1 | // $Id: equising.lib,v 1.2 2000-12-15 11:51:56 Singular Exp $ |
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2 | info=" |
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3 | LIBRARY: Equising.lib PROCEDURES FOR EQUISINGULARITY STRATA |
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4 | |
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5 | AUTHOR: Andrea Mindnich, e-mail:mindnich@mathematik.uni-kl.de |
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6 | PROCEDURES: |
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7 | esStratum computes the equisingularity stratum of a deformation |
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8 | isEquising tests, if a given deformation is equisingular |
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9 | " |
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10 | |
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11 | LIB "poly.lib"; |
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12 | LIB "elim.lib"; |
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13 | LIB "hnoether.lib"; |
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14 | |
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15 | |
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16 | /////////////////////////////////////////////////////////////////////////////// |
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17 | /////////////////////////////////////////////////////////////////////////////// |
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18 | |
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19 | |
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20 | // COMPUTES a weight vector. x and y get weight 1 and all other |
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21 | // variables get weight 0. |
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22 | static proc xyVector() |
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23 | { |
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24 | intvec iv ; |
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25 | iv[nvars(basering)]=0 ; |
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26 | iv[rvar(x)] =1; |
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27 | iv[rvar(y)] =1; |
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28 | return (iv); |
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29 | } |
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30 | |
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31 | |
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32 | /////////////////////////////////////////////////////////////////////////////// |
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33 | |
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34 | |
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35 | // exchanges the variables x and y in the polynomial p_f |
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36 | static proc swapXY(poly f) |
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37 | { |
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38 | def r_base = basering; |
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39 | ideal MI = maxideal(1); |
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40 | MI[rvar(x)]=y; |
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41 | MI[rvar(y)]=x; |
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42 | map phi = r_base, MI; |
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43 | f=phi(f); |
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44 | return (f); |
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45 | } |
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46 | |
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47 | |
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48 | /////////////////////////////////////////////////////////////////////////////// |
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49 | |
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50 | |
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51 | // ASSUME: p_mon is a monomial |
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52 | // and p_g is the product of the variables in p_mon. |
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53 | // COMPUTES the coefficient of p_mon in p_h. |
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54 | static proc coefficient(poly p_h, poly p_mon, poly p_g) |
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55 | { |
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56 | matrix coefMatrix = coef(p_h,p_g); |
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57 | int nc = ncols(coefMatrix); |
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58 | int ii=1; |
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59 | poly p_c=0; |
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60 | while(ii<=nc) |
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61 | { |
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62 | if (coefMatrix[1,ii] == p_mon) |
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63 | { |
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64 | p_c = coefMatrix[2,ii]; |
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65 | break; |
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66 | } |
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67 | ii++; |
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68 | } |
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69 | return (p_c); |
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70 | } |
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71 | |
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72 | |
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73 | /////////////////////////////////////////////////////////////////////////////// |
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74 | |
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75 | |
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76 | // in p_F the variable p_vari is substituted by the polynomial p_g. |
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77 | static proc eSubst(poly p_F, poly p_vari, poly p_g) |
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78 | { |
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79 | def r_base = basering; |
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80 | ideal MI; |
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81 | map phi; |
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82 | int ii = rvar(p_vari); |
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83 | if (ii != 0) |
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84 | { |
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85 | MI = maxideal(1); |
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86 | MI[ii] = p_g; |
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87 | phi = r_base, MI; |
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88 | p_F = phi(p_F); |
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89 | } |
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90 | return (p_F); |
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91 | } |
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92 | |
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93 | |
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94 | /////////////////////////////////////////////////////////////////////////////// |
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95 | |
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96 | |
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97 | // All ring variables of p_F which occur in (the set of generators of) the |
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98 | // ideal Id, are substituted by 0 |
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99 | static proc SimplifyF(poly p_F, ideal Id) |
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100 | { |
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101 | int i=1; |
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102 | int si = size(Id); |
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103 | for (i=1; i <= si; i++) |
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104 | { |
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105 | if (rvar(Id[i])) |
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106 | { |
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107 | p_F = subst(p_F, Id[i], 0); |
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108 | } |
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109 | } |
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110 | return(p_F); |
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111 | } |
