1 | /////////////////////////////////////////////////////////////////////////// |
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2 | version="version redcgs.lib 4.2.0.0 Dec_2020 "; // $Id$ |
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3 | category="General purpose"; |
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4 | info=" |
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5 | LIBRARY: redcgs.lib Reduced Comprehensive Groebner Systems. |
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6 | |
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7 | OVERVIEW: |
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8 | Comprehensive Groebner Systems. Canonical Forms. |
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9 | The library contains Monte's algorithms to compute disjoint, reduced |
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10 | Comprehensive Groebner Systems (CGS). A CGS is a set of pairs of |
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11 | (segment,basis). The segments S_i are subsets of the parameter space, |
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12 | and the bases B_i are sets of polynomials specializing to Groebner |
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13 | bases of the specialized ideal for every point in S_i. |
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14 | |
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15 | The purpose of the routines in this library is to obtain CGS with |
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16 | better properties, namely disjoint segments forming a partition of |
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17 | the parameter space and reduced bases. Reduced bases are sets of |
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18 | polynomials that specialize to the reduced Groebner basis of the |
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19 | specialized ideal preserving the leading power products (lpp). |
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20 | The lpp characterize the type of solution in each segment. |
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21 | |
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22 | A further objective is to summarize as much as possible the segments |
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23 | with the same lpp into a single segment, and if possible to obtain |
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24 | a final result that is canonical, i.e. independent of the algorithm |
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25 | and only attached to the given ideal. |
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26 | |
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27 | There are three fundamental routines in the library: mrcgs, rcgs and |
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28 | crcgs. mrcgs (Minimal Reduced CGS) is an algorithm that packs so |
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29 | much as it is able to do (using algorithms adhoc) the segments with |
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30 | the same lpp, obtaining the minimal number of segments. The hypothesis |
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31 | is that the result is also canonical, but for the moment there is no |
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32 | proof of the uniqueness of this minimal packing. Moreover, the |
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33 | segments that are obtained are not locally closed, i.e. there are not |
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34 | difference of two varieties. |
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35 | |
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36 | On the other side, Michael Wibmer has proved that for homogeneous ideals, |
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37 | all the segments with reduced bases having the same lpp admit a unique |
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38 | basis specializing well. For this purpose it is necessary to extend the |
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39 | description of the elements of the bases to functions, forming sheaves |
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40 | of polynomials instead of simple polynomials, so that the polynomials in |
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41 | a sheaf either preserve the lpp of the corresponding polynomial of |
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42 | the specialized Groebner basis (and then it specializes well) or |
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43 | it specializes to 0. Moreover, in a sheaf, for every point in the |
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44 | corresponding segment, at least one of the polynomials specializes well. |
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45 | specializes well. And moreover Wibmer's Theorem ensures that the packed |
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46 | segments are locally closed, that is can be described as the difference of |
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47 | two varieties. |
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48 | |
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49 | Using Wibmer's Theorem we proved that an affine ideal can be homogenized, |
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50 | than discussed by mrcgs and finally de-homogenized. The bases so obtained |
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51 | can be reduced and specialize well in the segment. If the theoretic |
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52 | objective is reached, and all the segments of the homogenized ideal |
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53 | have been packed, locally closed segments will be obtained. |
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54 | |
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55 | If we only homogenize the given basis of the ideal, then we cannot ensure |
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56 | the canonicity of the partition obtained, because there are many different |
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57 | bases of the given ideal that can be homogenized, and the homogenized ideals |
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58 | are not identical. This corresponds to the algorithm rcgs and is recommended |
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59 | as the most practical routine. It provides locally closed segments and |
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60 | is usually faster than mrcgs and crcgs. But the given partition is not |
