1 | version="$Id: equising.lib,v 1.7 2001-02-05 12:01:37 lossen Exp $"; |
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2 | category="Singularities"; |
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3 | info=" |
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4 | LIBRARY: equising.lib Equisingularity Stratum of a Family of Plane Curves |
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5 | AUTHOR: Andrea Mindnich, mindnich@mathematik.uni-kl.de |
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6 | |
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7 | PROCEDURES: |
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8 | esStratum(F[,m]); computes the equisingularity stratum of a deformation |
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9 | isEquising(F[,m]); tests if a given deformation is equisingular |
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10 | "; |
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11 | |
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12 | LIB "poly.lib"; |
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13 | LIB "elim.lib"; |
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14 | LIB "hnoether.lib"; |
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15 | /////////////////////////////////////////////////////////////////////////////// |
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16 | // COMPUTES a weight vector. x and y get weight 1 and all other |
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17 | // variables get weight 0. |
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18 | static proc xyVector() |
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19 | { |
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20 | intvec iv ; |
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21 | iv[nvars(basering)]=0 ; |
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22 | iv[rvar(x)] =1; |
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23 | iv[rvar(y)] =1; |
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24 | return (iv); |
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25 | } |
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26 | /////////////////////////////////////////////////////////////////////////////// |
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27 | // exchanges the variables x and y in the polynomial p_f |
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28 | static proc swapXY(poly f) |
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29 | { |
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30 | def r_base = basering; |
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31 | ideal MI = maxideal(1); |
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32 | MI[rvar(x)]=y; |
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33 | MI[rvar(y)]=x; |
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34 | map phi = r_base, MI; |
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35 | f=phi(f); |
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36 | return (f); |
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37 | } |
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38 | /////////////////////////////////////////////////////////////////////////////// |
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39 | // ASSUME: p_mon is a monomial |
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40 | // and p_g is the product of the variables in p_mon. |
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41 | // COMPUTES the coefficient of p_mon in p_h. |
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42 | static proc coefficient(poly p_h, poly p_mon, poly p_g) |
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43 | { |
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44 | matrix coefMatrix = coef(p_h,p_g); |
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45 | int nc = ncols(coefMatrix); |
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46 | int ii=1; |
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47 | poly p_c=0; |
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48 | while(ii<=nc) |
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49 | { |
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50 | if (coefMatrix[1,ii] == p_mon) |
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51 | { |
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52 | p_c = coefMatrix[2,ii]; |
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53 | break; |
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54 | } |
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55 | ii++; |
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56 | } |
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57 | return (p_c); |
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58 | } |
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59 | /////////////////////////////////////////////////////////////////////////////// |
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60 | // in p_F the variable p_vari is substituted by the polynomial p_g. |
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61 | static proc eSubst(poly p_F, poly p_vari, poly p_g) |
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62 | { |
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63 | def r_base = basering; |
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64 | ideal MI; |
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65 | map phi; |
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66 | int ii = rvar(p_vari); |
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67 | if (ii != 0) |
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68 | { |
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69 | MI = maxideal(1); |
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70 | MI[ii] = p_g; |
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71 | phi = r_base, MI; |
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72 | p_F = phi(p_F); |
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73 | } |
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74 | return (p_F); |
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75 | } |
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76 | /////////////////////////////////////////////////////////////////////////////// |
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77 | // All ring variables of p_F which occur in (the set of generators of) the |
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78 | // ideal Id, are substituted by 0 |
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79 | static proc SimplifyF(poly p_F, ideal Id) |
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80 | { |
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81 | int i=1; |
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82 | int si = size(Id); |
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83 | for (i=1; i <= si; i++) |
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84 | { |
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85 | if (rvar(Id[i])) |
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86 | { |
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87 | p_F = subst(p_F, Id[i], 0); |
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88 | } |
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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 | // Checks, if the basering is admissible. |
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98 | static proc checkBasering () |
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99 | { |
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100 | int error = 0; |
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101 | if(find(charstr(basering),"real")) |
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102 | { |
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103 | ERROR ("cannot compute esStratum with 'real' as coefficient field"); |
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104 | } |
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105 | if (nvars(basering) <= 2) |
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106 | { |
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107 | ERROR ("there are to less ringvariables to compute esStratum") |