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112 | |
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113 | |
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114 | /////////////////////////////////////////////////////////////////////////////// |
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115 | |
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116 | |
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117 | // Checks, if the basering is admissible. |
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118 | static proc checkBasering () |
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119 | { |
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120 | int error = 0; |
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121 | if(find(charstr(basering),"real")) |
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122 | { |
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123 | ERROR ("cannot compute esStratum with 'real' as coefficient field"); |
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124 | } |
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125 | if (nvars(basering) <= 2) |
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126 | { |
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127 | ERROR ("there are to less ringvariables to compute esStratum") |
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128 | } |
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129 | error = checkQIdeal(ideal(basering)); |
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130 | return(error); |
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131 | } |
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132 | |
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133 | /////////////////////////////////////////////////////////////////////////////// |
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134 | |
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135 | |
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136 | static proc getInput (list #) |
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137 | { |
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138 | def r_base = basering; |
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139 | |
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140 | int maxStep = -1; |
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141 | |
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142 | if (size(#) >= 1) |
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143 | { |
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144 | if (typeof(#[1]) == "int") |
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145 | { |
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146 | maxStep = #[1]; |
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147 | } |
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148 | |
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149 | else |
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150 | { |
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151 | ERROR("expected esStratum('poly', 'int') "); |
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152 | } |
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153 | } |
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154 | |
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155 | return(maxStep); |
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156 | } |
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157 | |
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158 | |
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159 | ////////////////////////////////////////////////////////////////////////////// |
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160 | |
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161 | |
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162 | // RETURNS: 0, if the ideal cond only depends on the deformation parameters |
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163 | // 1, otherwise. |
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164 | static proc checkQIdeal (ideal cond) |
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165 | { |
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166 | def r_base = basering; |
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167 | |
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168 | int i_print = printlevel-voice + 4; |
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169 | int i_nvars = nvars(basering); |
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170 | |
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171 | ideal id_help = subst(cond,var(i_nvars),0,var(i_nvars-1),0) - cond; |
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172 | |
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173 | // cond depends only on the first i_nvars-2 variables <=> |
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174 | // id_help == <0> |
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175 | if ( simplify(id_help, 2) != 0) |
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176 | { |
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177 | dbprint(i_print, |
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178 | "ideal(basering) must only depend on the deformation parameters"); |
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179 | return(1); |
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180 | } |
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181 | |
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182 | return(0); |
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183 | } |
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184 | |
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185 | |
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186 | /////////////////////////////////////////////////////////////////////////////// |
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187 | |
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188 | |
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189 | // COMPUTES string(minpoly) and substitutes the parameter by newParName |
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190 | static proc makeMinPolyString (string newParName) |
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191 | { |
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192 | int i; |
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193 | string parName = parstr(basering); |
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194 | int parNameSize = size(parName); |
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195 | |
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196 | string oldMinPolyStr = string (minpoly); |
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197 | int minPolySize = size(oldMinPolyStr); |
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198 | |
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199 | string newMinPolyStr = ""; |
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200 | |
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201 | for (i=1;i <= minPolySize; i++) |
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202 | { |
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203 | if (oldMinPolyStr[i,parNameSize] == parName) |
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204 | { |
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205 | newMinPolyStr = newMinPolyStr + newParName; |
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206 | i = i + parNameSize-1; |
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207 | } |
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208 | else |
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209 | { |
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210 | newMinPolyStr = newMinPolyStr + oldMinPolyStr[i]; |
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211 | } |
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212 | } |
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213 | |
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214 | return(newMinPolyStr); |
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215 | } |
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216 | |
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217 | |
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218 | /////////////////////////////////////////////////////////////////////////////// |
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219 | |
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220 | |
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221 | // Defines a new ring without deformation-parameters. |
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222 | static proc createHNERing() |