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61 | always canonical. |
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62 | |
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63 | Finally it is possible to homogenize the whole affine ideal, and then |
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64 | the packing algorithm will provide canonical segments by dehomogenizing. |
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65 | This corresponds to crcgs routine. It provides the best description |
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66 | of the segments and bases. In contrast crcgs algorithm is usually much |
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67 | more time consuming and it will not always finish in a reasonable time. |
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68 | Moreover it will contain more segments than mrcgs and possibly also more |
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69 | than rcgs. |
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70 | |
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71 | But the actual algorithms in the library to pack segments have some lacks. |
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72 | They are not theoretically always able to pack the segments that we know |
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73 | that can be packed. Nevertheless, thanks to Wibmer's Theorem, the |
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74 | algorithms rcgs and crcgs are able to detect if the objective has not been |
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75 | reached, and if so, to give a Warning. The warning does not invalidate the |
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76 | output, but it only recognizes that the theoretical objective is not |
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77 | completely reached by the actual computing methods and that some segments |
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78 | that can be packed have not been packed with a single basis. |
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79 | |
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80 | The routine buildtree is the first algorithm used in all the previous |
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81 | methods providing a first disjoint CGS, and can be used if none of the |
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82 | three fundamental algorithms of the library finishes in a reasonable time. |
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83 | |
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84 | There are also routines to visualize better the output of the previous |
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85 | algorithms: |
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86 | finalcases can be applied to the list provided by buildtree to obtain the |
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87 | CGS. The list provided by buildtree contains the whole discussion, and |
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88 | finalcases extracts the CGS. |
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89 | The output of buildtree can also be transformed into a file using |
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90 | buildtreetoMaple routine that can be read in Maple. Using Monte's dpgb |
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91 | library in Maple the output can be plotted (with the routine tplot). |
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92 | To plot the output of mrcgs, rcgs or crcgs in Maple, the library also |
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93 | provides the routine cantreetoMaple. The file written using it |
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94 | and read in Maple can then be plotted with the command plotcantree and |
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95 | printed with printcantree from the Monte's dpgb library in Maple. |
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96 | The output of mrcgs, rcgs and crcgs is given in form of tree using |
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97 | prime ideals in a canonical form that is described in the papers. |
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98 | Nevertheless this canonical form is somewhat uncomfortable to be |
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99 | interpreted. When the segments are all locally closed (and this is |
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100 | always the case for rcgs and crcgs) the routine cantodiffcgs transforms |
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101 | the output into a simpler form having only one list element for |
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102 | each segment and providing the two varieties whose difference represent |
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103 | the segment also in a canonical form. |
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104 | |
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105 | AUTHORS: Antonio Montes , Hans Schoenemann. |
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106 | OVERVIEW: see \"Minimal Reduced Comprehensive Groebner Systems\" |
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107 | by Antonio Montes. (http://www-ma2.upc.edu/~montes/). |
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108 | |
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109 | NOTATIONS: All given and determined polynomials and ideals are in the |
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110 | @* basering K[a][x]; (a=parameters, x=variables) |
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111 | @* After defining the ring and calling setglobalrings(); the rings |
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112 | @* @R (K[a][x]), |
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113 | @* @P (K[a]), |
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114 | @* @RP (K[x,a]) are defined globally |
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115 | @* They are used internally and can also be used by the user. |
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116 | @* The fundamental routines are: buildtree, mrcgs, rcgs and crcgs |
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117 | |
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118 | PROCEDURES: |
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119 | |
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120 | setglobalrings(); It is called by the fundamental routines of the library: |
