))+". |
---|
1020 | D^0 [4]:z2+x2y+xy2."); |
---|
1021 | |
---|
1022 | } |
---|
1023 | if(k==6) |
---|
1024 | { |
---|
1025 | return("The given function defines an isolated Singularity. |
---|
1026 | The Tjurina number is "+string(tjurina(f))+". |
---|
1027 | D^1 [4]:z2+x2y+xy2+xyz."); |
---|
1028 | |
---|
1029 | } |
---|
1030 | } |
---|
1031 | if((size(P[1])==3)&&(size(P[2])==3)) |
---|
1032 | { |
---|
1033 | |
---|
1034 | poly q=BlowUpO(f); |
---|
1035 | |
---|
1036 | if((tjurina(f)-tjurina(q))==4) |
---|
1037 | { |
---|
1038 | list Q=whichSUR(f); |
---|
1039 | |
---|
1040 | j=subst(Q[2],leadmonom(M[1][2]),0); |
---|
1041 | poly j1=jet(j,3); |
---|
1042 | list L=factorize(j1); |
---|
1043 | |
---|
1044 | if((size(L[1])==4)&&(size(L[2])==4)) |
---|
1045 | { |
---|
1046 | return("The given function defines an isolated Singularity. |
---|
1047 | The Tjurina number is "+string(tjurina(f))+". |
---|
1048 | D^0 ["+string(tjurina(f) div 2)+"]:z2+x2y+xy"+string(tjurina(f) div 4)+". "); |
---|
1049 | |
---|
1050 | } |
---|
1051 | if((size(L[1])==3)&&(size(L[2])==3)) |
---|
1052 | { |
---|
1053 | return("The given function defines an isolated Singularity. |
---|
1054 | The Tjurina number is "+string(tjurina(f))+". |
---|
1055 | D^0 ["+string((tjurina(f) div 2)+1)+"]:z2+x2y+y"+string(tjurina(f) div 4)+"z."); |
---|
1056 | |
---|
1057 | } |
---|
1058 | } |
---|
1059 | if((tjurina(f)-tjurina(q))==2) |
---|
1060 | { |
---|
1061 | |
---|
1062 | list Q=whichSUR(f); |
---|
1063 | if(Q[1]==8){Q[2]=BlowUpO(Q[2]);} |
---|
1064 | j=subst(Q[2],leadmonom(M[1][2]),0); |
---|
1065 | poly j1=jet(j,3); |
---|
1066 | |
---|
1067 | list L=factorize(j1); |
---|
1068 | if((size(L[1])==4)&&(size(L[2])==4)) |
---|
1069 | { |
---|
1070 | return("The given function defines an isolated Singularity. |
---|
1071 | The Tjurina number is "+string(tjurina(f))+". |
---|
1072 | D^"+string(findSUR(f))+" ["+string((tjurina(f)+2*findSUR(f)) div 2)+"]:z2+x2y+xy^"+string((tjurina(f)+2*findSUR(f)) div 4)+"+ xy"+string(((tjurina(f)+2*findSUR(f)) div 4)-findSUR(f))+" z."); |
---|
1073 | |
---|
1074 | } |
---|
1075 | if((size(L[1])==3)&&(size(L[2])==3)) |
---|
1076 | { |
---|
1077 | return("The given function defines an isolated Singularity. |
---|
1078 | The Tjurina number is "+string(tjurina(f))+". |
---|
1079 | D^"+string(findRSUR(f))+" ["+string(((tjurina(f)+2*findRSUR(f)) div 2)+1)+"]:z2+x2y+y"+string((tjurina(f)+2*findRSUR(f)) div 4)+"z+xy"+string(((tjurina(f)+2*findRSUR(f)) div 4)-findRSUR(f))+"z."); |
---|
1080 | |
---|
1081 | } |
---|
1082 | } |
---|
1083 | } |
---|
1084 | } |
---|
1085 | ///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1086 | static proc formf(poly f) |
---|
1087 | //=====Transform jet(f,2) in to the polynomial var(3)^2. |
---|
1088 | //=====This is the case when We are in Dk and Ek surface singularaties. |
---|
1089 | { |
---|
1090 | |
---|
1091 | poly j=jet(f,2); |
---|
1092 | list L=factorize(j); |
---|
1093 | def r=basering; |
---|
1094 | if(leadmonom(L[1][2])==var(1)) |
---|
1095 | { |
---|
1096 | f=subst(f,var(1),L[1][2]); |
---|
1097 | map phi=r,var(3),var(2),var(1); |
---|
1098 | return(phi(f)); |
---|
1099 | |
---|
1100 | } |
---|
1101 | if(leadmonom(L[1][2])==var(2)) |
---|
1102 | { |
---|
1103 | f=subst(f,var(2),L[1][2]); |
---|
1104 | map phi=r,var(1),var(3),var(2); |
---|
1105 | return(phi(f)); |
---|
1106 | |
---|
1107 | } |
---|
1108 | if(leadmonom(L[1][2])==var(3)) |
---|
1109 | { |
---|
1110 | return(f); |
---|
1111 | } |
---|
1112 | } |
---|
1113 | /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1114 | static proc whichSUR(poly f) |
---|
1115 | //====This procedure is required to separate the Surface Case D_2m from D_2m+1 as discribes in [2]. |
---|
1116 | { |
---|
1117 | int d=tjurina(f); |
---|
1118 | list L; |
---|
1119 | while(1) |
---|
1120 | { |
---|
1121 | f=BlowUpO(f); |
---|
1122 | d=tjurina(f); |
---|
1123 | |
---|
1124 | if((d==6)||(d==8)) |
---|
1125 | { |
---|
1126 | L[1]=d; |
---|
1127 | L[2]=f; |
---|
1128 | |
---|
1129 | return(L); |
---|
1130 | } |
---|
1131 | } |
---|
1132 | return(d); |
---|
1133 | } |
---|
1134 | /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1135 | static proc findSUR(poly f)//D2m,r |
---|
1136 | //====Number of blowup required in order either to get the difference equal to 4 or the Tjurina number is less than 6. |
---|
1137 | { |
---|
1138 | |
---|
1139 | int a, r,b; |
---|
1140 | while(1) |
---|
1141 | { |