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223 | { |
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224 | string str; |
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225 | string minpolyStr = string(minpoly); |
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226 | |
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227 | str = " ring HNERing = (" + charstr(basering) + "), (x,y), ls;"; |
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228 | execute (str); |
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229 | |
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230 | str = "minpoly ="+ minpolyStr+";"; |
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231 | execute(str); |
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232 | |
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233 | keepring(HNERing); |
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234 | } |
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235 | |
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236 | |
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237 | /////////////////////////////////////////////////////////////////////////////// |
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238 | |
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239 | |
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240 | // RETURNS: 1, if p_f = 0 or char(basering) divides the order of p_f |
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241 | // or p_f is not squarefree. |
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242 | // 0, otherwise |
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243 | static proc checkPoly (poly p_f) |
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244 | { |
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245 | int i_print = printlevel - voice + 3; |
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246 | int i_ord; |
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247 | |
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248 | if (p_f == 0) |
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249 | { |
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250 | dbprint(i_print,"The Input is a 'deformation' of the zero polynomial"); |
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251 | return(1); |
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252 | } |
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253 | |
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254 | i_ord = mindeg1(p_f); |
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255 | |
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256 | if (number(i_ord) == 0) |
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257 | { |
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258 | dbprint(i_print, |
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259 | "The characteristic of the coefficient field |
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260 | divides the order of the equation"); |
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261 | return(1); |
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262 | } |
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263 | |
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264 | if (squarefree(p_f) != p_f) |
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265 | { |
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266 | dbprint(i_print, "The curve is reducible"); |
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267 | return(1); |
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268 | } |
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269 | |
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270 | return(0); |
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271 | } |
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272 | |
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273 | |
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274 | /////////////////////////////////////////////////////////////////////////////// |
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275 | |
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276 | |
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277 | // COMPUTES the multiplicity sequence of p_f |
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278 | static proc calcMultSequence (poly p_f) |
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279 | { |
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280 | int i_print = printlevel-voice + 3; |
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281 | intvec multSeq=0; |
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282 | list hneList; |
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283 | int xNotTransversal; |
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284 | int fIrreducible = 1; |
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285 | |
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286 | // if basering = (p,a) or (p,a(1..s)), |
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287 | // p prime, a algebraic, a(1..s) transcendent use reddevelop |
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288 | // otherwise use develop |
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289 | if (char(basering) != 0 |
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290 | && npars(basering) !=0 |
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291 | && charstr(basering) == string(char(basering)) + "," + parstr(basering)) |
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292 | { |
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293 | hneList = reddevelop(p_f, -1); |
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294 | if ( size(hneList)>=2) |
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295 | { |
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296 | fIrreducible = 0; |
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297 | dbprint(i_print, "The curve is reducible"); |
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298 | return(multSeq, xNotTransversal, fIrreducible); |
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299 | } |
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300 | |
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301 | hneList = hneList[1]; |
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302 | |
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303 | xNotTransversal= hneList[3]; |
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304 | } |
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305 | else |
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306 | { |
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307 | hneList = develop(p_f, -1); |
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308 | |
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309 | xNotTransversal= hneList[3]; |
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310 | fIrreducible = hneList[5]; |
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311 | } |
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312 | |
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313 | if ( ! fIrreducible) |
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314 | { |
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315 | dbprint(i_print, "The curve is reducible"); |
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316 | return(multSeq, xNotTransversal, fIrreducible); |
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317 | } |
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318 | |
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319 | multSeq = multsequence (hneList); |
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320 | return(multSeq, xNotTransversal, fIrreducible); |
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321 | } |
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322 | |
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323 | |
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324 | /////////////////////////////////////////////////////////////////////////////// |
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325 | |
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326 | |
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327 | // ASSUME: The basering is no qring and has at least 3 variables |
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328 | // DEFINES: A new basering, "myRing", |
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329 | // with new names for the parameters and variables. |
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330 | // The new names for the parameters are a(1..k), |