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121 | (buildtree, mrcgs, rcgs, crcgs). |
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122 | After calling it, the rings @R, @P and @RP are defined |
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123 | globally. |
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124 | memberpos(f,J); Returns the list of two integers: the value 0 or 1 depending |
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125 | on if f belongs to J or not, and the position in J (0 if it |
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126 | does not belong). |
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127 | subset(F,G); If all elements of F belong to the ideal G it returns 1, |
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128 | and 0 otherwise. |
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129 | pdivi2(f,F); Pseudodivision of a polynomial f by an ideal F in @R. Returns a |
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130 | list (r,q,m) such that m*f=r+sum(q.G). |
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131 | facvar(ideal J) Returns all the free-square factors of the elements |
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132 | of ideal J (non repeated). Integer factors are ignored, |
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133 | even 0 is ignored. It can be called from ideal @R, but |
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134 | the given ideal J must only contain polynomials in the |
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135 | parameters. |
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136 | redspec(N,W); Given null and non-null conditions depending only on the |
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137 | parameters it returns a red-specification. |
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138 | pnormalform(f,N,W); Reduces the polynomial f w.r.t. to the null condition ideal N and the |
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139 | non-null condition ideal W (both depending on the parameters). |
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140 | buildtree(F); Returns a list T describing a first reduced CGS of the ideal |
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141 | F in K[a][x]. |
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142 | buildtreetoMaple(T); Writes into a file the output of buildtree in Maple readable |
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143 | form. |
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144 | finalcases(T); From the output of buildtree it provides the list |
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145 | of its terminal vertices. That list represents the dichotomic, |
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146 | reduced CGS obtained by buildtree. |
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147 | mrcgs(F); Returns a list T describing the Minimal Reduced CGS of the |
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148 | ideal F of K[a][x] |
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149 | rcgs(F); Returns a list T describing the Reduced CGS of the ideal F |
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150 | of K[a][x] obtained by direct homogenizing and de-homogenizing |
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151 | the basis of the given ideal. |
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152 | crcgs(F); Returns a list T describing the Canonical Reduced CGS of the |
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153 | ideal F of K[a][x] obtained by homogenizing and de-homogenizing |
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154 | the initial ideal. |
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155 | cantreetoMaple)(M); Writes into a file the output of mrcgs, rcgs or crcgs in Maple |
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156 | readable form. |
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157 | cantodiffcgs(list L);From the output of rcgs or crcgs (or even of mrcgs when |
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158 | it is possible) it returns a simpler list where the segments |
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159 | are given as difference of varieties. |
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160 | |
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161 | SEE ALSO: compregb_lib |
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162 | "; |
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163 | |
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164 | // ************ Begin of the redCGS library ********************* |
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165 | // Library redCGS |
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166 | // (Reduced Comprehesive Groebner Systems): |
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167 | // Initial data: 21-1-2008 |
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168 | // Release 1: |
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169 | // Final data: 3_7-2008 |
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170 | // All given and determined polynomials and ideals are in the |
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171 | // basering K[a][x]; |
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172 | // After calling setglobalrings(); the rings |
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173 | // @R (K[a][x]), |
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174 | // @P (K[a]), |
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175 | // @RP (K[x,a]) are globally defined |
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176 | // They are used internally and can also be called by the user; |
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177 | // setglobalrings() is called by buildtree, so it is not required to |
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178 | // call setglobalrings before using |
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179 | // the fundamental routines of the library. |
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180 | |
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181 | // ************ Begin of buildtree ****************************** |
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182 | |
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183 | LIB "primdec.lib"; |
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184 | |
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185 | proc setglobalrings() |
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186 | "USAGE: setglobalrings(); |