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108 | } |
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109 | error = checkQIdeal(ideal(basering)); |
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110 | return(error); |
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111 | } |
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112 | /////////////////////////////////////////////////////////////////////////////// |
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113 | static proc getInput (list #) |
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114 | { |
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115 | def r_base = basering; |
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116 | |
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117 | int maxStep = -1; |
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118 | |
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119 | if (size(#) >= 1) |
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120 | { |
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121 | if (typeof(#[1]) == "int") |
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122 | { |
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123 | maxStep = #[1]; |
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124 | } |
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125 | |
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126 | else |
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127 | { |
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128 | ERROR("expected esStratum('poly', 'int') "); |
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129 | } |
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130 | } |
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131 | |
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132 | return(maxStep); |
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133 | } |
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134 | ////////////////////////////////////////////////////////////////////////////// |
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135 | // RETURNS: 0, if the ideal cond only depends on the deformation parameters |
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136 | // 1, otherwise. |
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137 | static proc checkQIdeal (ideal cond) |
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138 | { |
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139 | def r_base = basering; |
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140 | |
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141 | int i_print = printlevel-voice + 4; |
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142 | int i_nvars = nvars(basering); |
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143 | |
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144 | ideal id_help = subst(cond,var(i_nvars),0,var(i_nvars-1),0) - cond; |
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145 | |
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146 | // cond depends only on the first i_nvars-2 variables <=> |
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147 | // id_help == <0> |
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148 | if ( simplify(id_help, 2) != 0) |
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149 | { |
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150 | dbprint(i_print, |
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151 | "ideal(basering) must only depend on the deformation parameters"); |
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152 | return(1); |
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153 | } |
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154 | |
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155 | return(0); |
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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 | // COMPUTES string(minpoly) and substitutes the parameter by newParName |
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161 | static proc makeMinPolyString (string newParName) |
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162 | { |
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163 | int i; |
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164 | string parName = parstr(basering); |
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165 | int parNameSize = size(parName); |
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166 | |
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167 | string oldMinPolyStr = string (minpoly); |
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168 | int minPolySize = size(oldMinPolyStr); |
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169 | |
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170 | string newMinPolyStr = ""; |
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171 | |
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172 | for (i=1;i <= minPolySize; i++) |
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173 | { |
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174 | if (oldMinPolyStr[i,parNameSize] == parName) |
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175 | { |
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176 | newMinPolyStr = newMinPolyStr + newParName; |
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177 | i = i + parNameSize-1; |
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178 | } |
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179 | else |
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180 | { |
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181 | newMinPolyStr = newMinPolyStr + oldMinPolyStr[i]; |
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182 | } |
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183 | } |
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184 | |
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185 | return(newMinPolyStr); |
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186 | } |
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187 | |
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188 | |
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189 | /////////////////////////////////////////////////////////////////////////////// |
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190 | // Defines a new ring without deformation-parameters. |
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191 | static proc createHNERing() |
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192 | { |
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193 | string str; |
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194 | string minpolyStr = string(minpoly); |
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195 | |
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196 | str = " ring HNERing = (" + charstr(basering) + "), (x,y), ls;"; |
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197 | execute (str); |
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198 | |
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199 | str = "minpoly ="+ minpolyStr+";"; |
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200 | execute(str); |
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201 | |
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202 | keepring(HNERing); |
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203 | } |
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204 | /////////////////////////////////////////////////////////////////////////////// |
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205 | // RETURNS: 1, if p_f = 0 or char(basering) divides the order of p_f |
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206 | // or p_f is not squarefree. |
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207 | // 0, otherwise |
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208 | static proc checkPoly (poly p_f) |
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209 | { |
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210 | int i_print = printlevel - voice + 3; |
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211 | int i_ord; |
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212 | |
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213 | if (p_f == 0) |