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331 | // and t(1..s),x,y for the variables |
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332 | // The ring ordering is ordStr. |
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333 | // NOTE: This proc uses 'execute'. |
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334 | static proc createMyRing(poly p_F, string ordStr ) |
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335 | { |
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336 | def r_old = basering; |
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337 | |
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338 | int chara = char(basering); |
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339 | string charaStr; |
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340 | int i; |
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341 | string minPolyStr = ""; |
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342 | string helpStr; |
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343 | int nDefParams = nvars(r_old)-2; |
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344 | |
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345 | |
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346 | ideal qIdeal = ideal(basering); |
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347 | |
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348 | if (npars(basering) == 0) |
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349 | { |
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350 | helpStr = "ring myRing =" |
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351 | + string(chara)+ ", (t(1..nDefParams), x, y),"+ ordStr +";"; |
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352 | execute(helpStr); |
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353 | } |
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354 | |
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355 | else |
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356 | { |
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357 | charaStr = charstr(basering); |
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358 | if (charaStr == string(chara) + "," + parstr(basering)) |
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359 | { |
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360 | if (minpoly !=0) |
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361 | { |
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362 | minPolyStr = makeMinPolyString("a"); |
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363 | |
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364 | helpStr = "ring myRing = |
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365 | (" + string(chara) + ",a), |
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366 | (t(1..nDefParams), x, y)," + ordStr + ";"; |
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367 | execute(helpStr); |
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368 | |
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369 | helpStr = "minpoly =" + minPolyStr + ";"; |
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370 | execute (helpStr); |
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371 | } |
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372 | else |
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373 | { |
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374 | helpStr = "ring myRing = |
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375 | (" + string(chara) + ",a(1..npars(basering)) ), |
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376 | (t(1..nDefParams), x, y)," + ordStr + ";"; |
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377 | execute(helpStr); |
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378 | } |
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379 | } |
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380 | else |
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381 | { |
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382 | i = find (charaStr,","); |
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383 | |
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384 | helpStr = " ring myRing = |
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385 | (" + charaStr[1,i] + "a), |
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386 | (t(1..nDefParams), x, y)," + ordStr + ";"; |
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387 | |
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388 | execute (helpStr); |
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389 | } |
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390 | } |
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391 | |
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392 | ideal qIdeal = fetch(r_old, qIdeal); |
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393 | |
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394 | if(qIdeal != 0) |
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395 | { |
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396 | def r_base = basering; |
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397 | kill myRing; |
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398 | qring myRing = std(qIdeal); |
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399 | } |
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400 | |
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401 | poly p_F = fetch(r_old, p_F); |
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402 | ideal ES; |
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403 | |
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404 | keepring(myRing); |
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405 | } |
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406 | |
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407 | |
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408 | /////////////////////////////////////////////////////////////////////////////// |
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409 | /////////// procedures to compute the equisingularity stratum ///////////////// |
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410 | /////////////////////////////////////////////////////////////////////////////// |
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411 | |
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412 | |
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413 | // DEFINES a new basering, myRing, which has one variable |
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414 | // more than the old ring. |
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415 | // The name for the new variable is "H(nhelpV)". |
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416 | // p_F and ES are "imaped" into the new ring. |
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417 | static proc extendRing (poly p_F, ideal ES, int nHelpV, ideal HCond) |
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418 | { |
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419 | def r_old = basering; |
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420 | |
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421 | string helpStr; |
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422 | string minPolyStr = ""; |
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423 | |
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424 | ideal qIdeal = ideal(basering); |
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425 | |
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426 | if (minpoly != 0) |
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427 | { |
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428 | if (charstr(basering) == string(char(basering)) + "," + parstr(basering)) |
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429 | { |
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430 | minPolyStr = string(minpoly); |
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431 | } |
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432 | } |
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433 | |
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434 | string str = |
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435 | "ring myRing = (" + charstr(r_old) + "), |
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436 | (H(" + string(nHelpV)+ ")," + string(maxideal(1)) + "), |
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437 | (dp(" + string(nHelpV) + "),dp);"; |
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438 | execute (str); |
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439 | |