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187 | No arguments |
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188 | RETURN: After its call the rings @R=K[a][x], @P=K[a], @RP=K[x,a] are |
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189 | defined as global variables. |
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190 | NOTE: It is called by the fundamental routines of the library. |
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191 | The user does not need to call it, except when none of |
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192 | the fundamental routines have been called and some |
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193 | other routines of the library are used. |
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194 | The basering R, must be of the form K[a][x], a=parameters, |
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195 | x=variables, and should be defined previously. |
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196 | KEYWORDS: ring, rings |
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197 | EXAMPLE: setglobalrings; shows an example" |
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198 | { |
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199 | def @R=basering; // must be of the form K[a][x], a=parameters, x=variables |
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200 | def Rx=ringlist(@R); |
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201 | def @P=ring(Rx[1]); |
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202 | list Lx; |
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203 | Lx[1]=0; |
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204 | Lx[2]=Rx[2]+Rx[1][2]; |
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205 | Lx[3]=Rx[1][3]; |
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206 | Lx[4]=Rx[1][4]; |
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207 | //def @K=ring(Lx); |
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208 | //exportto(Top,@K); //global ring K[x,a] with the order of x extended to x,a |
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209 | Rx[1]=0; |
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210 | def D=ring(Rx); |
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211 | def @RP=D+@P; |
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212 | exportto(Top,@R); // global ring K[a][x] |
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213 | exportto(Top,@P); // global ring K[a] |
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214 | exportto(Top,@RP); // global ring K[x,a] with product order |
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215 | setring(@R); |
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216 | } |
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217 | example |
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218 | { "EXAMPLE:"; echo = 2; |
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219 | ring R=(0,a,b),(x,y,z),dp; |
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220 | setglobalrings(); |
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221 | @R; |
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222 | @P; |
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223 | @RP; |
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224 | } |
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225 | |
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226 | //*************Auxiliary routines************** |
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227 | |
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228 | // cld : clears denominators of an ideal and normalizes to content 1 |
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229 | // can be used in @R or @P or @RP |
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230 | // input: |
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231 | // ideal J (J can be also poly), but the output is an ideal; |
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232 | // output: |
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233 | // ideal Jc (the new form of ideal J without denominators and |
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234 | // normalized to content 1) |
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235 | proc cld(ideal J) |
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236 | { |
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237 | if (size(J)==0){return(ideal(0));} |
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238 | def RR=basering; |
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239 | setring(@RP); |
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240 | def Ja=imap(RR,J); |
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241 | ideal Jb; |
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242 | if (size(Ja)==0){return(ideal(0));} |
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243 | int i; |
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244 | def j=0; |
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245 | for (i=1;i<=ncols(Ja);i++){if (size(Ja[i])!=0){j++; Jb[j]=cleardenom(Ja[i]);}} |
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246 | setring(RR); |
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247 | def Jc=imap(@RP,Jb); |
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248 | return(Jc); |
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249 | } |
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250 | |
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251 | proc memberpos(def f,def J) |
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252 | "USAGE: memberpos(f,J); |
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253 | (f,J) expected (polynomial,ideal) |
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254 | or (int,list(int)) |
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255 | or (int,intvec) |
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256 | or (intvec,list(intvec)) |
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257 | or (list(int),list(list(int))) |
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258 | or (ideal,list(ideal)) |
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259 | or (list(intvec), list(list(intvec))). |
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260 | The ring can be @R or @P or @RP or any other. |
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261 | RETURN: The list (t,pos) t int; pos int; |
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262 | t is 1 if f belongs to J and 0 if not. |
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263 | pos gives the position in J (or 0 if f does not belong). |