---|
1142 | r++; |
---|
1143 | a=tjurina(f); |
---|
1144 | if(a<=1) |
---|
1145 | { |
---|
1146 | return(r); |
---|
1147 | } |
---|
1148 | f=BlowUpO(f); |
---|
1149 | b=tjurina(f); |
---|
1150 | if((a-b)==4) |
---|
1151 | { |
---|
1152 | if(b==2) |
---|
1153 | { |
---|
1154 | return(r); |
---|
1155 | } |
---|
1156 | return(r-1); |
---|
1157 | } |
---|
1158 | if(b<6) |
---|
1159 | { |
---|
1160 | return(r-1); |
---|
1161 | } |
---|
1162 | |
---|
1163 | } |
---|
1164 | } |
---|
1165 | ///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1166 | static proc findRSUR(poly f)//D2m+1,r |
---|
1167 | //====Calculate the number of blow ups required by a polynomial in order to get the difference 4. |
---|
1168 | { |
---|
1169 | |
---|
1170 | int a, r,b; |
---|
1171 | while(1) |
---|
1172 | { |
---|
1173 | r++; |
---|
1174 | a=tjurina(f); |
---|
1175 | |
---|
1176 | if(a==2) |
---|
1177 | { |
---|
1178 | return(r); |
---|
1179 | } |
---|
1180 | |
---|
1181 | f=BlowUpO(f); |
---|
1182 | b=tjurina(f); |
---|
1183 | |
---|
1184 | if((a-b)==4) |
---|
1185 | { |
---|
1186 | return(r-1); |
---|
1187 | } |
---|
1188 | if(b<6) |
---|
1189 | { |
---|
1190 | return(r); |
---|
1191 | } |
---|
1192 | } |
---|
1193 | } |
---|
1194 | ///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1195 | static proc BlowUpO(poly f) |
---|
1196 | //====Gives the procedure of Blowing up ingeneral |
---|
1197 | { |
---|
1198 | //=== input f smooth or isolated singularity at zero |
---|
1199 | //=== output var(1) or poly with isolated singularity at zero, the transformation |
---|
1200 | //=== of the singularity of the blowing up to zero |
---|
1201 | def R=basering; |
---|
1202 | def S=changeord(list(list("ds",1:nvars(basering)))); |
---|
1203 | setring S; |
---|
1204 | int n=nvars(basering); |
---|
1205 | int i,t,c,d,e,j,k; |
---|
1206 | poly f=imap(R,f); |
---|
1207 | if(deg(lead(f))<=1){setring R;return(var(1));} |
---|
1208 | poly p; |
---|
1209 | def T=changeord(list(list("lp",1:nvars(basering)))); |
---|
1210 | setring T; |
---|
1211 | list L; |
---|
1212 | map phi,psi; |
---|
1213 | ideal mphi, mpsi,sing; |
---|
1214 | poly p,q,m,l; |
---|
1215 | poly h=var(1); |
---|
1216 | //=========== blow up======================================================= |
---|
1217 | for(i=1;i<=n;i++) |
---|
1218 | { |
---|
1219 | mphi=var(i)*maxideal(1); |
---|
1220 | mphi[i]=var(i); |
---|
1221 | phi=S,mphi; |
---|
1222 | p=phi(f); |
---|
1223 | q=p/var(i); |
---|
1224 | while(size(p)==size(q)) |
---|
1225 | { |
---|
1226 | p=q; |
---|
1227 | q=q/var(i); |
---|
1228 | } |
---|
1229 | //=============== p is the strict transform var(i) exceptional divisor ==== |
---|
1230 | //=============== analysis of singularities ================================ |
---|
1231 | sing=jacob(p),p,var(i); |
---|
1232 | sing=radical(sing); |
---|
1233 | option(redSB); |
---|
1234 | sing=std(sing); |
---|
1235 | if(dim(sing)>0){ERROR("not simple");} |
---|
1236 | sing=std(simplify(sing,1)); |
---|
1237 | if(dim(sing)==0) |
---|
1238 | { |
---|
1239 | if(vdim(sing)==1) |
---|
1240 | { |
---|
1241 | mpsi=std(maxideal(1)); |
---|
1242 | for(k=1;k<=n;k++){mpsi[k]=var(k)-sing[n-k+1][2];} |
---|
1243 | psi=T,mpsi; |
---|
1244 | p=psi(p); |
---|
1245 | } |
---|
1246 | else |
---|
1247 | { //this can only happen in case of a D-singularity |
---|
1248 | L=minAssGTZ(sing); |
---|
1249 | d=0; |
---|
1250 | for(j=1;j<=size(L);j++) |
---|
1251 | { |
---|
1252 | sing=std(L[j]); |
---|
1253 | sing=std(simplify(sing,1)); |
---|
1254 | if(vdim(sing)!=1){ERROR("something is wrong in blowUpO");} |
---|
1255 | mpsi=std(maxideal(1)); |
---|
1256 | for(k=1;k<=n;k++){mpsi[k]=var(k)-sing[n-k+1][2];} |
---|
1257 | psi=T,mpsi; |
---|
1258 | m=psi(p); |
---|
1259 | setring S; |
---|
1260 | p=imap(T,m); |
---|
1261 | e=tjurina(p); |
---|
1262 | setring T; |
---|
1263 | if(e>d) |
---|
1264 | { |
---|
1265 | d=e; |
---|
1266 | l=m; |
---|
1267 | } |
---|
1268 | } |
---|
1269 | p=l; |
---|
1270 | } |
---|
1271 | setring S; |
---|
1272 | p=imap(T,p); |
---|
1273 | c=tjurina(p); |
---|
1274 | setring T; |
---|
1275 | if(c>t) |
---|
1276 | { |
---|
1277 | t=c; |
---|
1278 | h=p; |
---|
1279 | } |
---|
1280 | } |
---|
1281 | } |
---|
1282 | setring R; |
---|
1283 | poly h=imap(T,h); |
---|
1284 | return(h); |
---|
1285 | } |
---|
1286 | //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1287 | static proc lengthBL(poly f) |
---|