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440 | if (minPolyStr != "") |
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441 | { |
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442 | helpStr = "minpoly =" + minPolyStr + ";"; |
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443 | execute(helpStr); |
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444 | } |
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445 | |
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446 | ideal qIdeal = imap(r_old, qIdeal); |
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447 | if(qIdeal != 0) |
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448 | { |
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449 | def r_base = basering; |
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450 | |
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451 | kill myRing; |
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452 | |
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453 | attrib(qIdeal,"isSB",1); |
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454 | qring myRing = qIdeal; |
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455 | } |
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456 | |
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457 | |
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458 | poly p_F = imap(r_old, p_F); |
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459 | ideal ES = imap(r_old, ES); |
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460 | ideal HCond = imap(r_old, HCond); |
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461 | |
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462 | keepring(myRing); |
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463 | } |
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464 | |
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465 | |
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466 | /////////////////////////////////////////////////////////////////////////////// |
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467 | |
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468 | |
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469 | // COMPUTES an ideal equimultCond, such that F_(n-1) mod equimultCond =0, |
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470 | // where F_(n-1) is the (nNew-1)-jet of p_F with respect to x,y. |
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471 | static proc calcEquimultCond(poly p_F, int nNew) |
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472 | { |
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473 | ideal equimultCond = 0; |
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474 | poly p_FnMinus1; |
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475 | matrix coefMatrix; |
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476 | int nc; |
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477 | int ii = 1; |
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478 | |
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479 | p_FnMinus1 = jet(p_F, nNew-1, xyVector()); |
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480 | |
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481 | coefMatrix = coef(p_FnMinus1, xy); |
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482 | |
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483 | nc = ncols(coefMatrix); |
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484 | |
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485 | for (ii=1; ii<=nc; ii++) |
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486 | { |
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487 | equimultCond[ii] = NF(coefMatrix[2,ii],std(0)); |
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488 | } |
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489 | |
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490 | p_F = p_F - p_FnMinus1; |
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491 | p_F = SimplifyF(p_F, equimultCond); |
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492 | |
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493 | return(equimultCond, p_F); |
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494 | } |
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495 | |
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496 | |
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497 | /////////////////////////////////////////////////////////////////////////////// |
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498 | |
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499 | |
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500 | // COMPUTES smallest integer >= nNew/nOld -1 |
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501 | static proc calcNZeroSteps (int nOld,int nNew) |
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502 | { |
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503 | int nZeroSteps; |
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504 | |
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505 | if (nOld mod nNew == 0) |
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506 | { |
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507 | nZeroSteps = nOld div nNew -1; |
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508 | } |
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509 | |
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510 | else |
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511 | { |
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512 | nZeroSteps = nOld div nNew; |
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513 | } |
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514 | |
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515 | return(nZeroSteps); |
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516 | } |
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517 | |
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518 | |
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519 | /////////////////////////////////////////////////////////////////////////////// |
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520 | |
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521 | |
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522 | // ASSUME: ord_(X,Y)(F)=nNew |
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523 | // COMPUTES an ideal I such that (p_F mod I)_nNew = p_c*y^nNew. |
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524 | static proc purePowerOfY (poly p_F, int nNew) |
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525 | { |
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526 | ideal id_help = 0; |
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527 | poly p_Fn; |
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528 | matrix coefMatrix; |
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529 | int nc; |
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530 | poly p_c; |
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531 | int ii=1; |
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532 | |
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533 | p_Fn = jet(p_F, nNew, xyVector()); |
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534 | |
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535 | coefMatrix = coef(p_Fn, xy); |
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536 | nc = ncols(coefMatrix); |
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537 | |
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538 | p_c = coefMatrix[2,nc]; |
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539 | |
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540 | for (ii=1; ii <= nc-1; ii++) |
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541 | { |
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542 | id_help[ii] = NF(coefMatrix[2,ii],std(0)); |
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543 | } |
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544 | |
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545 | p_F = p_F - p_Fn + p_c*y^nNew; |
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546 | |
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547 | p_F = SimplifyF(p_F, id_help); |
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548 | |
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549 | return(id_help, p_F, p_c); |
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550 | } |
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551 | |
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552 | |
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553 | /////////////////////////////////////////////////////////////////////////////// |