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264 | EXAMPLE: memberpos; shows an example" |
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265 | { |
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266 | int pos=0; |
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267 | int i=1; |
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268 | int j; |
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269 | int t=0; |
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270 | int nt; |
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271 | if (typeof(J)=="ideal"){nt=ncols(J);} |
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272 | else{nt=size(J);} |
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273 | if ((typeof(f)=="poly") or (typeof(f)=="int")) |
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274 | { // (poly,ideal) or |
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275 | // (poly,list(poly)) |
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276 | // (int,list(int)) or |
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277 | // (int,intvec) |
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278 | i=1; |
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279 | while(i<=nt) |
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280 | { |
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281 | if (f==J[i]){return(list(1,i));} |
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282 | i++; |
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283 | } |
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284 | return(list(0,0)); |
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285 | } |
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286 | else |
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287 | { |
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288 | if ((typeof(f)=="intvec") or ((typeof(f)=="list") and (typeof(f[1])=="int"))) |
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289 | { // (intvec,list(intvec)) or |
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290 | // (list(int),list(list(int))) |
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291 | i=1; |
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292 | t=0; |
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293 | pos=0; |
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294 | while((i<=nt) and (t==0)) |
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295 | { |
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296 | t=1; |
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297 | j=1; |
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298 | if (size(f)!=size(J[i])){t=0;} |
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299 | else |
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300 | { |
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301 | while ((j<=size(f)) and t) |
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302 | { |
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303 | if (f[j]!=J[i][j]){t=0;} |
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304 | j++; |
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305 | } |
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306 | } |
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307 | if (t){pos=i;} |
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308 | i++; |
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309 | } |
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310 | if (t){return(list(1,pos));} |
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311 | else{return(list(0,0));} |
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312 | } |
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313 | else |
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314 | { |
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315 | if (typeof(f)=="ideal") |
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316 | { // (ideal,list(ideal)) |
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317 | i=1; |
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318 | t=0; |
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319 | pos=0; |
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320 | while((i<=nt) and (t==0)) |
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321 | { |
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322 | t=1; |
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323 | j=1; |
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324 | if (ncols(f)!=ncols(J[i])){t=0;} |
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325 | else |
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326 | { |
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327 | while ((j<=ncols(f)) and t) |
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328 | { |
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329 | if (f[j]!=J[i][j]){t=0;} |
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330 | j++; |
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331 | } |
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332 | } |
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333 | if (t){pos=i;} |
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334 | i++; |
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335 | } |
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336 | if (t){return(list(1,pos));} |
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337 | else{return(list(0,0));} |
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338 | } |
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339 | else |
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340 | { |
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341 | if ((typeof(f)=="list") and (typeof(f[1])=="intvec")) |
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342 | { // (list(intvec),list(list(intvec))) |
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343 | i=1; |
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344 | t=0; |
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345 | pos=0; |
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346 | while((i<=nt) and (t==0)) |
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347 | { |
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348 | t=1; |
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349 | j=1; |
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350 | if (size(f)!=size(J[i])){t=0;} |
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351 | else |
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352 | { |
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353 | while ((j<=size(f)) and t) |
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354 | { |
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355 | if (f[j]!=J[i][j]){t=0;} |