1288 | //====Return the list of Tjurina numbers of each blowing up in the resolution before it becomes smooth. |
---|
1289 | { |
---|
1290 | list L; |
---|
1291 | int i=1; |
---|
1292 | int d=tjurina(f); |
---|
1293 | while(d>=2) |
---|
1294 | { |
---|
1295 | f=BlowUpO(f); |
---|
1296 | L[i]=d; |
---|
1297 | d=tjurina(f); |
---|
1298 | |
---|
1299 | i=i+1; |
---|
1300 | } |
---|
1301 | return(L); |
---|
1302 | } |
---|
1303 | ///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1304 | static proc curCeq2(poly f) |
---|
1305 | //====Classification of Curves in Case when p=2. |
---|
1306 | { |
---|
1307 | |
---|
1308 | int c; |
---|
1309 | int k; |
---|
1310 | number m,a,b; |
---|
1311 | ideal I; |
---|
1312 | int d; |
---|
1313 | k=tjurina(f); |
---|
1314 | if(k==-1) |
---|
1315 | { |
---|
1316 | return("The given function defines not an isolated singularity"); |
---|
1317 | |
---|
1318 | } |
---|
1319 | if(deg(lead(f))==2) |
---|
1320 | { |
---|
1321 | poly f1=jet(f,2); |
---|
1322 | list T=factorize(f1); |
---|
1323 | if(size(T[1])==3) |
---|
1324 | { |
---|
1325 | return("The given function defines an isolated Singularity. |
---|
1326 | The Tjurina number is "+string(tjurina(f))+". |
---|
1327 | A1:x2+xy."); |
---|
1328 | |
---|
1329 | } |
---|
1330 | if(size(T[1])==2) |
---|
1331 | { |
---|
1332 | poly g=BlowUpO(f); |
---|
1333 | if((tjurina(f)-tjurina(g))==4) |
---|
1334 | { |
---|
1335 | return("The given function defines an isolated Singularity. |
---|
1336 | The Tjurina number is "+string(tjurina(f))+". |
---|
1337 | A["+string(k div 2)+"]:x2+y "+string(tjurina(f) div 2+1)+". where r=0. "); |
---|
1338 | } |
---|
1339 | else |
---|
1340 | { |
---|
1341 | d=whichtru(f); |
---|
1342 | if(d==1) |
---|
1343 | { |
---|
1344 | if((k mod 4)==0) |
---|
1345 | { |
---|
1346 | |
---|
1347 | return("The given function defines an isolated Singularity. |
---|
1348 | The Tjurina number is "+string(tjurina(f))+". |
---|
1349 | A["+string(2*(tjurina(f) div 2)-1)+"]:x2+xy"+string(tjurina(f) div 2)+"."); |
---|
1350 | } |
---|
1351 | else |
---|
1352 | { |
---|
1353 | return("The given function defines an isolated Singularity. |
---|
1354 | The Tjurina number is "+string(tjurina(f))+". |
---|
1355 | A["+string((2*(tjurina(f)+1) div 2)-1)+"]:x2+xy"+string((tjurina(f)+1) div 2)+"."); |
---|
1356 | |
---|
1357 | } |
---|
1358 | } |
---|
1359 | if(d==0) |
---|
1360 | { |
---|
1361 | if((findR(f) mod 2)==0) |
---|
1362 | { |
---|
1363 | return("The given function defines an isolated Singularity. |
---|
1364 | The Tjurina number is "+string(tjurina(f))+". |
---|
1365 | A^"+string(findR(f))+" ["+string(2*((tjurina(f)+2*findR(f)) div 4))+"]:x2+y"+string(((tjurina(f)+2*findR(f)) div 2)+1)+"+xy"+string(((tjurina(f)+2*findR(f)) div 2)-findR(f))+"."); |
---|
1366 | } |
---|
1367 | if((findR(f) mod 2)!=0) |
---|
1368 | { |
---|
1369 | return("The given function defines an isolated Singularity. |
---|
1370 | The Tjurina number is "+string(tjurina(f))+". |
---|
1371 | A^"+string(findR(f))+" ["+string(2*(((tjurina(f)+1)+2*findR(f)) div 4))+"]:x2+y"+string((((tjurina(f)+1)+2*findR(f)) div 2)+1)+"+xy"+string((((tjurina(f)+1)+2*findR(f)) div 2)-findR(f))+"."); |
---|
1372 | |
---|
1373 | } |
---|
1374 | } |
---|
1375 | |
---|
1376 | } |
---|
1377 | |
---|
1378 | } |
---|
1379 | |
---|
1380 | } |
---|
1381 | if(deg(lead(f))==3) |
---|
1382 | { |
---|
1383 | if(jet(f,3)==0) |
---|
1384 | {return("The given function defines not a simple Singularity. |
---|
1385 | The Tjurina number is "+string(tjurina(f))+".")} |
---|
1386 | poly f1=jet(f,3); |
---|
1387 | list L=factorize(f1); |
---|
1388 | if(size(L[1])==4) |
---|
1389 | { |
---|
1390 | return("The given function defines an isolated Singularity. |
---|
1391 | The Tjurina number is "+string(tjurina(f))+". |
---|
1392 | D[4]:x2y+xy2."); |
---|
1393 | |
---|
1394 | } |
---|
1395 | if(size(L[1])==3) |
---|
1396 | { |
---|
1397 | poly g=BlowUpO(f); |
---|
1398 | if((tjurina(f)-tjurina(g))==8) |
---|
1399 | { |
---|
1400 | return("The given function defines an isolated Singularity. |
---|
1401 | The Tjurina number is "+string(tjurina(f))+". |
---|
1402 | D^0["+string((tjurina(f) div 2)+1)+"]:x2y+y"+string(tjurina(f) div 2)+". where m="+string(tjurina(f)/4)+"."); |
---|
1403 | |
---|
1404 | } |
---|
1405 | else |
---|
1406 | { |
---|
1407 | if(whichtru(f)==1) |
---|
1408 | { |
---|
1409 | return("The given function defines an isolated Singularity. |
---|