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214 | { |
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215 | dbprint(i_print,"The Input is a 'deformation' of the zero polynomial"); |
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216 | return(1); |
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217 | } |
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218 | |
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219 | i_ord = mindeg1(p_f); |
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220 | |
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221 | if (number(i_ord) == 0) |
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222 | { |
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223 | dbprint(i_print, |
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224 | "The characteristic of the coefficient field |
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225 | divides the order of the equation"); |
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226 | return(1); |
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227 | } |
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228 | |
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229 | if (squarefree(p_f) != p_f) |
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230 | { |
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231 | dbprint(i_print, "The curve is reducible"); |
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232 | return(1); |
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233 | } |
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234 | |
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235 | return(0); |
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236 | } |
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237 | /////////////////////////////////////////////////////////////////////////////// |
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238 | // COMPUTES the multiplicity sequence of p_f |
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239 | static proc calcMultSequence (poly p_f) |
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240 | { |
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241 | int i_print = printlevel-voice + 3; |
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242 | intvec multSeq=0; |
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243 | list hneList; |
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244 | int xNotTransversal; |
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245 | int fIrreducible = 1; |
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246 | |
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247 | // if basering = (p,a) or (p,a(1..s)), |
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248 | // p prime, a algebraic, a(1..s) transcendent use reddevelop |
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249 | // otherwise use develop |
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250 | if (char(basering) != 0 |
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251 | && npars(basering) !=0 |
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252 | && charstr(basering) == string(char(basering)) + "," + parstr(basering)) |
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253 | { |
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254 | hneList = reddevelop(p_f, -1); |
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255 | if ( size(hneList)>=2) |
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256 | { |
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257 | fIrreducible = 0; |
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258 | dbprint(i_print, "The curve is reducible"); |
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259 | return(multSeq, xNotTransversal, fIrreducible); |
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260 | } |
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261 | |
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262 | hneList = hneList[1]; |
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263 | |
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264 | xNotTransversal= hneList[3]; |
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265 | } |
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266 | else |
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267 | { |
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268 | hneList = develop(p_f, -1); |
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269 | |
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270 | xNotTransversal= hneList[3]; |
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271 | fIrreducible = hneList[5]; |
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272 | } |
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273 | |
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274 | if ( ! fIrreducible) |
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275 | { |
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276 | dbprint(i_print, "The curve is reducible"); |
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277 | return(multSeq, xNotTransversal, fIrreducible); |
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278 | } |
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279 | |
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280 | multSeq = multsequence (hneList); |
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281 | return(multSeq, xNotTransversal, fIrreducible); |
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282 | } |
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283 | /////////////////////////////////////////////////////////////////////////////// |
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284 | // ASSUME: The basering is no qring and has at least 3 variables |
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285 | // DEFINES: A new basering, "myRing", |
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286 | // with new names for the parameters and variables. |
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287 | // The new names for the parameters are a(1..k), |
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288 | // and t(1..s),x,y for the variables |
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289 | // The ring ordering is ordStr. |
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290 | // NOTE: This proc uses 'execute'. |
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291 | static proc createMyRing(poly p_F, string ordStr ) |
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292 | { |
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293 | def r_old = basering; |
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294 | |
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295 | int chara = char(basering); |
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296 | string charaStr; |
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297 | int i; |
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298 | string minPolyStr = ""; |
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299 | string helpStr; |
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300 | int nDefParams = nvars(r_old)-2; |
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301 | |
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302 | |
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303 | ideal qIdeal = ideal(basering); |
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304 | |
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305 | if (npars(basering) == 0) |
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306 | { |
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307 | helpStr = "ring myRing =" |
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308 | + string(chara)+ ", (t(1..nDefParams), x, y),"+ ordStr +";"; |
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309 | execute(helpStr); |
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310 | } |
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311 | |
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312 | else |
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313 | { |
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314 | charaStr = charstr(basering); |
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315 | if (charaStr == string(chara) + "," + parstr(basering)) |