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356 | j++; |
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357 | } |
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358 | } |
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359 | if (t){pos=i;} |
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360 | i++; |
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361 | } |
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362 | if (t){return(list(1,pos));} |
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363 | else{return(list(0,0));} |
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364 | } |
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365 | } |
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366 | } |
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367 | } |
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368 | } example |
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369 | { "EXAMPLE:"; echo = 2; |
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370 | list L=(7,4,5,1,1,4,9); |
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371 | memberpos(1,L); |
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372 | } |
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373 | |
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374 | |
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375 | proc subset(def J,def K) |
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376 | "USAGE: subset(J,K); |
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377 | (J,K) expected (ideal,ideal) |
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378 | or (list, list) |
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379 | RETURN: 1 if all the elements of J are in K, 0 if not. |
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380 | EXAMPLE: subset; shows an example;" |
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381 | { |
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382 | int i=1; |
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383 | int nt; |
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384 | if (typeof(J)=="ideal"){nt=ncols(J);} |
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385 | else{nt=size(J);} |
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386 | if (size(J)==0){return(1);} |
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387 | while(i<=nt) |
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388 | { |
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389 | if (memberpos(J[i],K)[1]){i++;} |
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390 | else {return(0);} |
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391 | } |
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392 | return(1); |
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393 | } |
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394 | example |
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395 | { "EXAMPLE:"; echo = 2; |
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396 | list J=list(7,3,2); |
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397 | list K=list(1,2,3,5,7,8); |
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398 | subset(J,K); |
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399 | } |
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400 | |
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401 | //*************Auxiliary routines************** |
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402 | |
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403 | |
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404 | // elimintfromideal: elimine the constant numbers from the ideal |
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405 | // (designed for W, nonnull conditions) |
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406 | // input: ideal J in the ring @P |
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407 | // output:ideal K with the elements of J that are non constants, in the ring @P |
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408 | proc elimintfromideal(ideal J) |
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409 | { |
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410 | int i; |
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411 | int j=0; |
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412 | ideal K; |
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413 | if (size(J)==0){return(ideal(0));} |
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414 | for (i=1;i<=ncols(J);i++){if (size(variables(J[i])) !=0){j++; K[j]=J[i];}} |
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415 | return(K); |
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416 | } |
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417 | |
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418 | // simpqcoeffs : simplifies a quotient of two polynomials of @R |
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419 | // for ring @R |
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420 | // input: two coefficients (or terms) of @R (that are considered as quotients) |
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421 | // output: the two coefficients reduced without common factors |
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422 | proc simpqcoeffs(poly n,poly m) |
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423 | { |
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424 | def nc=content(n); |
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425 | def mc=content(m); |
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426 | def gc=gcd(nc,mc); |
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427 | ideal s=n/gc,m/gc; |
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428 | return (s); |
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429 | } |
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430 | |
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431 | |
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432 | // pdivi2 : pseudodivision of a polynomial f by an ideal F in @R |
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433 | // in the ring @R |
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434 | // input: |
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435 | // poly f0 (given in the ring @R) |
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436 | // ideal F0 (given in the ring @R) |
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437 | // output: |
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438 | // list (poly r, ideal q, poly mu) |
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439 | proc pdivi2(poly f,ideal F) |