1410 | The Tjurina number is "+string(tjurina(f))+". |
---|
1411 | D["+string(tjurina(f))+"]:x2y+xy"+string(tjurina(f) div 2)+". "); |
---|
1412 | } |
---|
1413 | if(whichtru(f)==0) |
---|
1414 | { |
---|
1415 | if((findRD(f) mod 2)==0) |
---|
1416 | { |
---|
1417 | return("The given function defines an isolated Singularity. |
---|
1418 | The Tjurina number is "+string(tjurina(f))+". |
---|
1419 | D^"+string(findRD(f))+"["+string(2*((tjurina(f) div 2+findRD(f)) div 2)+1)+"]:x2y+y"+string((tjurina(f) div 2+findRD(f)))+"+xy"+string((tjurina(f) div 2+findRD(f))-findRD(f))+" where even r="+string(findRD(f))+"."); |
---|
1420 | } |
---|
1421 | else |
---|
1422 | { |
---|
1423 | return("The given function defines an isolated Singularity. |
---|
1424 | The Tjurina number is "+string(tjurina(f))+". |
---|
1425 | D^"+string(findRD(f))+"["+string(2*(((tjurina(f)+1) div 2+findRD(f)) div 2)+1)+"]:x2y+y"+string(((tjurina(f)+1) div 2+findRD(f)))+"+xy"+string(((tjurina(f)+1) div 2+findRD(f))-findRD(f))+",odd r="+string(findRD(f))+"."); |
---|
1426 | } |
---|
1427 | } |
---|
1428 | } |
---|
1429 | } |
---|
1430 | a=leadcoef(f); |
---|
1431 | b=leadcoef(f-lead(f)); |
---|
1432 | f=subst(f,var(1),1/a*(var(1)-a*b*var(2))); |
---|
1433 | I=var(1),var(2)^2; |
---|
1434 | I=std(I^3); |
---|
1435 | if(reduce(f,I)!=0) |
---|
1436 | { |
---|
1437 | if(k==6) |
---|
1438 | { |
---|
1439 | return("The given function defines an isolated Singularity. |
---|
1440 | The Tjurina number is "+string(tjurina(f))+". |
---|
1441 | E^1[6]:x3+y4+xy3."); |
---|
1442 | |
---|
1443 | } |
---|
1444 | if(k==7) |
---|
1445 | { |
---|
1446 | return("The given function defines an isolated Singularity. |
---|
1447 | The Tjurina number is "+string(tjurina(f))+". |
---|
1448 | E[7]:x3+xy3."); |
---|
1449 | } |
---|
1450 | if(k==8) |
---|
1451 | { |
---|
1452 | poly t=BlowUpO(f); |
---|
1453 | if(tjurina(t)==0) |
---|
1454 | { |
---|
1455 | return("The given function defines an isolated Singularity. |
---|
1456 | The Tjurina number is "+string(tjurina(f))+". |
---|
1457 | E^0[6]:x3+y4."); |
---|
1458 | } |
---|
1459 | else |
---|
1460 | { |
---|
1461 | return("The given function defines an isolated Singularity. |
---|
1462 | The Tjurina number is "+string(tjurina(f))+". |
---|
1463 | E[8]:x3+y5."); |
---|
1464 | } |
---|
1465 | } |
---|
1466 | } |
---|
1467 | else |
---|
1468 | {return("The given function defines not a simple singularity. |
---|
1469 | The Tjurina number is "+string(tjurina(f))+".");} |
---|
1470 | } |
---|
1471 | |
---|
1472 | if(deg(lead(f))>3) |
---|
1473 | { |
---|
1474 | |
---|
1475 | return("The given function defines not a simple singularity. |
---|
1476 | The Tjurina number is "+string(tjurina(f))+"."); |
---|
1477 | |
---|
1478 | } |
---|
1479 | } |
---|
1480 | example |
---|
1481 | { |
---|
1482 | "EXAMPLE:"; echo=2; |
---|
1483 | ring R=2,(x,y),Ds; |
---|
1484 | curCeq2(x3+x2y+xy2+y3+xy3+y4); |
---|
1485 | |
---|
1486 | } |
---|
1487 | /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1488 | static proc whichtru(poly f) |
---|
1489 | //====Gives the information that we do successive blowups and the last singularaty before becoming smmoth has Tjurina |
---|
1490 | //====number 1 if not than returns 0.(returns either a singularaty or smooth surve)============ |
---|
1491 | { |
---|
1492 | int d=tjurina(f); |
---|
1493 | while(1) |
---|
1494 | { |
---|
1495 | f=BlowUpO(f); |
---|
1496 | d=tjurina(f); |
---|
1497 | if(d==1) |
---|
1498 | { |
---|
1499 | return(d); |
---|
1500 | } |
---|
1501 | if(d==0) |
---|
1502 | { |
---|
1503 | return(d); |
---|
1504 | } |
---|
1505 | } |
---|
1506 | return(d); |
---|
1507 | } |
---|
1508 | /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1509 | static proc findR(poly f) |
---|
1510 | //====Find the number of blowups required by polynomial such that |
---|
1511 | //====difference between two consecutive blowups become 4 |
---|
1512 | //====this procedure find the value of r in case of Ak singularaties as in [1]. |
---|
1513 | { |
---|
1514 | int a, r,b; |
---|
1515 | while(1) |
---|
1516 | { |
---|
1517 | r++; |
---|
1518 | a=tjurina(f); |
---|
1519 | if(a<=1) |
---|
1520 | { |
---|
1521 | return(-1); |
---|
1522 | } |
---|
1523 | f=BlowUpO(f); |
---|
1524 | |
---|
1525 | b=tjurina(f); |
---|
1526 | |
---|
1527 | if((a-b)==4) |
---|
1528 | { |
---|
1529 | return(r-1); |
---|
1530 | } |
---|
1531 | } |
---|
1532 | } |
---|