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316 | { |
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317 | if (minpoly !=0) |
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318 | { |
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319 | minPolyStr = makeMinPolyString("a"); |
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320 | |
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321 | helpStr = "ring myRing = |
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322 | (" + string(chara) + ",a), |
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323 | (t(1..nDefParams), x, y)," + ordStr + ";"; |
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324 | execute(helpStr); |
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325 | |
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326 | helpStr = "minpoly =" + minPolyStr + ";"; |
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327 | execute (helpStr); |
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328 | } |
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329 | else |
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330 | { |
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331 | helpStr = "ring myRing = |
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332 | (" + string(chara) + ",a(1..npars(basering)) ), |
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333 | (t(1..nDefParams), x, y)," + ordStr + ";"; |
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334 | execute(helpStr); |
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335 | } |
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336 | } |
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337 | else |
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338 | { |
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339 | i = find (charaStr,","); |
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340 | |
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341 | helpStr = " ring myRing = |
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342 | (" + charaStr[1,i] + "a), |
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343 | (t(1..nDefParams), x, y)," + ordStr + ";"; |
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344 | |
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345 | execute (helpStr); |
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346 | } |
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347 | } |
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348 | |
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349 | ideal qIdeal = fetch(r_old, qIdeal); |
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350 | |
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351 | if(qIdeal != 0) |
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352 | { |
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353 | def r_base = basering; |
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354 | kill myRing; |
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355 | qring myRing = std(qIdeal); |
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356 | } |
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357 | |
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358 | poly p_F = fetch(r_old, p_F); |
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359 | ideal ES; |
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360 | |
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361 | keepring(myRing); |
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362 | } |
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363 | |
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364 | /////////////////////////////////////////////////////////////////////////////// |
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365 | /////////// procedures to compute the equisingularity stratum ///////////////// |
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366 | /////////////////////////////////////////////////////////////////////////////// |
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367 | // DEFINES a new basering, myRing, which has one variable |
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368 | // more than the old ring. |
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369 | // The name for the new variable is "H(nhelpV)". |
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370 | // p_F and ES are "imaped" into the new ring. |
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371 | static proc extendRing (poly p_F, ideal ES, int nHelpV, ideal HCond) |
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372 | { |
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373 | def r_old = basering; |
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374 | |
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375 | string helpStr; |
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376 | string minPolyStr = ""; |
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377 | |
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378 | ideal qIdeal = ideal(basering); |
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379 | |
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380 | if (minpoly != 0) |
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381 | { |
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382 | if (charstr(basering) == string(char(basering)) + "," + parstr(basering)) |
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383 | { |
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384 | minPolyStr = string(minpoly); |
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385 | } |
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386 | } |
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387 | |
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388 | string str = |
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389 | "ring myRing = (" + charstr(r_old) + "), |
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390 | (H(" + string(nHelpV)+ ")," + string(maxideal(1)) + "), |
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391 | (dp(" + string(nHelpV) + "),dp);"; |
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392 | execute (str); |
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393 | |
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394 | if (minPolyStr != "") |
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395 | { |
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396 | helpStr = "minpoly =" + minPolyStr + ";"; |
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397 | execute(helpStr); |
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398 | } |
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399 | |
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400 | ideal qIdeal = imap(r_old, qIdeal); |
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401 | if(qIdeal != 0) |
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402 | { |
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403 | def r_base = basering; |
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404 | |
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405 | kill myRing; |
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406 | |
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407 | attrib(qIdeal,"isSB",1); |
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408 | qring myRing = qIdeal; |
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409 | } |
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410 | |
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411 | |
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412 | poly p_F = imap(r_old, p_F); |
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413 | ideal ES = imap(r_old, ES); |
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414 | ideal HCond = imap(r_old, HCond); |
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415 | |
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416 | keepring(myRing); |
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417 | } |
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418 | /////////////////////////////////////////////////////////////////////////////// |
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419 | // COMPUTES an ideal equimultCond, such that F_(n-1) mod equimultCond =0, |