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554 | |
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555 | |
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556 | // ASSUME: ord_(X,Y)(F)=nNew |
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557 | // COMPUTES an ideal I such that p_Fn mod I = p_c*(y-p_a*x)^nNew, |
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558 | // where p_Fn is the homogeneous part of p_F of order nNew. |
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559 | static proc purePowerOfLin (poly p_F, ideal HCond, int nNew, int nHelpV) |
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560 | { |
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561 | ideal id_help = 0; |
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562 | poly p_Fn; |
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563 | matrix coefMatrix; |
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564 | poly p_c; |
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565 | poly p_ca; |
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566 | poly p_help; |
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567 | poly p_a; |
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568 | int ii; |
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569 | int jj; |
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570 | int bico; |
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571 | |
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572 | p_Fn = jet(p_F, nNew, xyVector()); |
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573 | |
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574 | coefMatrix = coeffs(subst(p_Fn,x,1),y); |
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575 | |
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576 | p_c = coefMatrix[nNew+1,1]; |
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577 | p_ca = coefMatrix[nNew,1]/(-nNew); |
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578 | |
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579 | if (npars(basering)==1 && charstr(basering) != string(char(basering)) + "," + parstr(basering)) |
---|
580 | { |
---|
581 | p_a = H(nHelpV); |
---|
582 | HCond = HCond + ideal(p_ca - p_a*p_c); |
---|
583 | } |
---|
584 | else |
---|
585 | { |
---|
586 | p_help = p_ca/p_c; |
---|
587 | if (p_help * p_c == p_ca) |
---|
588 | { |
---|
589 | p_a = p_help; |
---|
590 | } |
---|
591 | else |
---|
592 | { |
---|
593 | p_a = H(nHelpV); |
---|
594 | HCond = HCond + ideal(p_ca - p_a*p_c); |
---|
595 | } |
---|
596 | } |
---|
597 | |
---|
598 | bico = (nNew*(nNew-1))/2; |
---|
599 | |
---|
600 | for (ii = 2; ii <= nNew ; ii++) |
---|
601 | { |
---|
602 | |
---|
603 | if (coefMatrix[nNew+1-ii,1] == 0) |
---|
604 | { |
---|
605 | if (number(bico) != 0) |
---|
606 | // Then a^i=0 since c is a unit |
---|
607 | { |
---|
608 | id_help = id_help + ideal(NF(p_a^(ii),std(0))); |
---|
609 | for (jj = ii+1; jj <= nNew; jj++) |
---|
610 | // the remaining coefficients (of y^(nnew-k)*x^k with k>i) |
---|
611 | // are also zero. |
---|
612 | { |
---|
613 | id_help = id_help |
---|
614 | + ideal(NF(coefMatrix[nNew+1-jj,1],std(0))); |
---|
615 | } |
---|
616 | break; |
---|
617 | } |
---|
618 | } |
---|
619 | |
---|
620 | else |
---|
621 | { |
---|
622 | id_help = id_help + |
---|
623 | ideal(NF(coefMatrix[nNew+1-ii,1] - bico*p_c*(-p_a)^ii,std(0))); |
---|
624 | } |
---|
625 | |
---|
626 | bico = (bico*(nNew-ii))/(ii+1); |
---|
627 | } |
---|
628 | |
---|
629 | p_F = SimplifyF(p_F, id_help); |
---|
630 | |
---|
631 | return(id_help, HCond, p_F, p_c, p_a); |
---|
632 | } |
---|
633 | |
---|
634 | |
---|
635 | /////////////////////////////////////////////////////////////////////////////// |
---|
636 | |
---|
637 | |
---|
638 | // eliminates the variables H(1),..,H(nHelpV) from the ideal ES + HCond |
---|
639 | static proc helpVarElim(ideal ES, ideal HCond, int nHelpV) |
---|
640 | { |
---|
641 | ES = ES + HCond; |
---|
642 | ES = std(ES); |
---|
643 | ES = nselect(ES,1,nHelpV); |
---|
644 | |
---|
645 | return(ES); |
---|
646 | } |
---|
647 | |
---|
648 | |
---|
649 | /////////////////////////////////////////////////////////////////////////////// |
---|
650 | |
---|
651 | |
---|
652 | // ASSUME: ord(F)=nNew and p_c(y-p_a*x)^n is the nNew-jet of F with respect |
---|
653 | // to X,Y |
---|
654 | // COMPUTES F(x,yx+a*x)/x^n |
---|
655 | static proc formalBlowUp(poly p_F, poly p_c, poly p_a, int nNew) |
---|
656 | { |
---|
657 | |
---|
658 | p_F = p_F - jet(p_F, nNew, xyVector()); |
---|
659 | |
---|
660 | if (p_a != 0) |
---|
661 | { |
---|
662 | p_F = eSubst(p_F, y , yx + p_a*x); |
---|
663 | } |
---|
664 | else |
---|
665 | { |
---|
666 | p_F = subst(p_F, y, xy); |
---|
667 | } |
---|
668 | |
---|
669 | p_F = p_F/(x^nNew); |
---|
670 | |
---|
671 | p_F = p_F + p_c * y^nNew; |
---|
672 | |
---|
673 | return (p_F); |
---|
674 | } |
---|
675 | |
---|
676 | |
---|
677 | /////////////////////////////////////////////////////////////////////////////// |
---|
678 | |
---|
679 | |
---|
680 | // ASSUME: p_F in K[t(1)..t(s),x,y] |
---|
681 | // COMPUTES the minimal ideal ES, such that the deformation p_F mod ES is |
---|
682 | // equisingular. |
---|
683 | // The computation is done up to iteration step 'maxstep'. |
---|
684 | // RETURNS: list l, such that |
---|
685 | // l[1]=1 if some error has occured, |
---|
686 | // l[1]=0 otherwise and then l[2] = es_cond. |
---|
687 | static proc calcEsCond(poly p_F, intvec multSeq, int maxStep) |
---|
688 | { |
---|
689 | def r_old = basering; |
---|
690 | |
---|
691 | ideal ES = 0; |
---|
692 | |
---|
693 | int ii; |
---|
694 | int step = 1; |
---|
695 | int nNew = multSeq[step]; |
---|
696 | int nOld = nNew; |
---|
697 | int nZeroSteps; |
---|
698 | int nHelpV = 1; |
---|
699 | ideal HCond = 0; |
---|
700 | int maxDeg = 0; |
---|
701 | int z = printlevel - voice + 3; |
---|
702 | string str; |
---|
703 | |
---|
704 | extendRing(p_F, ES, nHelpV, HCond); |
---|
705 | |
---|
706 | poly p_c; |
---|
707 | poly p_a; |
---|
708 | ideal I; |
---|
709 | |
---|
710 | for (ii = 1; ii <= size(multSeq); ii++) |
---|
711 | { |
---|
712 | maxDeg = maxDeg + multSeq[ii]; |
---|
713 | } |
---|
714 | |
---|
715 | while (step <= maxStep) |
---|
716 | { |
---|
717 | |
---|
718 | nOld = nNew; |
---|
719 | nNew = multSeq[step]; |
---|
720 | |
---|
721 | p_F = jet(p_F, maxDeg, xyVector()); |
---|
722 | maxDeg = maxDeg - nNew; |
---|
723 | |
---|
724 | if (nNew<nOld) |
---|
725 | //start a new line in the HNE of F |
---|
726 | // _ _ |
---|
727 | // for the next | nold/nnew -1 | iteration steps the coefficient 'a' |
---|
728 | // in the leading form Fn = c(y-ax)^n should be zero. |
---|
729 | { |
---|
730 | p_F = swapXY(p_F); |
---|
731 | nZeroSteps = calcNZeroSteps (nOld, nNew); |
---|
732 | } |
---|
733 | |
---|
734 | I, p_F = calcEquimultCond (p_F, nNew); |
---|
735 | ES = ES + I; |
---|
736 | |
---|
737 | if (z>1) |
---|
738 | { |
---|
739 | "// conditions for equimultiplicity in step:", step; |
---|
740 | I; |
---|
741 | if (nHelpV >1) |
---|
742 | { |
---|
743 | str = string(nHelpV); |
---|
744 | "// conditions for help variables H(1),..,H("+str+"):"; |
---|
745 | HCond; |
---|
746 | } |
---|
747 | pause("press <return> to continue"); |
---|
748 | } |
---|
749 | |
---|
750 | if (nZeroSteps > 0) |
---|
751 | { |
---|
752 | nZeroSteps--; |
---|
753 | |
---|
754 | // compute conditions, s.th. Fn = c*y^nnew ? |
---|
755 | I, p_F, p_c = purePowerOfY (p_F, nNew); |
---|
756 | ES = ES + I; |
---|
757 | |
---|
758 | if (z>1) |
---|
759 | { |
---|
760 | "// conditions for pure power in step:", step; |
---|
761 | I; |
---|
762 | if (nHelpV > 1) |
---|
763 | { |
---|
764 | str = string(nHelpV); |
---|
765 | "// conditions for help variables H(1),..,H("+str+"):"; |
---|
766 | HCond; |
---|
767 | } |
---|
768 | pause("press <return> to continue"); |
---|
769 | } |
---|
770 | p_a =0; |
---|
771 | } |
---|
772 | |
---|
773 | else |
---|
774 | { |
---|
775 | I, HCond, p_F, p_c, p_a = purePowerOfLin (p_F, HCond, nNew, nHelpV); |
---|
776 | |
---|
777 | ES = ES + I; |
---|
778 | |
---|
779 | if (z>1) |
---|
780 | { |
---|
781 | "// conditions for pure power in step:", step; |
---|
782 | I; |
---|
783 | str = string(nHelpV); |
---|
784 | "// conditions for help variables H(1),..,H("+str+"):"; |
---|
785 | HCond; |
---|
786 | pause("press <return> to continue"); |
---|
787 | } |
---|
788 | } |
---|
789 | |
---|
790 | p_F = formalBlowUp (p_F, p_c, p_a, nNew); |
---|
791 | |
---|
792 | if (p_a == H(nHelpV)) |
---|
793 | { |
---|
794 | nHelpV++; |
---|
795 | |
---|
796 | def r_base = basering; |
---|
797 | kill myRing; |
---|
798 | |
---|
799 | extendRing(p_F, ES, nHelpV, HCond); |
---|
800 | |
---|
801 | kill r_base; |
---|
802 | |