1533 | //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1534 | static proc findRD(poly p) |
---|
1535 | //=====Find the number of blowups required by polynomial such |
---|
1536 | //=====that difference between two consecutive blowups become 4 |
---|
1537 | //=====this procedure find the value of r in case of Dk singularaties as in [1]. |
---|
1538 | { |
---|
1539 | |
---|
1540 | int k=tjurina(p); |
---|
1541 | poly q=BlowUpO(p); |
---|
1542 | int t=tjurina(q); |
---|
1543 | int r; |
---|
1544 | |
---|
1545 | if((k mod 4)==0) |
---|
1546 | { |
---|
1547 | r=findR(BlowUpO(p))+2; |
---|
1548 | return(r); |
---|
1549 | } |
---|
1550 | if((k mod 4 !=0)&&(t mod 4==0)) |
---|
1551 | { |
---|
1552 | if(t!=0) |
---|
1553 | { |
---|
1554 | r=findR(BlowUpO(p))+1; |
---|
1555 | return(r); |
---|
1556 | } |
---|
1557 | } |
---|
1558 | if((k mod 2)==0) |
---|
1559 | { |
---|
1560 | r=findR(BlowUpO(p))+2; |
---|
1561 | return(r); |
---|
1562 | } |
---|
1563 | |
---|
1564 | } |
---|
1565 | ////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1566 | static proc splitting_SQUA(poly p, int b) |
---|
1567 | { |
---|
1568 | // assumes that p=x_i^2+h, h of order >=3 |
---|
1569 | // returns q(x_1,??,x_n) such that x_i^2 +q is right equivalent to p mod |
---|
1570 | // <x_1,??,x_n>^b and x_i is only linear in q |
---|
1571 | def R=basering; int n=nvars(R); |
---|
1572 | ideal M=maxideal(1); |
---|
1573 | int j; |
---|
1574 | map phi; |
---|
1575 | p=jet(p,b); |
---|
1576 | p=simplify(p,1); |
---|
1577 | list T=FindSQUA(p); |
---|
1578 | int i=T[1]; |
---|
1579 | poly q=p/var(i)^2; |
---|
1580 | while(q!=1) |
---|
1581 | { |
---|
1582 | for(j=1;j<=n;j++) |
---|
1583 | { |
---|
1584 | M[j]=var(j)*q; |
---|
1585 | } |
---|
1586 | phi=R,M; |
---|
1587 | p=phi(p); |
---|
1588 | p=p*inverseUnit(q,b); |
---|
1589 | p=jet(p,b); |
---|
1590 | q=p/var(i)^2; |
---|
1591 | } |
---|
1592 | return(p); |
---|
1593 | } |
---|
1594 | /////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1595 | static proc splitting_two2(poly p, int i, int j, int b) |
---|
1596 | { |
---|
1597 | // assumes that p=x_i*x_j+h, no x_i^2, no in h, h of order >=2 |
---|
1598 | // returns q(x_1,??x_i-1,x_i+1,...,x_j-1,x_j+1,...,x_n) such that x_i*x_j +q is right equivalent to p mod |
---|
1599 | // <x_1,??,x_n>^b |
---|
1600 | if(b<2){b=2;} |
---|
1601 | def R=basering; |
---|
1602 | poly q=jet((p-var(i)*var(j))/var(i),b); |
---|
1603 | while(q!=0) |
---|
1604 | { |
---|
1605 | p=quickSubst2(p,var(j)-q,j,b); |
---|
1606 | q=jet((p-var(i)*var(j))/var(i),b); |
---|
1607 | } |
---|
1608 | return(simplify(substitute(jet(p,b),var(j),0),1)); //make the leading coefficient 1 |
---|
1609 | |
---|
1610 | } |
---|
1611 | //////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1612 | static proc quickSubst2(poly h, poly r, int i, int b) |
---|
1613 | { |
---|
1614 | //=== assume h, r is in Q[x_1,...,x_n], computes jet(h(x_i=r),b) |
---|
1615 | h=jet(h,b); |
---|
1616 | r=jet(r,b); |
---|
1617 | matrix M=coef(h,var(i)); |
---|
1618 | poly q = 1; |
---|
1619 | int j,k,d; |
---|
1620 | intvec v; |
---|
1621 | d=deg(M[1,1]); |
---|
1622 | v[d+1]=1; |
---|
1623 | for(k = 2; k <= ncols(M); k++) |
---|
1624 | { |
---|
1625 | v[deg(M[1,k])+1]=1; |
---|
1626 | } |
---|
1627 | h=0; |
---|
1628 | for(k=1;k<=d+1;k++) |
---|
1629 | { |
---|
1630 | if(v[k]==1) |
---|
1631 | { |
---|
1632 | h=h+jet(q*M[2,ncols(M)-j],b); |
---|
1633 | j++; |
---|
1634 | } |
---|
1635 | q=jet(q*r,b); |
---|
1636 | } |
---|
1637 | return(h); |
---|
1638 | } |
---|
1639 | //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1640 | static proc FindPRO(poly f) |
---|
1641 | { |
---|
1642 | //input the polynomial |
---|
1643 | // output is a list T where T[1]=the first variable which appear as product in the quadraic form. |
---|
1644 | // T[2]=the second variable which appear as product with the var(T[1]) in the quadraic form. |
---|
1645 | def R=basering; |
---|
1646 | int n=nvars(R); |
---|
1647 | int i,j,k; |
---|
1648 | list T; |
---|
1649 | for(i=1;i<=n;i++) |
---|
1650 | { |
---|
1651 | for(k=i+1;k<=n;k++) |
---|
1652 | { |
---|
1653 | for(j=1;j<=size(f);j++) |
---|
1654 | { |
---|
1655 | if(leadmonom(f[j])==var(i)*var(k)) |
---|
1656 | { |
---|
1657 | |
---|
1658 | T[1]=i; |
---|
1659 | T[2]=k; |
---|
1660 | return(T); |
---|
1661 | } |
---|
1662 | } |
---|
1663 | } |
---|
1664 | |
---|
1665 | } |
---|
1666 | return(T); |
---|
1667 | } |
---|