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420 | // where F_(n-1) is the (nNew-1)-jet of p_F with respect to x,y. |
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421 | static proc calcEquimultCond(poly p_F, int nNew) |
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422 | { |
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423 | ideal equimultCond = 0; |
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424 | poly p_FnMinus1; |
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425 | matrix coefMatrix; |
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426 | int nc; |
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427 | int ii = 1; |
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428 | |
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429 | p_FnMinus1 = jet(p_F, nNew-1, xyVector()); |
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430 | |
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431 | coefMatrix = coef(p_FnMinus1, xy); |
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432 | |
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433 | nc = ncols(coefMatrix); |
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434 | |
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435 | for (ii=1; ii<=nc; ii++) |
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436 | { |
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437 | equimultCond[ii] = NF(coefMatrix[2,ii],std(0)); |
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438 | } |
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439 | |
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440 | p_F = p_F - p_FnMinus1; |
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441 | p_F = SimplifyF(p_F, equimultCond); |
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442 | |
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443 | return(equimultCond, p_F); |
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444 | } |
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445 | /////////////////////////////////////////////////////////////////////////////// |
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446 | // COMPUTES smallest integer >= nNew/nOld -1 |
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447 | static proc calcNZeroSteps (int nOld,int nNew) |
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448 | { |
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449 | int nZeroSteps; |
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450 | |
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451 | if (nOld mod nNew == 0) |
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452 | { |
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453 | nZeroSteps = nOld div nNew -1; |
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454 | } |
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455 | |
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456 | else |
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457 | { |
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458 | nZeroSteps = nOld div nNew; |
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459 | } |
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460 | |
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461 | return(nZeroSteps); |
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462 | } |
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463 | /////////////////////////////////////////////////////////////////////////////// |
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464 | // ASSUME: ord_(X,Y)(F)=nNew |
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465 | // COMPUTES an ideal I such that (p_F mod I)_nNew = p_c*y^nNew. |
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466 | static proc purePowerOfY (poly p_F, int nNew) |
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467 | { |
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468 | ideal id_help = 0; |
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469 | poly p_Fn; |
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470 | matrix coefMatrix; |
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471 | int nc; |
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472 | poly p_c; |
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473 | int ii=1; |
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474 | |
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475 | p_Fn = jet(p_F, nNew, xyVector()); |
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476 | |
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477 | coefMatrix = coef(p_Fn, xy); |
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478 | nc = ncols(coefMatrix); |
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479 | |
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480 | p_c = coefMatrix[2,nc]; |
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481 | |
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482 | for (ii=1; ii <= nc-1; ii++) |
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483 | { |
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484 | id_help[ii] = NF(coefMatrix[2,ii],std(0)); |
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485 | } |
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486 | |
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487 | p_F = p_F - p_Fn + p_c*y^nNew; |
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488 | |
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489 | p_F = SimplifyF(p_F, id_help); |
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490 | |
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491 | return(id_help, p_F, p_c); |
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492 | } |
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493 | /////////////////////////////////////////////////////////////////////////////// |
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494 | // ASSUME: ord_(X,Y)(F)=nNew |
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495 | // COMPUTES an ideal I such that p_Fn mod I = p_c*(y-p_a*x)^nNew, |
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496 | // where p_Fn is the homogeneous part of p_F of order nNew. |
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497 | static proc purePowerOfLin (poly p_F, ideal HCond, int nNew, int nHelpV) |
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498 | { |
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499 | ideal id_help = 0; |
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500 | poly p_Fn; |
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501 | matrix coefMatrix; |
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502 | poly p_c; |
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503 | poly p_ca; |
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504 | poly p_help; |
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505 | poly p_a; |
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506 | int ii; |
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507 | int jj; |
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508 | int bico; |
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509 | |
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510 | p_Fn = jet(p_F, nNew, xyVector()); |
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511 | |
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512 | coefMatrix = coeffs(subst(p_Fn,x,1),y); |
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513 | |
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514 | p_c = coefMatrix[nNew+1,1]; |
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515 | p_ca = coefMatrix[nNew,1]/(-nNew); |
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516 | |
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517 | if (npars(basering)==1 && charstr(basering) != string(char(basering)) + "," + parstr(basering)) |
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518 | { |
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519 | p_a = H(nHelpV); |
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520 | HCond = HCond + ideal(p_ca - p_a*p_c); |
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521 | } |
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522 | else |