---|
803 | poly p_a; |
---|
804 | poly p_c; |
---|
805 | ideal I; |
---|
806 | } |
---|
807 | step++; |
---|
808 | } |
---|
809 | if (nHelpV > 1) |
---|
810 | { |
---|
811 | ES = helpVarElim(ES, HCond, nHelpV); |
---|
812 | } |
---|
813 | |
---|
814 | if (nameof(basering)=="myRing") |
---|
815 | { |
---|
816 | setring r_old; |
---|
817 | ES = imap(myRing, ES); |
---|
818 | } |
---|
819 | |
---|
820 | return(ES); |
---|
821 | } |
---|
822 | |
---|
823 | |
---|
824 | /////////////////////////////////////////////////////////////////////////////// |
---|
825 | |
---|
826 | |
---|
827 | proc esStratum (poly p_F, list #) |
---|
828 | "USAGE: esStratum(F[,m]); F a polynomial, m an integer |
---|
829 | ASSUME: F defines a deformation of an irreducible bivariate polynomial f |
---|
830 | and that char(basering) does not divide mult(f). |
---|
831 | If nv is the number of variables of the basering, then the first nv-2 |
---|
832 | ringvariables are the deformation parameters. |
---|
833 | If the basering is a qring, ideal(basering) must only depend |
---|
834 | on the deformation parameters. |
---|
835 | RETURN: A list l of an ideal and an integer, where |
---|
836 | l[1] is the ideal in the deformation parameters, defining the |
---|
837 | equisingularity stratum of F, |
---|
838 | l[2] = 1 if some error has occured, l[2] = 0 otherwise. |
---|
839 | If m is given, the computation stops after m steps of the iteration. |
---|
840 | NOTE: printlevel > 0 displays comments and pauses after intermediate |
---|
841 | computations (default: printlevel=0) |
---|
842 | This proc uses 'execute' or calls a procedure using 'execute'. |
---|
843 | EXAMPLE: example esStratum; shows an example |
---|
844 | " |
---|
845 | { |
---|
846 | def r_user = basering; |
---|
847 | |
---|
848 | int ii = 1; |
---|
849 | int i_nvars = nvars(basering); |
---|
850 | int error = 0; |
---|
851 | int xNotTransversal; |
---|
852 | int fIrreducible; |
---|
853 | int maxStep; |
---|
854 | int userMaxStep; |
---|
855 | ideal cond; |
---|
856 | intvec multSeq; |
---|
857 | ideal ES = 0; |
---|
858 | |
---|
859 | error = checkBasering(); |
---|
860 | if (error) |
---|
861 | { |
---|
862 | return(list(ES,error)); |
---|
863 | } |
---|
864 | |
---|
865 | userMaxStep = getInput(#); |
---|
866 | |
---|
867 | // define a new basering "myRing" with new names for parameters |
---|
868 | // and variables. |
---|
869 | // The new names are 'a(1)', ..., 'a(npars)' for the parameters |
---|
870 | // and 't(1)', ..., 't(nvars-2)', 'x', 'y' for the variables. |
---|
871 | createMyRing(p_F, "dp"); |
---|
872 | |
---|
873 | // define a ring without deformation parameters, to compute the HNE |
---|
874 | // of F mod <t_1,..,t_s> |
---|
875 | createHNERing(); |
---|
876 | |
---|
877 | poly p_f = imap(myRing,p_F); |
---|
878 | |
---|
879 | error = checkPoly(p_f); |
---|
880 | if (error) |
---|
881 | { |
---|
882 | setring r_user; |
---|
883 | return(list( ideal(0),error)); |
---|
884 | } |
---|
885 | |
---|
886 | // compute the multiplicitysequence of p_f. |
---|
887 | multSeq, xNotTransversal, fIrreducible = calcMultSequence(p_f); |
---|
888 | |
---|
889 | if ( ! fIrreducible) |
---|
890 | { |
---|
891 | setring r_user; |
---|
892 | return(list(ideal(0),1)); |
---|
893 | } |
---|
894 | |
---|
895 | setring myRing; |
---|
896 | |
---|
897 | if (xNotTransversal) |
---|
898 | { |
---|
899 | p_F = swapXY(p_F); |
---|
900 | } |
---|
901 | |
---|
902 | if (userMaxStep != -1 && userMaxStep < size(multSeq)-1) |
---|
903 | { |
---|
904 | maxStep = userMaxStep; |
---|
905 | } |
---|
906 | else |
---|
907 | { |
---|
908 | maxStep = size(multSeq)-1; |
---|
909 | } |
---|
910 | |
---|
911 | ES = calcEsCond(p_F, multSeq, maxStep); |
---|
912 | |
---|
913 | setring r_user; |
---|
914 | ES = fetch(myRing, ES); |
---|
915 | |
---|
916 | return(list(ES, error)); |
---|
917 | } |
---|
918 | |
---|
919 | example |
---|
920 | { |
---|
921 | "EXAMPLE:"; echo=2; |
---|
922 | ring r = 11,(a,b,c,d,e,f,g,x,y),ds; |
---|
923 | poly F = |
---|
924 | xa+yb+x2+2xy+y2c+y^2+y3d+y4e+y5f+y6g+x^7; |
---|
925 | esStratum(F); |
---|
926 | esStratum(F,2); |
---|
927 | ideal I = f-fa,e+b; |
---|
928 | qring q = std(I); |
---|
929 | poly F = imap(r,F); |
---|
930 | esStratum(F); |
---|
931 | } |
---|
932 | |
---|
933 | /////////////////////////////////////////////////////////////////////////////// |
---|
934 | //////////////////// procedures for equisingularity test/////////////////////// |
---|
935 | /////////////////////////////////////////////////////////////////////////////// |
---|
936 | |
---|
937 | |
---|
938 | |
---|
939 | // DEFINES a new basering, myRing, which has one variable |
---|
940 | // more than the old ring. |
---|
941 | // The name for the new variable is "H(nhelpV)". |
---|
942 | static proc T_extendRing(poly p_F, int nHelpV, ideal HCond) |
---|
943 | { |
---|
944 | def r_old = basering; |
---|
945 | |
---|
946 | ideal qIdeal = ideal(basering); |
---|
947 | |
---|
948 | string helpStr; |
---|
949 | string minPolyStr = ""; |
---|
950 | |
---|
951 | if(minpoly != 0) |
---|
952 | { |
---|
953 | if (charstr(basering) == string(char(basering)) + "," + parstr(basering)) |
---|
954 | { |
---|
955 | minPolyStr = string(minpoly); |
---|
956 | } |
---|
957 | } |
---|
958 | |
---|
959 | string str = "ring myRing = |
---|
960 | (" + charstr(r_old) + "), |
---|
961 | (H(" + string( nHelpV)+ ")," + string(maxideal(1)) + "), |
---|
962 | (dp(" + string( nHelpV) + "), ds);"; |
---|
963 | execute (str); |
---|
964 | |
---|
965 | if (minPolyStr != "") |
---|
966 | { |
---|
967 | helpStr = "minpoly =" + minPolyStr + ";"; |
---|
968 | execute(helpStr); |
---|
969 | } |
---|
970 | |
---|
971 | ideal qIdeal = imap(r_old, qIdeal); |
---|
972 | if(qIdeal != 0) |
---|
973 | { |
---|
974 | def r_base = basering; |
---|
975 | kill myRing; |
---|
976 | qring myRing = std(qIdeal); |
---|
977 | } |
---|
978 | |
---|
979 | poly p_F =imap(r_old, p_F); |
---|
980 | ideal HCond = imap(r_old, HCond); |
---|
981 | |
---|
982 | keepring(myRing); |
---|
983 | } |
---|
984 | |
---|
985 | |
---|
986 | /////////////////////////////////////////////////////////////////////////////// |
---|
987 | |
---|
988 | |
---|
989 | // tests, if ord p_F = nNew. |
---|
990 | static proc equimultTest (poly p_F, int nHelpV, int nNew, ideal HCond) |
---|
991 | { |
---|
992 | poly p_FnMinus1; |
---|
993 | ideal id_help; |
---|
994 | matrix coefMatrix; |
---|
995 | int i; |
---|
996 | int nc; |
---|
997 | |
---|
998 | p_FnMinus1 = jet(p_F, nNew-1, xyVector()); |
---|
999 | |
---|
1000 | coefMatrix = coef(p_FnMinus1, xy); |
---|
1001 | nc = ncols(coefMatrix); |
---|
1002 | |
---|
1003 | for (i=1; i<=nc; i++) |
---|
1004 | { |
---|
1005 | id_help[i] = coefMatrix[2,i]; |
---|
1006 | } |
---|
1007 | |
---|
1008 | id_help = T_helpVarElim(id_help, HCond, nHelpV); |
---|
1009 | |
---|
1010 | if (reduce(id_help, std(0)) !=0 ) |
---|
1011 | { |
---|
1012 | return(0, p_F); |
---|
1013 | } |
---|
1014 | |
---|
1015 | p_F = p_F - p_FnMinus1; |
---|
1016 | |
---|
1017 | return(1, p_F); |
---|
1018 | } |
---|
1019 | |
---|
1020 | |
---|
1021 | /////////////////////////////////////////////////////////////////////////////// |
---|
1022 | |
---|
1023 | |
---|
1024 | // ASSUME: ord(p_F)=nNew |
---|
1025 | // tests, if p_F = p_c*y^nNew for some p_c. |
---|
1026 | static proc pPOfYTest (poly p_F, int nHelpV, int nNew, ideal HCond) |
---|
1027 | { |
---|
1028 | poly p_Fn; |
---|
1029 | poly p_c; |
---|
1030 | ideal id_help; |
---|
1031 | int nc; |
---|
1032 | int i=1; |
---|
1033 | matrix coefMatrix; |
---|
1034 | |
---|
1035 | p_Fn = jet(p_F, nNew, xyVector()); |
---|
1036 | |
---|
1037 | coefMatrix = coef(p_Fn, xy); |
---|
1038 | nc = ncols(coefMatrix); |
---|
1039 | |
---|
1040 | p_c = coefMatrix[2,1]; |
---|
1041 | |
---|
1042 | for (i = 2; i <= nc; i++) |
---|
1043 | { |
---|
1044 | id_help[i] = coefMatrix[2,i]; |
---|
1045 | } |
---|
1046 | |
---|
1047 | id_help = T_helpVarElim(id_help, HCond, nHelpV); |
---|
1048 | |
---|
1049 | if (reduce(id_help, std(0)) !=0 ) |
---|
1050 | { |
---|
1051 | return(0, p_c); |
---|
1052 | } |
---|
1053 | |
---|
1054 | return(1, p_c); |
---|
1055 | } |
---|
1056 | |
---|
1057 | |
---|
1058 | /////////////////////////////////////////////////////////////////////////////// |
---|
1059 | |
---|
1060 | |
---|
1061 | // ASSUME: ord(p_F)=nNew |
---|
1062 | // tests, if p_F = p_c*(y - p_a*x)^nNew for some p_a, p_c. |
---|
1063 | static proc pPOfLinTest(poly p_F, int nNew, int nHelpV, ideal HCond) |
---|
1064 | { |
---|
1065 | poly p_Fn; |
---|
1066 | poly p_c; |
---|
1067 | poly p_ca; |
---|
1068 | poly p_help; |
---|
1069 | poly p_a; |
---|
1070 | ideal id_help; |
---|
1071 | |
---|
1072 | p_Fn = jet(p_F, nNew, xyVector()); |
---|
1073 | |
---|
1074 | p_c = coefficient(p_Fn,y^nNew,y); |