1668 | /////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1669 | static proc FindSQUA(poly f) |
---|
1670 | { |
---|
1671 | //input the polynomial |
---|
1672 | // output is a list T where T[1]=the first variable which appears as a square in the quadraic part of f. |
---|
1673 | // T[2]= the number of squares appearing in the quadraic part of f. |
---|
1674 | def R=basering; |
---|
1675 | int n=nvars(R); |
---|
1676 | int i,j,k; |
---|
1677 | list T; |
---|
1678 | for(i=1;i<=n;i++) |
---|
1679 | { |
---|
1680 | for(j=1;j<=size(f);j++) |
---|
1681 | { |
---|
1682 | if(leadmonom(f[j])==var(i)^2) |
---|
1683 | { |
---|
1684 | k++; |
---|
1685 | T[1]=i; |
---|
1686 | T[2]=k; |
---|
1687 | return(T); |
---|
1688 | } |
---|
1689 | } |
---|
1690 | } |
---|
1691 | return(T); |
---|
1692 | } |
---|
1693 | ///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1694 | static proc findVAR(poly f) |
---|
1695 | { |
---|
1696 | //input: a poly f |
---|
1697 | //output a list L=v,k v an intvec, v[i]=0, if depends on var(i), v[i]=1 else |
---|
1698 | //k is the number of variables occurring in f |
---|
1699 | intvec v; |
---|
1700 | int i,k;int n=nvars(basering); |
---|
1701 | list L; |
---|
1702 | for(i=1;i<=n;i++) |
---|
1703 | { |
---|
1704 | if(f==subst(f,var(i),0)) |
---|
1705 | { |
---|
1706 | v[i]=1; |
---|
1707 | } |
---|
1708 | else |
---|
1709 | { |
---|
1710 | v[i]=0; |
---|
1711 | k++; |
---|
1712 | } |
---|
1713 | } |
---|
1714 | L=v,k; |
---|
1715 | return(L); |
---|
1716 | } |
---|
1717 | ////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1718 | static proc redvar2(poly p) |
---|
1719 | { |
---|
1720 | //input a polynomial depending on x_i_1,...,x_i_k |
---|
1721 | //output a new ring with k variables containing the polynomial |
---|
1722 | def R=basering; |
---|
1723 | int n=nvars(R); |
---|
1724 | list N=findVAR(p); |
---|
1725 | int m=N[2]; |
---|
1726 | intvec v=N[1]; |
---|
1727 | int l; |
---|
1728 | int i; |
---|
1729 | if(n==m) |
---|
1730 | { |
---|
1731 | |
---|
1732 | def S=R; |
---|
1733 | if(defined(h)) |
---|
1734 | { |
---|
1735 | kill h; |
---|
1736 | } |
---|
1737 | poly h; |
---|
1738 | export h; |
---|
1739 | |
---|
1740 | } |
---|
1741 | else |
---|
1742 | { |
---|
1743 | def S=defring("2",m,"u","ds"); |
---|
1744 | setring S; |
---|
1745 | ideal M; |
---|
1746 | for(i=1;i<=n;i++) |
---|
1747 | { |
---|
1748 | if(v[i]==1) |
---|
1749 | { |
---|
1750 | M[i]=0; |
---|
1751 | } |
---|
1752 | else |
---|
1753 | { |
---|
1754 | l++; |
---|
1755 | M[i]=var(l); |
---|
1756 | } |
---|
1757 | } |
---|
1758 | map phi=R,M; |
---|
1759 | poly p=phi(p); |
---|
1760 | export p; |
---|
1761 | setring R; |
---|
1762 | } |
---|
1763 | return(S); |
---|
1764 | } |
---|
1765 | //////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1766 | static proc inverseUnit(poly q,int b) |
---|
1767 | { |
---|
1768 | //input a polynomial q with non-zero constant part |
---|
1769 | //output the inverse of q up to order b as a power series |
---|
1770 | number c=leadcoef(q); |
---|
1771 | q=q/c; |
---|
1772 | int i; |
---|
1773 | poly u=1; |
---|
1774 | poly a=q-1; |
---|
1775 | poly s=-a; |
---|
1776 | for(i=1;i<=b;i++) |
---|
1777 | { |
---|
1778 | u=u+s; |
---|
1779 | s=s*a; |
---|
1780 | } |
---|
1781 | return(u/c); |
---|
1782 | } |
---|
1783 | ///////////////////////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1784 | static proc splittingLchar2(poly f) |
---|
1785 | { |
---|
1786 | //the quadratic part of the splitting lemma in characteristic 2 |
---|
1787 | //input: poly f |
---|
1788 | //output: a polynomial right equivalent to the input polynomial such that the quadratic part is in normal form |
---|
1789 | poly p=jet(f,2); |
---|
1790 | if(p==0){return(f);} |
---|
1791 | if(!homog(p)){return(jet(f,1));} |
---|
1792 | def R=basering; |
---|
1793 | def S=changeord(list(list("lp",1:nvars(basering)))); |
---|
1794 | setring S; |
---|
1795 | ideal ma=maxideal(1); |
---|
1796 | number a; |
---|
1797 | int c=ringlist(S)[1][1]; |
---|
1798 | poly f=imap(R,f); |
---|
1799 | poly p=imap(R,p); |
---|
1800 | poly h,q; |
---|
1801 | intvec v=leadexp(p); |
---|
1802 | int i,j,k; |
---|
1803 | intvec w; |
---|
1804 | for(k=1;k<=size(v);k++) |
---|
1805 | { |
---|
1806 | if(v[k]!=0){w[size(w)+1]=k;} |
---|
1807 | } |
---|
1808 | i=w[2]; |
---|
1809 | if(size(w)==3){j=w[3];} |
---|
1810 | if(j>0) |
---|
1811 | { |
---|
1812 | a=1/leadcoef(p); |
---|
1813 | ma[j]=a*(var(j)+(p-lead(p))/var(i)); |
---|