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523 | { |
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524 | p_help = p_ca/p_c; |
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525 | if (p_help * p_c == p_ca) |
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526 | { |
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527 | p_a = p_help; |
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528 | } |
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529 | else |
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530 | { |
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531 | p_a = H(nHelpV); |
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532 | HCond = HCond + ideal(p_ca - p_a*p_c); |
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533 | } |
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534 | } |
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535 | |
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536 | bico = (nNew*(nNew-1))/2; |
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537 | |
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538 | for (ii = 2; ii <= nNew ; ii++) |
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539 | { |
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540 | |
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541 | if (coefMatrix[nNew+1-ii,1] == 0) |
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542 | { |
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543 | if (number(bico) != 0) |
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544 | // Then a^i=0 since c is a unit |
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545 | { |
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546 | id_help = id_help + ideal(NF(p_a^(ii),std(0))); |
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547 | for (jj = ii+1; jj <= nNew; jj++) |
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548 | // the remaining coefficients (of y^(nnew-k)*x^k with k>i) |
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549 | // are also zero. |
---|
550 | { |
---|
551 | id_help = id_help |
---|
552 | + ideal(NF(coefMatrix[nNew+1-jj,1],std(0))); |
---|
553 | } |
---|
554 | break; |
---|
555 | } |
---|
556 | } |
---|
557 | |
---|
558 | else |
---|
559 | { |
---|
560 | id_help = id_help + |
---|
561 | ideal(NF(coefMatrix[nNew+1-ii,1] - bico*p_c*(-p_a)^ii,std(0))); |
---|
562 | } |
---|
563 | |
---|
564 | bico = (bico*(nNew-ii))/(ii+1); |
---|
565 | } |
---|
566 | |
---|
567 | p_F = SimplifyF(p_F, id_help); |
---|
568 | |
---|
569 | return(id_help, HCond, p_F, p_c, p_a); |
---|
570 | } |
---|
571 | /////////////////////////////////////////////////////////////////////////////// |
---|
572 | // eliminates the variables H(1),..,H(nHelpV) from the ideal ES + HCond |
---|
573 | static proc helpVarElim(ideal ES, ideal HCond, int nHelpV) |
---|
574 | { |
---|
575 | ES = ES + HCond; |
---|
576 | ES = std(ES); |
---|
577 | ES = nselect(ES,1,nHelpV); |
---|
578 | |
---|
579 | return(ES); |
---|
580 | } |
---|
581 | /////////////////////////////////////////////////////////////////////////////// |
---|
582 | // ASSUME: ord(F)=nNew and p_c(y-p_a*x)^n is the nNew-jet of F with respect |
---|
583 | // to X,Y |
---|
584 | // COMPUTES F(x,yx+a*x)/x^n |
---|
585 | static proc formalBlowUp(poly p_F, poly p_c, poly p_a, int nNew) |
---|
586 | { |
---|
587 | |
---|
588 | p_F = p_F - jet(p_F, nNew, xyVector()); |
---|
589 | |
---|
590 | if (p_a != 0) |
---|
591 | { |
---|
592 | p_F = eSubst(p_F, y , yx + p_a*x); |
---|
593 | } |
---|
594 | else |
---|
595 | { |
---|
596 | p_F = subst(p_F, y, xy); |
---|
597 | } |
---|
598 | |
---|
599 | p_F = p_F/(x^nNew); |
---|
600 | |
---|
601 | p_F = p_F + p_c * y^nNew; |
---|
602 | |
---|
603 | return (p_F); |
---|
604 | } |
---|
605 | /////////////////////////////////////////////////////////////////////////////// |
---|
606 | // ASSUME: p_F in K[t(1)..t(s),x,y] |
---|
607 | // COMPUTES the minimal ideal ES, such that the deformation p_F mod ES is |
---|
608 | // equisingular. |
---|
609 | // The computation is done up to iteration step 'maxstep'. |
---|
610 | // RETURNS: list l, such that |
---|
611 | // l[1]=1 if some error has occured, |
---|
612 | // l[1]=0 otherwise and then l[2] = es_cond. |
---|
613 | static proc calcEsCond(poly p_F, intvec multSeq, int maxStep) |
---|
614 | { |
---|
615 | def r_old = basering; |
---|
616 | |
---|
617 | ideal ES = 0; |
---|
618 | |
---|
619 | int ii; |
---|
620 | int step = 1; |
---|
621 | int nNew = multSeq[step]; |
---|
622 | int nOld = nNew; |
---|
623 | int nZeroSteps; |
---|
624 | int nHelpV = 1; |
---|
625 | ideal HCond = 0; |
---|
626 | int maxDeg = 0; |
---|
627 | int z = printlevel - voice + 3; |
---|
628 | string str; |
---|
629 | |
---|
630 | extendRing(p_F, ES, nHelpV, HCond); |
---|
631 | |
---|
632 | poly p_c; |
---|
633 | poly p_a; |
---|
634 | ideal I; |
---|
635 | |
---|
636 | for (ii = 1; ii <= size(multSeq); ii++) |
---|
637 | { |
---|
638 | maxDeg = maxDeg + multSeq[ii]; |
---|
639 | } |
---|
640 | |
---|
641 | while (step <= maxStep) |
---|
642 | { |
---|
643 | |
---|
644 | nOld = nNew; |
---|
645 | nNew = multSeq[step]; |
---|
646 | |
---|
647 | p_F = jet(p_F, maxDeg, xyVector()); |
---|
648 | maxDeg = maxDeg - nNew; |
---|
649 | |
---|
650 | if (nNew<nOld) |
---|
651 | //start a new line in the HNE of F |
---|
652 | // _ _ |
---|
653 | // for the next | nold/nnew -1 | iteration steps the coefficient 'a' |
---|
654 | // in the leading form Fn = c(y-ax)^n should be zero. |
---|
655 | { |
---|
656 | p_F = swapXY(p_F); |
---|
657 | nZeroSteps = calcNZeroSteps (nOld, nNew); |
---|
658 | } |
---|
659 | |
---|
660 | I, p_F = calcEquimultCond (p_F, nNew); |
---|
661 | ES = ES + I; |
---|
662 | |
---|
663 | if (z>1) |
---|
664 | { |
---|
665 | "// conditions for equimultiplicity in step:", step; |
---|
666 | I; |
---|
667 | if (nHelpV >1) |
---|
668 | { |
---|
669 | str = string(nHelpV); |
---|
670 | "// conditions for help variables H(1),..,H("+str+"):"; |
---|
671 | HCond; |
---|
672 | } |
---|
673 | pause("press <return> to continue"); |
---|
674 | } |
---|
675 | |
---|
676 | if (nZeroSteps > 0) |
---|
677 | { |
---|
678 | nZeroSteps--; |
---|
679 | |
---|
680 | // compute conditions, s.th. Fn = c*y^nnew ? |
---|
681 | I, p_F, p_c = purePowerOfY (p_F, nNew); |
---|
682 | ES = ES + I; |
---|
683 | |
---|
684 | if (z>1) |
---|
685 | { |
---|
686 | "// conditions for pure power in step:", step; |
---|
687 | I; |
---|
688 | if (nHelpV > 1) |
---|
689 | { |
---|
690 | str = string(nHelpV); |
---|
691 | "// conditions for help variables H(1),..,H("+str+"):"; |
---|
692 | HCond; |
---|
693 | } |
---|
694 | pause("press <return> to continue"); |
---|
695 | } |
---|
696 | p_a =0; |
---|
697 | } |
---|
698 | |
---|
699 | else |
---|
700 | { |
---|
701 | I, HCond, p_F, p_c, p_a = purePowerOfLin (p_F, HCond, nNew, nHelpV); |
---|
702 | |
---|
703 | ES = ES + I; |
---|
704 | |
---|
705 | if (z>1) |
---|
706 | { |
---|
707 | "// conditions for pure power in step:", step; |
---|
708 | I; |
---|
709 | str = string(nHelpV); |
---|
710 | "// conditions for help variables H(1),..,H("+str+"):"; |
---|
711 | HCond; |
---|
712 | pause("press <return> to continue"); |
---|
713 | } |
---|
714 | } |
---|
715 | |
---|
716 | p_F = formalBlowUp (p_F, p_c, p_a, nNew); |
---|
717 | |
---|
718 | if (p_a == H(nHelpV)) |
---|
719 | { |
---|
720 | nHelpV++; |
---|
721 | |
---|
722 | def r_base = basering; |
---|
723 | kill myRing; |
---|
724 | |
---|
725 | extendRing(p_F, ES, nHelpV, HCond); |
---|
726 | |
---|
727 | kill r_base; |
---|
728 | |
---|
729 | poly p_a; |
---|
730 | poly p_c; |
---|
731 | ideal I; |
---|
732 | } |
---|
733 | step++; |
---|
734 | } |
---|
735 | if (nHelpV > 1) |
---|
736 | { |
---|
737 | ES = helpVarElim(ES, HCond, nHelpV); |
---|
738 | } |
---|
739 | |
---|
740 | if (nameof(basering)=="myRing") |
---|
741 | { |
---|
742 | setring r_old; |
---|
743 | ES = imap(myRing, ES); |
---|
744 | } |
---|
745 | |
---|
746 | return(ES); |
---|
747 | } |
---|
748 | |
---|
749 | /////////////////////////////////////////////////////////////////////////////// |
---|
750 | // main procedure |
---|
751 | /////////////////////////////////////////////////////////////////////////////// |
---|
752 | |
---|
753 | proc esStratum (poly p_F, list #) |
---|
754 | "USAGE: esStratum(F[,m]); F poly, m int |