---|
1075 | p_ca = coefficient(p_Fn,y^(nNew-1)*x,xy)/-nNew; |
---|
1076 | |
---|
1077 | if (npars(basering)==1 |
---|
1078 | && charstr(basering) != string(char(basering)) + "," + parstr(basering)) |
---|
1079 | { |
---|
1080 | p_a = H(nHelpV); |
---|
1081 | HCond = HCond + ideal(p_ca - p_a*p_c); |
---|
1082 | } |
---|
1083 | else |
---|
1084 | { |
---|
1085 | p_help = p_ca/p_c; |
---|
1086 | if (p_help * p_c == p_ca) |
---|
1087 | { |
---|
1088 | p_a = p_help; |
---|
1089 | } |
---|
1090 | else |
---|
1091 | { |
---|
1092 | p_a = H(nHelpV); |
---|
1093 | HCond = HCond + ideal(p_ca - p_a*p_c); |
---|
1094 | } |
---|
1095 | } |
---|
1096 | |
---|
1097 | id_help = ideal(p_Fn - p_c *(y - p_a *x)^nNew); |
---|
1098 | id_help = T_helpVarElim(id_help, HCond, nHelpV); |
---|
1099 | |
---|
1100 | if (reduce(id_help, std(0)) != 0 ) |
---|
1101 | { |
---|
1102 | return(0, p_F, p_c, p_a, HCond); |
---|
1103 | } |
---|
1104 | |
---|
1105 | return(1, p_F, p_c, p_a, HCond); |
---|
1106 | } |
---|
1107 | |
---|
1108 | |
---|
1109 | ////////////////////////////////////////////////////////////////////////////// |
---|
1110 | |
---|
1111 | |
---|
1112 | // eliminates the variables H(1),..,H(nHelpV) from the ideal ES + HCond |
---|
1113 | static proc T_helpVarElim(ideal ES, ideal HCond, int nHelpV) |
---|
1114 | { |
---|
1115 | |
---|
1116 | def r_old = basering; |
---|
1117 | |
---|
1118 | ideal qIdeal = ideal(basering); |
---|
1119 | |
---|
1120 | string helpStr; |
---|
1121 | string minPolyStr = ""; |
---|
1122 | |
---|
1123 | if(minpoly != 0) |
---|
1124 | { |
---|
1125 | if (charstr(basering) == string(char(basering)) + "," + parstr(basering)) |
---|
1126 | { |
---|
1127 | minPolyStr = string(minpoly); |
---|
1128 | } |
---|
1129 | } |
---|
1130 | |
---|
1131 | string str = "ring myRing = |
---|
1132 | (" + charstr(r_old) + "),(" + string(maxideal(1)) + "), |
---|
1133 | (dp(" + string( nHelpV) + "), dp);"; |
---|
1134 | execute (str); |
---|
1135 | |
---|
1136 | if (minPolyStr != "") |
---|
1137 | { |
---|
1138 | helpStr = "minpoly =" + minPolyStr + ";"; |
---|
1139 | execute(helpStr); |
---|
1140 | } |
---|
1141 | |
---|
1142 | ideal qIdeal = imap(r_old, qIdeal); |
---|
1143 | if(qIdeal != 0) |
---|
1144 | { |
---|
1145 | def r_base = basering; |
---|
1146 | kill myRing; |
---|
1147 | qring myRing = std(qIdeal); |
---|
1148 | } |
---|
1149 | |
---|
1150 | ideal ES = imap(r_old, ES); |
---|
1151 | ideal HCond = imap(r_old, HCond); |
---|
1152 | |
---|
1153 | ES = ES + HCond; |
---|
1154 | ES = std(ES); |
---|
1155 | ES = nselect(ES,1,nHelpV); |
---|
1156 | |
---|
1157 | setring r_old; |
---|
1158 | ES = imap (myRing,ES); |
---|
1159 | |
---|
1160 | return(ES); |
---|
1161 | } |
---|
1162 | /////////////////////////////////////////////////////////////////////////////// |
---|
1163 | |
---|
1164 | |
---|
1165 | // ASSUME: F in K[t(1)..t(s),x,y], |
---|
1166 | // the ringordering is ds |
---|
1167 | // RETURNS: list l, such that |
---|
1168 | // l[1]=1 if some error has occured, |
---|
1169 | // l[1]=0 otherwise and then |
---|
1170 | // l[2] = 1, if the deformation is equisingular and |
---|
1171 | // l[2] = 0 otherwise. |
---|
1172 | static proc equisingTest (poly p_F, intvec multSeq, int maxStep) |
---|
1173 | { |
---|
1174 | def r_old = basering; |
---|
1175 | |
---|
1176 | ideal id_Es = 0; |
---|
1177 | |
---|
1178 | int isES = 1; |
---|
1179 | int step = 1; |
---|
1180 | int nNew = multSeq[step]; |
---|
1181 | int nOld = nNew; |
---|
1182 | int zeroSteps; |
---|
1183 | ideal HCond = 0; |
---|
1184 | int nHelpV = 1; |
---|
1185 | |
---|
1186 | T_extendRing (p_F, nHelpV, HCond); |
---|
1187 | |
---|
1188 | poly p_c; |
---|
1189 | poly p_a; |
---|
1190 | |
---|
1191 | while (step <= maxStep) |
---|
1192 | { |
---|
1193 | nOld = nNew; |
---|
1194 | nNew = multSeq[step]; |
---|
1195 | |
---|
1196 | if (nNew < nOld) |
---|
1197 | //start a new line in the HNE of F |
---|
1198 | // _ _ |
---|
1199 | // for the next | nold/nnew -1 | iteration steps the coefficient 'a' |
---|
1200 | // in the leading form Fn = c(y-ax) should be zero |
---|
1201 | { |
---|
1202 | p_F = swapXY(p_F); |
---|
1203 | zeroSteps = calcNZeroSteps (nOld, nNew); |
---|
1204 | } |
---|
1205 | |
---|
1206 | isES, p_F = equimultTest (p_F, nHelpV, nNew, HCond); |
---|
1207 | |
---|
1208 | if (! isES) |
---|
1209 | { |
---|
1210 | return(0); |
---|
1211 | } |
---|
1212 | |
---|
1213 | if (zeroSteps > 0) |
---|
1214 | { |
---|
1215 | zeroSteps--; |
---|
1216 | |
---|
1217 | isES, p_c = pPOfYTest (p_F, nHelpV, nNew, HCond); |
---|
1218 | p_a = 0; |
---|
1219 | } |
---|
1220 | else |
---|
1221 | { |
---|
1222 | isES, p_F, p_c, p_a, HCond = pPOfLinTest (p_F, nNew, nHelpV, HCond); |
---|
1223 | } |
---|
1224 | |
---|
1225 | if (! isES) |
---|
1226 | { |
---|
1227 | return(0); |
---|
1228 | } |
---|
1229 | |
---|
1230 | p_F = formalBlowUp (p_F, p_c, p_a, nNew); |
---|
1231 | |
---|
1232 | if (p_a == H(nHelpV)) |
---|
1233 | { |
---|
1234 | nHelpV++; |
---|
1235 | |
---|
1236 | def r_base = basering; |
---|
1237 | kill myRing; |
---|
1238 | |
---|
1239 | T_extendRing(p_F, nHelpV, HCond); |
---|
1240 | |
---|
1241 | kill r_base; |
---|
1242 | |
---|
1243 | poly p_a; |
---|
1244 | poly p_c; |
---|
1245 | } |
---|
1246 | |
---|
1247 | step++; |
---|
1248 | } |
---|
1249 | |
---|
1250 | return(1); |
---|
1251 | } |
---|
1252 | |
---|
1253 | /////////////////////////////////////////////////////////////////////////////// |
---|
1254 | |
---|
1255 | |
---|
1256 | proc isEquising (poly p_F, list #) |
---|
1257 | "USAGE: esStratum(F[,m]); F a polynomial, m an integer |
---|
1258 | ASSUME: F defines a deformation of an irreducible bivariate polynomial f |
---|
1259 | and that char(basering) does not divide mult(f). |
---|
1260 | If nv is the number of variables of the basering, then the first nv-2 |
---|
1261 | ringvariables are the deformation parameters. |
---|
1262 | If the basering is a qring, ideal(basering) must only depend |
---|
1263 | on the deformation parameters. |
---|
1264 | RETURN: A list l of two integers, where |
---|
1265 | l[1] = 1 if F is an equisingular deformation,l[1] = 0 otherwise. |
---|
1266 | l[2] = 1 if some error has occured, l[2] = 0 otherwise. |
---|
1267 | If m is given, the computation stops after m steps of the iteration. |
---|
1268 | NOTE: This proc uses 'execute' or calls a procedure using 'execute'. |
---|
1269 | EXAMPLE: example isEquising; shows an example |
---|
1270 | " |
---|
1271 | { |
---|
1272 | def r_user = basering; |
---|
1273 | |
---|
1274 | int ii = 1; |
---|
1275 | int i_nvars = nvars(basering); |
---|
1276 | int error = 0; |
---|
1277 | int maxStep; |
---|
1278 | int userMaxStep; |
---|
1279 | int xNotTransversal = 0; |
---|
1280 | int fIrreducible = 1; |
---|
1281 | intvec multSeq; |
---|
1282 | ideal isES = 1; |
---|
1283 | |
---|
1284 | error = checkBasering(); |
---|
1285 | if (error) |
---|
1286 | { |
---|
1287 | return(0,1); |
---|
1288 | } |
---|
1289 | |
---|
1290 | userMaxStep = getInput(#); |
---|
1291 | |
---|
1292 | // define a new basering "myRing" with new names for parameters |
---|
1293 | // and variables. |
---|
1294 | // The new names are 'a(1)', ..., 'a(npars)' for the parameters |
---|
1295 | // and 't(1)', ..., 't(nvars-2)', 'x', 'y' for the variables. |
---|
1296 | createMyRing(p_F, "ds"); |
---|
1297 | |
---|
1298 | createHNERing(); |
---|
1299 | |
---|
1300 | poly p_f = imap(myRing,p_F); |
---|
1301 | |
---|
1302 | error = checkPoly(p_f); |
---|
1303 | if (error) |
---|
1304 | { |
---|
1305 | return(0,1); |
---|
1306 | } |
---|
1307 | |
---|
1308 | // compute the multiplicity sequence of p_f. |
---|
1309 | multSeq, xNotTransversal, fIrreducible = calcMultSequence(p_f); |
---|
1310 | |
---|
1311 | if ( ! fIrreducible) |
---|
1312 | { |
---|
1313 | return(list(0,1)); |
---|
1314 | } |
---|
1315 | |
---|
1316 | setring myRing; |
---|
1317 | |
---|
1318 | if (xNotTransversal) |
---|
1319 | { |
---|
1320 | p_F = swapXY(p_F); |
---|
1321 | } |
---|
1322 | |
---|
1323 | if (userMaxStep != -1 && userMaxStep < size(multSeq)-1) |
---|
1324 | { |
---|
1325 | maxStep = userMaxStep; |
---|
1326 | } |
---|
1327 | else |
---|
1328 | { |
---|
1329 | maxStep = size(multSeq)-1; |
---|
1330 | } |
---|
1331 | |
---|
1332 | int isES = equisingTest(p_F, multSeq, maxStep); |
---|
1333 | |
---|
1334 | return(list(isES, error)); |
---|
1335 | } |
---|
1336 | |
---|
1337 | example |
---|
1338 | { |
---|
1339 | "EXAMPLE:"; echo=2; |
---|
1340 | ring r = 11,(T,x,y),ds; |
---|
1341 | poly F = (x+y)^2+y^3*T+x^7; |
---|
1342 | isEquising(F); |
---|
1343 | isEquising(F,1); |
---|
1344 | isEquising(F,2); |
---|
1345 | ideal I = ideal(T); |
---|
1346 | qring q = std(I); |
---|
1347 | poly F = imap(r,F); |
---|
1348 | isEquising(F,2); |
---|
1349 | } |
---|
1350 | |
---|