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440 | "USAGE: pdivi2(f,F); |
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441 | poly f: the polynomial to be divided |
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442 | ideal F: the divisor ideal |
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443 | RETURN: A list (poly r, ideal q, poly m). r is the remainder of the |
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444 | pseudodivision, q is the ideal of quotients, and m is the |
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445 | factor by which f is to be multiplied. |
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446 | NOTE: Pseudodivision of a polynomial f by an ideal F in @R. Returns a |
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447 | list (r,q,m) such that m*f=r+sum(q.G). |
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448 | KEYWORDS: division, reduce |
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449 | EXAMPLE: example pdivi2; shows an example" |
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450 | { |
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451 | int i; |
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452 | int j; |
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453 | poly r=0; |
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454 | poly mu=1; |
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455 | def p=f; |
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456 | ideal q; |
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457 | for (i=1; i<=size(F); i++){q[i]=0;} |
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458 | ideal lpf; |
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459 | ideal lcf; |
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460 | for (i=1;i<=size(F);i++){lpf[i]=leadmonom(F[i]);} |
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461 | for (i=1;i<=size(F);i++){lcf[i]=leadcoef(F[i]);} |
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462 | poly lpp; |
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463 | poly lcp; |
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464 | poly qlm; |
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465 | poly nu; |
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466 | poly rho; |
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467 | int divoc=0; |
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468 | ideal qlc; |
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469 | while (p!=0) |
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470 | { |
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471 | i=1; |
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472 | divoc=0; |
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473 | lpp=leadmonom(p); |
---|
474 | lcp=leadcoef(p); |
---|
475 | while (divoc==0 and i<=size(F)) |
---|
476 | { |
---|
477 | qlm=lpp/lpf[i]; |
---|
478 | if (qlm!=0) |
---|
479 | { |
---|
480 | qlc=simpqcoeffs(lcp,lcf[i]); |
---|
481 | nu=qlc[2]; |
---|
482 | mu=mu*nu; |
---|
483 | rho=qlc[1]*qlm; |
---|
484 | p=nu*p-rho*F[i]; |
---|
485 | r=nu*r; |
---|
486 | for (j=1;j<=size(F);j++){q[j]=nu*q[j];} |
---|
487 | q[i]=q[i]+rho; |
---|
488 | divoc=1; |
---|
489 | } |
---|
490 | else {i++;} |
---|
491 | } |
---|
492 | if (divoc==0) |
---|
493 | { |
---|
494 | r=r+lcp*lpp; |
---|
495 | p=p-lcp*lpp; |
---|
496 | } |
---|
497 | } |
---|
498 | list res=r,q,mu; |
---|
499 | return(res); |
---|
500 | } |
---|
501 | example |
---|
502 | { "EXAMPLE:"; echo = 2; |
---|
503 | ring R=(0,a,b,c),(x,y),dp; |
---|
504 | setglobalrings(); |
---|
505 | poly f=(ab-ac)*xy+(ab)*x+(5c); |
---|
506 | ideal F=ax+b,cy+a; |
---|
507 | def r=pdivi2(f,F); |
---|
508 | r; |
---|
509 | r[3]*f-(r[2][1]*F[1]+r[2][2]*F[2])-r[1]; |
---|
510 | } |
---|
511 | |
---|
512 | // pspol : S-poly of two polynomials in @R |
---|
513 | // @R |
---|
514 | // input: |
---|
515 | // poly f (given in the ring @R) |
---|
516 | // poly g (given in the ring @R) |
---|
517 | // output: |
---|
518 | // list (S, red): S is the S-poly(f,g) and red is a Boolean variable |
---|
519 | // if red==1 then S reduces by Buchberger 1st criterion (not used) |
---|
520 | proc pspol(poly f,poly g) |
---|
521 | { |
---|
522 | def lcf=leadcoef(f); |
---|
523 | def lcg=leadcoef(g); |
---|
524 | def lpf=leadmonom(f); |
---|
525 | def lpg=leadmonom(g); |
---|
526 | def v=gcd(lpf,lpg); |
---|
527 | def s=simpqcoeffs(lcf,lcg); |
---|
528 | def vf=lpf/v; |
---|
529 | def vg=lpg/v; |
---|
530 | poly S=s[2]*vg*f-s[1]*vf*g; |
---|
531 | return(S); |
---|
532 | } |
---|
533 | |
---|
534 | // facvar: Returns all the free-square factors of the elements |
---|
535 | // of ideal J (non repeated). Integer factors are ignored, |
---|
536 | // even 0 is ignored. It can be called from ideal @R, but |
---|
537 | // the given ideal J must only contain poynomials in the |
---|
538 | // parameters. |
---|
539 | // Operates in the ring @P, but can be called from ring @R. |
---|
540 | // input: ideal J |
---|
541 | // output: ideal Jc: Returns all the free-square factors of the elements |
---|
542 | // of ideal J (non repeated). Integer factors are ignored, |
---|
543 | // even 0 is ignored. It can be called from ideal @R, but |
---|
544 | // the given ideal J must only contain poynomials in the |
---|
545 | // parameters. |
---|
546 | proc facvar(ideal J) |
---|
547 | "USAGE: facvar(J); |
---|
548 | J: an ideal in the parameters |
---|
549 | RETURN: all the free-square factors of the elements |
---|
550 | of ideal J (non repeated). Integer factors are ignored, |
---|
551 | even 0 is ignored. It can be called from ideal @R, but |
---|
552 | the given ideal J must only contain poynomials in the |
---|
553 | parameters. |
---|
554 | NOTE: Operates in the ring @P, and the ideal J must contain only |
---|
555 | polynomials in the parameters, but can be called from ring @R. |
---|
556 | KEYWORDS: factor |
---|
557 | EXAMPLE: facvar; shows an example" |
---|
558 | { |
---|
559 | int i; |
---|
560 | def RR=basering; |
---|
561 | setring(@P); |
---|
562 | def Ja=imap(RR,J); |
---|
563 | if(size(Ja)==0){return(ideal(0));} |
---|
564 | Ja=elimintfromideal(Ja); // also in ideal @P |
---|
565 | ideal Jb; |
---|
566 | if (size(Ja)==0){Jb=ideal(0);} |
---|
567 | else |
---|
568 | { |
---|
569 | for (i=1;i<=ncols(Ja);i++){if(size(Ja[i])!=0){Jb=Jb,factorize(Ja[i],1);}} |
---|
570 | Jb=simplify(Jb,2+4+8); |
---|
571 | Jb=cld(Jb); |
---|
572 | Jb=elimintfromideal(Jb); // also in ideal @P |
---|
573 | } |
---|
574 | setring(RR); |
---|
575 | def Jc=imap(@P,Jb); |
---|
576 | return(Jc); |
---|
577 | } |
---|
578 | example |
---|
579 | { "EXAMPLE:"; echo = 2; |
---|
580 | ring R=(0,a,b,c),(x,y,z),dp; |
---|