---|
755 | ASSUME: F defines a deformation of an irreducible bivariate polynomial f |
---|
756 | and the characteristic of the basering does not divide mult(f). @* |
---|
757 | If nv is the number of variables of the basering, then the first nv-2 |
---|
758 | variables are the deformation parameters. @* |
---|
759 | If the basering is a qring, ideal(basering) must only depend |
---|
760 | on the deformation parameters. |
---|
761 | RETURN: list l of an ideal and an integer, where |
---|
762 | @format |
---|
763 | l[1] is the ideal in the deformation parameters, defining the ES-stratum of F, |
---|
764 | l[2]=1 if some error has occured, l[2]=0 otherwise. |
---|
765 | @end format |
---|
766 | NOTE: If m is given, the computation stops after m steps of the iteration. @* |
---|
767 | printlevel > 0 displays comments and pauses after intermediate |
---|
768 | computations (default: printlevel=0) @* |
---|
769 | This procedure uses @code{execute} or calls a procedure using |
---|
770 | @code{execute}. |
---|
771 | EXAMPLE: example esStratum; shows an example |
---|
772 | " |
---|
773 | { |
---|
774 | def r_user = basering; |
---|
775 | |
---|
776 | int ii = 1; |
---|
777 | int i_nvars = nvars(basering); |
---|
778 | int error = 0; |
---|
779 | int xNotTransversal; |
---|
780 | int fIrreducible; |
---|
781 | int maxStep; |
---|
782 | int userMaxStep; |
---|
783 | ideal cond; |
---|
784 | intvec multSeq; |
---|
785 | ideal ES = 0; |
---|
786 | |
---|
787 | error = checkBasering(); |
---|
788 | if (error) |
---|
789 | { |
---|
790 | return(list(ES,error)); |
---|
791 | } |
---|
792 | |
---|
793 | userMaxStep = getInput(#); |
---|
794 | |
---|
795 | // define a new basering "myRing" with new names for parameters |
---|
796 | // and variables. |
---|
797 | // The new names are 'a(1)', ..., 'a(npars)' for the parameters |
---|
798 | // and 't(1)', ..., 't(nvars-2)', 'x', 'y' for the variables. |
---|
799 | createMyRing(p_F, "dp"); |
---|
800 | |
---|
801 | // define a ring without deformation parameters, to compute the HNE |
---|
802 | // of F mod <t_1,..,t_s> |
---|
803 | createHNERing(); |
---|
804 | |
---|
805 | poly p_f = imap(myRing,p_F); |
---|
806 | |
---|
807 | error = checkPoly(p_f); |
---|
808 | if (error) |
---|
809 | { |
---|
810 | setring r_user; |
---|
811 | return(list( ideal(0),error)); |
---|
812 | } |
---|
813 | |
---|
814 | // compute the multiplicitysequence of p_f. |
---|
815 | multSeq, xNotTransversal, fIrreducible = calcMultSequence(p_f); |
---|
816 | |
---|
817 | if ( ! fIrreducible) |
---|
818 | { |
---|
819 | setring r_user; |
---|
820 | return(list(ideal(0),1)); |
---|
821 | } |
---|
822 | |
---|
823 | setring myRing; |
---|
824 | |
---|
825 | if (xNotTransversal) |
---|
826 | { |
---|
827 | p_F = swapXY(p_F); |
---|
828 | } |
---|
829 | |
---|
830 | if (userMaxStep != -1 && userMaxStep < size(multSeq)-1) |
---|
831 | { |
---|
832 | maxStep = userMaxStep; |
---|
833 | } |
---|
834 | else |
---|
835 | { |
---|
836 | maxStep = size(multSeq)-1; |
---|
837 | } |
---|
838 | |
---|
839 | ES = calcEsCond(p_F, multSeq, maxStep); |
---|
840 | |
---|
841 | setring r_user; |
---|
842 | ES = fetch(myRing, ES); |
---|
843 | |
---|
844 | return(list(ES, error)); |
---|
845 | } |
---|
846 | |
---|
847 | example |
---|
848 | { |
---|
849 | "EXAMPLE:"; echo=2; |
---|
850 | ring r = 11,(a,b,c,d,e,f,g,x,y),ds; |
---|
851 | poly F = (x2+2xy+y2+x5)+ax+by+cx2+dxy+ey2+fx3+gx4; |
---|
852 | esStratum(F); |
---|
853 | esStratum(F,2); |
---|
854 | ideal I = f-fa,e+b; |
---|
855 | qring q = std(I); |
---|
856 | poly F = imap(r,F); |
---|
857 | esStratum(F); |
---|
858 | } |
---|
859 | |
---|
860 | /////////////////////////////////////////////////////////////////////////////// |
---|
861 | // procedures for equisingularity test |
---|
862 | /////////////////////////////////////////////////////////////////////////////// |
---|
863 | |
---|
864 | // DEFINES a new basering, myRing, which has one variable |
---|
865 | // more than the old ring. |
---|
866 | // The name for the new variable is "H(nhelpV)". |
---|
867 | static proc T_extendRing(poly p_F, int nHelpV, ideal HCond) |
---|
868 | { |
---|
869 | def r_old = basering; |
---|
870 | |
---|
871 | ideal qIdeal = ideal(basering); |
---|
872 | |
---|
873 | string helpStr; |
---|
874 | string minPolyStr = ""; |
---|
875 | |
---|
876 | if(minpoly != 0) |
---|
877 | { |
---|
878 | if (charstr(basering) == string(char(basering)) + "," + parstr(basering)) |
---|
879 | { |
---|
880 | minPolyStr = string(minpoly); |
---|
881 | } |
---|
882 | } |
---|
883 | |
---|
884 | string str = "ring myRing = |
---|
885 | (" + charstr(r_old) + "), |
---|
886 | (H(" + string( nHelpV)+ ")," + string(maxideal(1)) + "), |
---|
887 | (dp(" + string( nHelpV) + "), ds);"; |
---|
888 | execute (str); |
---|
889 | |
---|
890 | if (minPolyStr != "") |
---|
891 | { |
---|
892 | helpStr = "minpoly =" + minPolyStr + ";"; |
---|
893 | execute(helpStr); |
---|
894 | } |
---|
895 | |
---|
896 | ideal qIdeal = imap(r_old, qIdeal); |
---|
897 | if(qIdeal != 0) |
---|
898 | { |
---|
899 | def r_base = basering; |
---|
900 | kill myRing; |
---|
901 | qring myRing = std(qIdeal); |
---|
902 | } |
---|
903 | |
---|
904 | poly p_F =imap(r_old, p_F); |
---|
905 | ideal HCond = imap(r_old, HCond); |
---|
906 | |
---|
907 | keepring(myRing); |
---|
908 | } |
---|
909 | /////////////////////////////////////////////////////////////////////////////// |
---|
910 | // tests, if ord p_F = nNew. |
---|
911 | static proc equimultTest (poly p_F, int nHelpV, int nNew, ideal HCond) |
---|
912 | { |
---|
913 | poly p_FnMinus1; |
---|
914 | ideal id_help; |
---|
915 | matrix coefMatrix; |
---|
916 | int i; |
---|
917 | int nc; |
---|
918 | |
---|
919 | p_FnMinus1 = jet(p_F, nNew-1, xyVector()); |
---|
920 | |
---|
921 | coefMatrix = coef(p_FnMinus1, xy); |
---|
922 | nc = ncols(coefMatrix); |
---|
923 | |
---|
924 | for (i=1; i<=nc; i++) |
---|
925 | { |
---|
926 | id_help[i] = coefMatrix[2,i]; |
---|
927 | } |
---|
928 | |
---|
929 | id_help = T_helpVarElim(id_help, HCond, nHelpV); |
---|
930 | |
---|
931 | if (reduce(id_help, std(0)) !=0 ) |
---|
932 | { |
---|
933 | return(0, p_F); |
---|
934 | } |
---|
935 | |
---|
936 | p_F = p_F - p_FnMinus1; |
---|
937 | |
---|
938 | return(1, p_F); |
---|
939 | } |
---|
940 | /////////////////////////////////////////////////////////////////////////////// |
---|
941 | // ASSUME: ord(p_F)=nNew |
---|
942 | // tests, if p_F = p_c*y^nNew for some p_c. |
---|
943 | static proc pPOfYTest (poly p_F, int nHelpV, int nNew, ideal HCond) |
---|
944 | { |
---|
945 | poly p_Fn; |
---|
946 | poly p_c; |
---|
947 | ideal id_help; |
---|
948 | int nc; |
---|
949 | int i=1; |
---|
950 | matrix coefMatrix; |
---|
951 | |
---|
952 | p_Fn = jet(p_F, nNew, xyVector()); |
---|
953 | |
---|
954 | coefMatrix = coef(p_Fn, xy); |
---|
955 | nc = ncols(coefMatrix); |
---|
956 | |
---|
957 | p_c = coefMatrix[2,1]; |
---|
958 | |
---|
959 | for (i = 2; i <= nc; i++) |
---|
960 | { |
---|
961 | id_help[i] = coefMatrix[2,i]; |
---|
962 | } |
---|
963 | |
---|
964 | id_help = T_helpVarElim(id_help, HCond, nHelpV); |
---|
965 | |
---|
966 | if (reduce(id_help, std(0)) !=0 ) |
---|
967 | { |
---|
968 | return(0, p_c); |
---|
969 | } |
---|
970 | |
---|
971 | return(1, p_c); |
---|
972 | } |
---|
973 | /////////////////////////////////////////////////////////////////////////////// |
---|
974 | // ASSUME: ord(p_F)=nNew |
---|
975 | // tests, if p_F = p_c*(y - p_a*x)^nNew for some p_a, p_c. |
---|
976 | static proc pPOfLinTest(poly p_F, int nNew, int nHelpV, ideal HCond) |
---|
977 | { |
---|
978 | poly p_Fn; |
---|
979 | poly p_c; |
---|
980 | poly p_ca; |
---|
981 | poly p_help; |
---|
982 | poly p_a; |
---|
983 | ideal id_help; |
---|
984 | |
---|
985 | p_Fn = jet(p_F, nNew, xyVector()); |
---|
986 | |
---|
987 | p_c = coefficient(p_Fn,y^nNew,y); |
---|
988 | p_ca = coefficient(p_Fn,y^(nNew-1)*x,xy)/-nNew; |
---|
989 | |
---|
990 | if (npars(basering)==1 |
---|
991 | && charstr(basering) != string(char(basering)) + "," + parstr(basering)) |
---|
992 | { |
---|
993 | p_a = H(nHelpV); |
---|
994 | HCond = HCond + ideal(p_ca - p_a*p_c); |
---|
995 | } |
---|
996 | else |
---|
997 | { |
---|
998 | p_help = p_ca/p_c; |
---|
999 | if (p_help * p_c == p_ca) |
---|
1000 | { |
---|
1001 | p_a = p_help; |
---|
1002 | } |
---|
1003 | else |
---|
1004 | { |
---|