1814 | map phi=S,ma; |
---|
1815 | p=phi(p); |
---|
1816 | f=phi(f); |
---|
1817 | q=(p-lead(p))/var(j); |
---|
1818 | ma=maxideal(1); |
---|
1819 | ma[i]=var(i)+q; |
---|
1820 | phi=S,ma; |
---|
1821 | f=phi(f); |
---|
1822 | h=splittingLchar2(f-var(i)*var(j)); |
---|
1823 | if(jet(h,2)==0) |
---|
1824 | { |
---|
1825 | setring R; |
---|
1826 | f=imap(S,f); |
---|
1827 | return(f); |
---|
1828 | } |
---|
1829 | v=leadexp(h); |
---|
1830 | for(k=1;k<=size(v);k++) |
---|
1831 | { |
---|
1832 | if(v[k]!=0){break;} |
---|
1833 | } |
---|
1834 | if(v[k]==2) |
---|
1835 | { |
---|
1836 | ma=maxideal(1); |
---|
1837 | ma[i]=var(j); |
---|
1838 | ma[j]=var(k); |
---|
1839 | ma[k]=var(i); |
---|
1840 | phi=S,ma; |
---|
1841 | h=phi(h); |
---|
1842 | f=h+var(j)*var(k); |
---|
1843 | } |
---|
1844 | else |
---|
1845 | { |
---|
1846 | f=h+var(i)*var(j); |
---|
1847 | } |
---|
1848 | setring R; |
---|
1849 | f=imap(S,f); |
---|
1850 | return(f); |
---|
1851 | } |
---|
1852 | |
---|
1853 | a= leadcoef(p)^(c div 2); |
---|
1854 | ma[i]=1/a*var(i); |
---|
1855 | map phi=S,ma; |
---|
1856 | p=phi(p); |
---|
1857 | f=phi(f); |
---|
1858 | q=p/var(i); |
---|
1859 | if(size(q)==1) |
---|
1860 | { |
---|
1861 | if(p==var(i)^2) |
---|
1862 | { |
---|
1863 | setring R; |
---|
1864 | f=imap(S,f); |
---|
1865 | return(f); |
---|
1866 | } |
---|
1867 | h=splittingLchar2(f-var(i)^2); |
---|
1868 | p=jet(h,2); |
---|
1869 | v=leadexp(p); |
---|
1870 | h=h+var(i)^2; |
---|
1871 | |
---|
1872 | for(k=1;k<=size(v);k++) |
---|
1873 | { |
---|
1874 | if(v[k]!=0){j=k;break;} |
---|
1875 | } |
---|
1876 | if(v[j]==2) |
---|
1877 | { |
---|
1878 | ma=maxideal(1); |
---|
1879 | ma[i]=var(i)+var(j); |
---|
1880 | phi=S,ma; |
---|
1881 | h=phi(h); |
---|
1882 | h=splittingLchar2(h); |
---|
1883 | } |
---|
1884 | setring R; |
---|
1885 | f=imap(S,h); |
---|
1886 | return(f); |
---|
1887 | } |
---|
1888 | else |
---|
1889 | { |
---|
1890 | q=q-lead(q); |
---|
1891 | v=leadexp(q); |
---|
1892 | for(k=1;k<=size(v);k++) |
---|
1893 | { |
---|
1894 | if(v[k]!=0){j=k;break;} |
---|
1895 | } |
---|
1896 | ma=maxideal(1); |
---|
1897 | ma[j]=1/leadcoef(q)*(var(j)+q-lead(q)); |
---|
1898 | phi=S,ma; |
---|
1899 | f=phi(f); |
---|
1900 | ma=maxideal(1); |
---|
1901 | ma[j]=var(i); |
---|
1902 | ma[i]=var(j); |
---|
1903 | phi=S,ma; |
---|
1904 | f=phi(f); |
---|
1905 | p=jet(f,2); |
---|
1906 | if(leadmonom(p)==var(i)^2) |
---|
1907 | { |
---|
1908 | f=phi(f); |
---|
1909 | p=jet(f,2); |
---|
1910 | ma=maxideal(1); |
---|
1911 | ma[i]=p/var(j)-var(j); |
---|
1912 | phi=S,ma; |
---|
1913 | f=phi(f); |
---|
1914 | h=splittingLchar2(f-var(i)^2-var(i)*var(j)-var(j)^2); |
---|
1915 | p=jet(h,2); |
---|
1916 | |
---|
1917 | f=var(i)^2+var(i)*var(j)+var(j)^2+h; |
---|
1918 | if(p!=0) |
---|
1919 | { |
---|
1920 | v=leadexp(p); |
---|
1921 | for(k=1;k<=size(v);k++) |
---|
1922 | { |
---|
1923 | if(v[k]!=0){break;} |
---|
1924 | } |
---|
1925 | if(v[k]==2) |
---|
1926 | { |
---|
1927 | ma=maxideal(1); |
---|
1928 | ma[k]=var(i)+var(k); |
---|
1929 | phi=S,ma; |
---|
1930 | f=phi(f); |
---|
1931 | setring R; |
---|
1932 | f=imap(S,f); |
---|
1933 | return(splittingLchar2(f)); |
---|
1934 | } |
---|
1935 | setring R; |
---|
1936 | f=imap(S,f); |
---|
1937 | return(f); |
---|
1938 | } |
---|
1939 | else //we return (x_i)^2+x_i*x_j+(x_j)^2 since we need a field extension |
---|
1940 | { //to transform it to x_i*x_j |
---|
1941 | setring R; |
---|
1942 | f=imap(S,f); |
---|
1943 | return(f); |
---|
1944 | } |
---|
1945 | } |
---|
1946 | setring R; |
---|
1947 | f=imap(S,f); |
---|
1948 | return(splittingLchar2(f)); |
---|
1949 | } |
---|
1950 | } |
---|
1951 | /////////////////////////////////////////////////////////////////////////////////////////////////////////// |
---|
1952 | /* |
---|
1953 | =============================== Examples for characteristic 2 ========================================== |
---|
1954 | ring r=2,(x,y,z,w,t,v),ds; |
---|
1955 | poly p=xy+zw+t5+v3; |
---|
1956 | classifyCeq(p); |
---|
1957 | |
---|
1958 | map phi=r,v,t,w,z,y,x; |
---|
1959 | poly c=1+x+y+z; |
---|
1960 | poly q=c*p; |
---|
1961 | q=phi(q); |
---|
1962 | q=zw+tv+x3+zw2+zwt+zwv+wtv+t2v+tv2+x3w+x3t+x3v+y5+y5w+y5t+y5v; |
---|
1963 | classifyCeq(q); |
---|
1964 | |
---|
1965 | poly p=ztv+x3+zw2+zwt+zwv+wtv+tv2+tv2+x3w+x3t+x3v+xy3+xy3w+xy3v; |
---|
1966 | classifyCeq(p); |
---|
1967 | |
---|
1968 | poly p=zw+tv+x3+zw2+zwt+wtv+t2v+tv2+x3w+x3t+x3v+y4+y4w+y4t+y4v; |