1005 | p_a = H(nHelpV); |
---|
1006 | HCond = HCond + ideal(p_ca - p_a*p_c); |
---|
1007 | } |
---|
1008 | } |
---|
1009 | |
---|
1010 | id_help = ideal(p_Fn - p_c *(y - p_a *x)^nNew); |
---|
1011 | id_help = T_helpVarElim(id_help, HCond, nHelpV); |
---|
1012 | |
---|
1013 | if (reduce(id_help, std(0)) != 0 ) |
---|
1014 | { |
---|
1015 | return(0, p_F, p_c, p_a, HCond); |
---|
1016 | } |
---|
1017 | |
---|
1018 | return(1, p_F, p_c, p_a, HCond); |
---|
1019 | } |
---|
1020 | ////////////////////////////////////////////////////////////////////////////// |
---|
1021 | // eliminates the variables H(1),..,H(nHelpV) from the ideal ES + HCond |
---|
1022 | static proc T_helpVarElim(ideal ES, ideal HCond, int nHelpV) |
---|
1023 | { |
---|
1024 | |
---|
1025 | def r_old = basering; |
---|
1026 | |
---|
1027 | ideal qIdeal = ideal(basering); |
---|
1028 | |
---|
1029 | string helpStr; |
---|
1030 | string minPolyStr = ""; |
---|
1031 | |
---|
1032 | if(minpoly != 0) |
---|
1033 | { |
---|
1034 | if (charstr(basering) == string(char(basering)) + "," + parstr(basering)) |
---|
1035 | { |
---|
1036 | minPolyStr = string(minpoly); |
---|
1037 | } |
---|
1038 | } |
---|
1039 | |
---|
1040 | string str = "ring myRing = |
---|
1041 | (" + charstr(r_old) + "),(" + string(maxideal(1)) + "), |
---|
1042 | (dp(" + string( nHelpV) + "), dp);"; |
---|
1043 | execute (str); |
---|
1044 | |
---|
1045 | if (minPolyStr != "") |
---|
1046 | { |
---|
1047 | helpStr = "minpoly =" + minPolyStr + ";"; |
---|
1048 | execute(helpStr); |
---|
1049 | } |
---|
1050 | |
---|
1051 | ideal qIdeal = imap(r_old, qIdeal); |
---|
1052 | if(qIdeal != 0) |
---|
1053 | { |
---|
1054 | def r_base = basering; |
---|
1055 | kill myRing; |
---|
1056 | qring myRing = std(qIdeal); |
---|
1057 | } |
---|
1058 | |
---|
1059 | ideal ES = imap(r_old, ES); |
---|
1060 | ideal HCond = imap(r_old, HCond); |
---|
1061 | |
---|
1062 | ES = ES + HCond; |
---|
1063 | ES = std(ES); |
---|
1064 | ES = nselect(ES,1,nHelpV); |
---|
1065 | |
---|
1066 | setring r_old; |
---|
1067 | ES = imap (myRing,ES); |
---|
1068 | |
---|
1069 | return(ES); |
---|
1070 | } |
---|
1071 | /////////////////////////////////////////////////////////////////////////////// |
---|
1072 | // ASSUME: F in K[t(1)..t(s),x,y], |
---|
1073 | // the ringordering is ds |
---|
1074 | // RETURNS: list l, such that |
---|
1075 | // l[1]=1 if some error has occured, |
---|
1076 | // l[1]=0 otherwise and then |
---|
1077 | // l[2] = 1, if the deformation is equisingular and |
---|
1078 | // l[2] = 0 otherwise. |
---|
1079 | static proc equisingTest (poly p_F, intvec multSeq, int maxStep) |
---|
1080 | { |
---|
1081 | def r_old = basering; |
---|
1082 | |
---|
1083 | ideal id_Es = 0; |
---|
1084 | |
---|
1085 | int isES = 1; |
---|
1086 | int step = 1; |
---|
1087 | int nNew = multSeq[step]; |
---|
1088 | int nOld = nNew; |
---|
1089 | int zeroSteps; |
---|
1090 | ideal HCond = 0; |
---|
1091 | int nHelpV = 1; |
---|
1092 | |
---|
1093 | T_extendRing (p_F, nHelpV, HCond); |
---|
1094 | |
---|
1095 | poly p_c; |
---|
1096 | poly p_a; |
---|
1097 | |
---|
1098 | while (step <= maxStep) |
---|
1099 | { |
---|
1100 | nOld = nNew; |
---|
1101 | nNew = multSeq[step]; |
---|
1102 | |
---|
1103 | if (nNew < nOld) |
---|
1104 | //start a new line in the HNE of F |
---|
1105 | // _ _ |
---|
1106 | // for the next | nold/nnew -1 | iteration steps the coefficient 'a' |
---|
1107 | // in the leading form Fn = c(y-ax) should be zero |
---|
1108 | { |
---|
1109 | p_F = swapXY(p_F); |
---|
1110 | zeroSteps = calcNZeroSteps (nOld, nNew); |
---|
1111 | } |
---|
1112 | |
---|
1113 | isES, p_F = equimultTest (p_F, nHelpV, nNew, HCond); |
---|
1114 | |
---|
1115 | if (! isES) |
---|
1116 | { |
---|
1117 | return(0); |
---|
1118 | } |
---|
1119 | |
---|
1120 | if (zeroSteps > 0) |
---|
1121 | { |
---|
1122 | zeroSteps--; |
---|
1123 | |
---|
1124 | isES, p_c = pPOfYTest (p_F, nHelpV, nNew, HCond); |
---|
1125 | p_a = 0; |
---|
1126 | } |
---|
1127 | else |
---|
1128 | { |
---|
1129 | isES, p_F, p_c, p_a, HCond = pPOfLinTest (p_F, nNew, nHelpV, HCond); |
---|
1130 | } |
---|
1131 | |
---|
1132 | if (! isES) |
---|
1133 | { |
---|
1134 | return(0); |
---|
1135 | } |
---|
1136 | |
---|
1137 | p_F = formalBlowUp (p_F, p_c, p_a, nNew); |
---|
1138 | |
---|
1139 | if (p_a == H(nHelpV)) |
---|
1140 | { |
---|
1141 | nHelpV++; |
---|
1142 | |
---|
1143 | def r_base = basering; |
---|
1144 | kill myRing; |
---|
1145 | |
---|
1146 | T_extendRing(p_F, nHelpV, HCond); |
---|
1147 | |
---|
1148 | kill r_base; |
---|
1149 | |
---|
1150 | poly p_a; |
---|
1151 | poly p_c; |
---|
1152 | } |
---|
1153 | |
---|
1154 | step++; |
---|
1155 | } |
---|
1156 | |
---|
1157 | return(1); |
---|
1158 | } |
---|
1159 | /////////////////////////////////////////////////////////////////////////////// |
---|
1160 | |
---|
1161 | proc isEquising (poly p_F, list #) |
---|
1162 | "USAGE: isEquising(F[,m]); F poly, m int |
---|
1163 | ASSUME: F defines a deformation of an irreducible bivariate polynomial f |
---|
1164 | and the characteristic of the basering does not divide mult(f). @* |
---|
1165 | If nv is the number of variables of the basering, then the first nv-2 |
---|
1166 | variables are the deformation parameters. @* |
---|
1167 | If the basering is a qring, ideal(basering) must only depend |
---|
1168 | on the deformation parameters. |
---|
1169 | RETURN: list l of two integers, where |
---|
1170 | @format |
---|
1171 | l[1]=1 if F is an equisingular deformation, l[1]=0 otherwise. |
---|
1172 | l[2]=1 if some error has occured, l[2]=0 otherwise. |
---|
1173 | @end format |
---|
1174 | NOTE: If m is given, the computation stops after m steps of the iteration. @* |
---|
1175 | This procedure uses @code{execute} or calls a procedure using |
---|
1176 | @code{execute}. |
---|
1177 | EXAMPLE: example isEquising; shows an example |
---|
1178 | " |
---|
1179 | { |
---|
1180 | def r_user = basering; |
---|
1181 | |
---|
1182 | int ii = 1; |
---|
1183 | int i_nvars = nvars(basering); |
---|
1184 | int error = 0; |
---|
1185 | int maxStep; |
---|
1186 | int userMaxStep; |
---|
1187 | int xNotTransversal = 0; |
---|
1188 | int fIrreducible = 1; |
---|
1189 | intvec multSeq; |
---|
1190 | ideal isES = 1; |
---|
1191 | |
---|
1192 | error = checkBasering(); |
---|
1193 | if (error) |
---|
1194 | { |
---|
1195 | return(0,1); |
---|
1196 | } |
---|
1197 | |
---|
1198 | userMaxStep = getInput(#); |
---|
1199 | |
---|
1200 | // define a new basering "myRing" with new names for parameters |
---|
1201 | // and variables. |
---|
1202 | // The new names are 'a(1)', ..., 'a(npars)' for the parameters |
---|
1203 | // and 't(1)', ..., 't(nvars-2)', 'x', 'y' for the variables. |
---|
1204 | createMyRing(p_F, "ds"); |
---|
1205 | |
---|
1206 | createHNERing(); |
---|
1207 | |
---|
1208 | poly p_f = imap(myRing,p_F); |
---|
1209 | |
---|
1210 | error = checkPoly(p_f); |
---|
1211 | if (error) |
---|
1212 | { |
---|
1213 | return(0,1); |
---|
1214 | } |
---|
1215 | |
---|
1216 | // compute the multiplicity sequence of p_f. |
---|
1217 | multSeq, xNotTransversal, fIrreducible = calcMultSequence(p_f); |
---|
1218 | |
---|
1219 | if ( ! fIrreducible) |
---|
1220 | { |
---|
1221 | return(list(0,1)); |
---|
1222 | } |
---|
1223 | |
---|
1224 | setring myRing; |
---|
1225 | |
---|
1226 | if (xNotTransversal) |
---|
1227 | { |
---|
1228 | p_F = swapXY(p_F); |
---|
1229 | } |
---|
1230 | |
---|
1231 | if (userMaxStep != -1 && userMaxStep < size(multSeq)-1) |
---|
1232 | { |
---|
1233 | maxStep = userMaxStep; |
---|
1234 | } |
---|
1235 | else |
---|
1236 | { |
---|
1237 | maxStep = size(multSeq)-1; |
---|
1238 | } |
---|
1239 | |
---|
1240 | int isES = equisingTest(p_F, multSeq, maxStep); |
---|
1241 | |
---|
1242 | return(list(isES, error)); |
---|
1243 | } |
---|
1244 | |
---|
1245 | example |
---|
1246 | { |
---|
1247 | "EXAMPLE:"; echo=2; |
---|
1248 | ring r = 11,(a,b,x,y),ds; |
---|
1249 | poly F = (x2+2xy+y2+x5)+ay3+bx5; |
---|
1250 | isEquising(F); |
---|
1251 | isEquising(F,1); |
---|
1252 | ideal I = ideal(a); |
---|
1253 | qring q = std(I); |
---|
1254 | poly F = imap(r,F); |
---|
1255 | isEquising(F); |
---|
1256 | } |
---|
1257 | /////////////////////////////////////////////////////////////////////////////// |
---|
1258 | /* |
---|
1259 | Weiter Beispiele aus Dipl. von A. Mindnich einfuegen |
---|
1260 | */ |
---|