---|
1969 | classifyCeq(p);//E^0[6] |
---|
1970 | |
---|
1971 | poly p=zw+tv+x3+zw2+zwt+zwv+wtv+t2v+tv2+x3w+x3t+x3v+xy3+y4+xy3w+xy3t+xy3v+y4w+y4t+y4v; |
---|
1972 | classifyCeq(p);//E^1[6] |
---|
1973 | |
---|
1974 | poly p=y2+zw+tv+y2w+y2t+y2v+zw2+zwt+zwv+wtv+t2v+tv2+x11+x11w+x11t+x11v; |
---|
1975 | classifyCeq(p);//A[10] r=0; |
---|
1976 | |
---|
1977 | poly p=zw+tv+xy2+zw2+zwv+zwt+wtv+t2v+tv2+xy2w+xy2t+xy2v+x14y+x14yw+x14yt+x14yv+x26+x26w+x26t+x26v; |
---|
1978 | classifyCeq(p); |
---|
1979 | |
---|
1980 | ring r=2,(x,y,z,t,w,u,v),Ds; |
---|
1981 | ideal I=z+t,x+t+w,v,y,u,t,w+z; |
---|
1982 | det(jacob(I)); |
---|
1983 | map phi=r,I; |
---|
1984 | poly a=1+x+u+v; |
---|
1985 | poly p=xy+zt+wu+v19; |
---|
1986 | poly q=a*p; |
---|
1987 | poly j=phi(q); |
---|
1988 | classifyCeq(j); |
---|
1989 | classifyCeq(p); |
---|
1990 | |
---|
1991 | poly p=x2+yz+tw+u3+v2x; |
---|
1992 | classifyCeq(p); |
---|
1993 | |
---|
1994 | poly q=a*p; |
---|
1995 | poly j=phi(q); |
---|
1996 | classifyCeq(j); |
---|
1997 | |
---|
1998 | poly p=xy+zw+t2+u2v+v19u; |
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1999 | classifyCeq(p); |
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2000 | |
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2001 | poly q=a*p; |
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2002 | poly j=phi(q); |
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2003 | classifyCeq(j); |
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2004 | |
---|
2005 | poly p=xy+zw+t2+u2v+v5t+uvt; |
---|
2006 | classifyCeq(p); |
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2007 | |
---|
2008 | poly q=a*p; |
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2009 | poly j=phi(q); |
---|
2010 | classifyCeq(j); |
---|
2011 | |
---|
2012 | poly p=xy+zw+t2+u2v+v5u+uvt; |
---|
2013 | classifyCeq(p); |
---|
2014 | |
---|
2015 | poly q=a*p; |
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2016 | poly j=phi(q); |
---|
2017 | classifyCeq(j); |
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2018 | |
---|
2019 | ring r=2,(x,y,z,t,w,u,v,e),Ds; |
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2020 | ideal I=x+y+z+t+w,y+z+t+w,e,v,u,t,w,z; |
---|
2021 | map si=r,I; |
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2022 | |
---|
2023 | poly p=xy+zt+wu+v2+e81; |
---|
2024 | poly j=si(p); |
---|
2025 | classifyCeq(j); |
---|
2026 | |
---|
2027 | poly p=xy+zt+wu+v2+e81+e^(80-12)*v; |
---|
2028 | classifyCeq(p); |
---|
2029 | poly j=si(p); |
---|
2030 | classifyCeq(j); |
---|
2031 | |
---|
2032 | poly p=xy+zt+wu+v2e+e60+e31v; |
---|
2033 | classifyCeq(p); |
---|
2034 | |
---|
2035 | poly p=xy+zt+wu+v2e+e60+e59v; |
---|
2036 | poly j=si(p); |
---|
2037 | classifyCeq(j); |
---|
2038 | =============================== Examples for characteristic > 2 ======================================== |
---|
2039 | ring R=3,(x,y,z,u,v,w,s,t),ds; |
---|
2040 | ideal M=maxideal(1); |
---|
2041 | M[3]=x-y+z+u+w-s+t; |
---|
2042 | poly f=u3+v4+u2v2+x2+y2+z2+s2+t2+w2; |
---|
2043 | classifyCeq(f); |
---|
2044 | |
---|
2045 | map phi=R,M; |
---|
2046 | f=phi(f); |
---|
2047 | classifyCeq(f); |
---|
2048 | |
---|
2049 | f=u2+y32+s2+t2+v2+x2+z2+w2; |
---|
2050 | classifyCeq(f); |
---|
2051 | |
---|
2052 | map phi=R,M; |
---|
2053 | f=phi(f); |
---|
2054 | classifyCeq(f); |
---|
2055 | |
---|
2056 | |
---|
2057 | f=v2w+w61+x2+y2+z2+t2+z2+s2+u2; |
---|
2058 | classifyCeq(f); |
---|
2059 | |
---|
2060 | f=phi(f); |
---|
2061 | classifyCeq(f); |
---|
2062 | |
---|
2063 | f=u2+v2+w2+s2+t2-x2y+xy2+z6+y11+xy11+x14+z53; // f is of corank=3 |
---|
2064 | classifyCeq(f); |
---|
2065 | |
---|
2066 | f=x3+y4+xy5+s2+t2; |
---|
2067 | classifyCeq(f); |
---|
2068 | |
---|
2069 | f=u3+v5+u2v2+x2+y2+z2+t2+s2+w2; |
---|
2070 | classifyCeq(f); |
---|
2071 | |
---|
2072 | ideal M=maxideal(1); |
---|
2073 | M[2]=x+y+2s+t; |
---|
2074 | map phi=R,M; |
---|
2075 | f=phi(f); |
---|
2076 | classifyCeq(f); |
---|
2077 | |
---|
2078 | ring R=3,(u,v,w,s,t),ds; |
---|
2079 | poly f=w2s+s21+u2+v2+t2; |
---|
2080 | classifyCeq(f); |
---|
2081 | |
---|
2082 | ideal M=maxideal(1); |
---|
2083 | M[3]=u+2v+w+2t+2s; |
---|
2084 | map phi=R,M; |
---|
2085 | f=phi(f); |
---|
2086 | classifyCeq(f); |
---|
2087 | |
---|
2088 | ring R=5,(x,y,z,u,v,w,s,t),dp; |
---|
2089 | poly f=v2w+w61+x2+y2+z2+t2+z2+s2+u2; |
---|
2090 | classifyCeq(f); |
---|
2091 | |
---|
2092 | */ |
---|