581 | setglobalrings(); |
---|
582 | ideal J=a2-b2,a2-2ab+b2,abc-bc; |
---|
583 | facvar(J); |
---|
584 | } |
---|
585 | |
---|
586 | // Wred: eliminate the factors in the polynom f that are in W |
---|
587 | // in ring @RP |
---|
588 | // input: |
---|
589 | // poly f: |
---|
590 | // ideal W of non-null conditions (already supposed that it is facvar) |
---|
591 | // output: |
---|
592 | // poly f2 where the non-null conditions in W have been dropped from f |
---|
593 | proc Wred(poly f, ideal W) |
---|
594 | { |
---|
595 | if (f==0){return(f);} |
---|
596 | def RR=basering; |
---|
597 | setring(@RP); |
---|
598 | def ff=imap(RR,f); |
---|
599 | def RPW=imap(RR,W); |
---|
600 | def l=factorize(ff,2); |
---|
601 | int i; |
---|
602 | poly f1=1; |
---|
603 | for(i=1;i<=size(l[1]);i++) |
---|
604 | { |
---|
605 | if ((memberpos(l[1][i],RPW)[1]) or (memberpos(-l[1][i],RPW)[1])){;} |
---|
606 | else{f1=f1*((l[1][i])^(l[2][i]));} |
---|
607 | } |
---|
608 | setring(RR); |
---|
609 | def f2=imap(@RP,f1); |
---|
610 | return(f2); |
---|
611 | } |
---|
612 | |
---|
613 | // pnormalform: reduces a polynomial w.r.t. a red-spec dividing by N and eliminating factors in W. |
---|
614 | // called in the ring @R |
---|
615 | // operates in the ring @RP |
---|
616 | // input: |
---|
617 | // poly f |
---|
618 | // ideal N (depends only on the parameters) |
---|
619 | // ideal W (depends only on the parameters) |
---|
620 | // (N,W) must be a red-spec (depends only on the parameters) |
---|
621 | // output: poly f2 reduced w.r.t. to the red-spec (N,W) |
---|
622 | // note: for security a lot of work is done. If (N,W) is already a red-spec it should be simplified |
---|
623 | proc pnormalform(poly f, ideal N, ideal W) |
---|
624 | "USAGE: pnormalform(f,N,W); |
---|
625 | f: the polynomial to be reduced modulo N,W (in parameters and |
---|
626 | variables) |
---|
627 | N: the null conditions ideal |
---|
628 | W: the non-null conditions (set of irreducible polynomials, ideal) |
---|
629 | RETURN: a reduced polynomial g of f, whose coefficients are reduced |
---|
630 | modulo N and having no factor in W. |
---|
631 | NOTE: Should be called from ring @R. Ideals N and W must be polynomials |
---|
632 | in the parameters forming a red-specification (see definition) the papers). |
---|
633 | KEYWORDS: division, pdivi2, reduce |
---|
634 | EXAMPLE: pnormalform; shows an example" |
---|
635 | { |
---|
636 | def RR=basering; |
---|
637 | setring(@RP); |
---|
638 | def fa=imap(RR,f); |
---|
639 | def Na=imap(RR,N); |
---|
640 | def Wa=imap(RR,W); |
---|
641 | option(redSB); |
---|
642 | Na=groebner(Na); |
---|
643 | def r=cld(reduce(fa,Na)); |
---|
644 | def f1=Wred(r[1],Wa); |
---|
645 | setring(RR); |
---|
646 | def f2=imap(@RP,f1); |
---|
647 | return(f2) |
---|
648 | } |
---|
649 | example |
---|
650 | { "EXAMPLE:"; echo = 2; |
---|
651 | ring R=(0,a,b,c),(x,y),dp; |
---|
652 | setglobalrings(); |
---|
653 | poly f=(b^2-1)*x^3*y+(c^2-1)*x*y^2+(c^2*b-b)*x+(a-bc)*y; |
---|
654 | ideal N=(ab-c)*(a-b),(a-bc)*(a-b); |
---|
655 | ideal W=a^2-b^2,bc; |
---|
656 | def r=redspec(N,W); |
---|
657 | pnormalform(f,r[1],r[2]); |
---|
658 | } |
---|
659 | |
---|
660 | // idint: ideal intersection |
---|
661 | // in the ring @P. |
---|
662 | // it works in an extended ring |
---|
663 | // input: two ideals in the ring @P |
---|
664 | // output the intersection of both (is not a GB) |
---|
665 | proc idint(ideal I, ideal J) |
---|
666 | { |
---|
667 | def RR=basering; |
---|
668 | ring T=0,t,lp; |
---|
669 | def K=T+RR; |
---|
670 | setring(K); |
---|
671 | def Ia=imap(RR,I); |
---|
672 | def Ja=imap(RR,J); |
---|
673 | ideal IJ; |
---|
674 | int i; |
---|
675 | for(i=1;i<=size(Ia);i++){IJ[i]=t*Ia[i];} |
---|
676 | for(i=1;i<=size(Ja);i++){IJ[size(Ia)+i]=(1-t)*Ja[i];} |
---|
677 | ideal eIJ=eliminate(IJ,t); |
---|
678 | setring(RR); |
---|
679 | return(imap(K,eIJ)); |
---|
680 | } |
---|
681 | |
---|
682 | |
---|
683 | // redspec: generates a red-specification |
---|
684 | // called in any ring |
---|
685 | // it changes to the ring @P |
---|
686 | // input: |
---|
687 | // ideal N : the ideal of null-conditions |
---|
688 | // ideal W : set of non-null polynomials: if W corresponds to no non null conditions then W=ideal(0) |
---|
689 | // otherwise it should be given as an ideal. |
---|
690 | // returns: list (Na,Wa,DGN) |
---|
691 | // the completely reduced specification: |
---|
692 | // Na = ideal reduced and radical of the red-spec |
---|
693 | // facvar(Wa) = ideal the reduced non-null set of polynomials of the red-spec. |
---|
694 | // if it corresponds to no non null conditions then it is ideal(0) |
---|
695 | // otherwise the ideal is returned. |
---|
696 | // DGN = the list of prime ideals associated to Na (uses primASSGTZ in "primdec.lib") |
---|
697 | // none of the polynomials in facvar(Wa) are contained in none of the ideals in DGN |
---|
698 | // If the given conditions are not compatible, then N=ideal(1) and DGN=list(ideal(1)) |
---|
699 | proc redspec(ideal Ni, ideal Wi) |
---|
700 | "USAGE: redspec(N,W); |
---|
701 | N: null conditions ideal |
---|
702 | W: set of non-null polynomials (ideal) |
---|
703 | RETURN: a list (N1,W1,L1) containing a red-specification of the segment (N,W). |
---|
704 | N1 is the radical reduced ideal characterizing the segment. |
---|
705 | V(N1) is the Zarisky closure of the segment (N,W). |
---|
706 | The segment S=V(N1) \ V(h), where h=prod(w in W1) |
---|
707 | N1 is uniquely determined and no prime component of N1 contains none of |
---|
708 | the polynomials in W1. The polynomials in W1 are prime and reduced |
---|
709 | w.r.t. N1, and are considered non-null on the segment. |
---|
710 | L1 contains the list of prime components of N1. |
---|
711 | NOTE: can be called from ring @R but it works in ring @P. |
---|
712 | KEYWORDS: specification |
---|
713 | EXAMPLE: redspec; shows an example" |
---|
714 | { |
---|
715 | ideal Nc; |
---|
716 | ideal Wc; |
---|
717 | def RR=basering; |
---|
718 | setring(@P); |
---|
719 | def N=imap(RR,Ni); |
---|
720 | def W=imap(RR,Wi); |
---|
721 | ideal Wa; |
---|
722 | ideal Wb; |
---|
723 | if(size(W)==0){Wa=ideal(0);} |
---|
724 | //when there are no non-null conditions then W=ideal(0) |
---|
725 | else |
---|
726 | { |
---|
727 | Wa=facvar(W); |
---|
728 | } |
---|
729 | if (size(N)==0) |
---|
730 | { |
---|
731 | setring(RR); |
---|
732 | Wc=imap(@P,Wa); |
---|
733 | return(list(ideal(0), Wc, list(ideal(0)))); |
---|
734 | } |
---|
735 | int i; |
---|
736 | list LNb; |
---|
737 | list LNa; |
---|
738 | def LN=minAssGTZ(N); |
---|
739 | for (i=1;i<=size(LN);i++) |
---|
740 | { |
---|
741 | option(redSB); |
---|
742 | LNa[i]=groebner(LN[i]); |
---|
743 | } |
---|
744 | poly h=1; |
---|
745 | if (size(Wa)!=0) |
---|
746 | { |
---|
747 | for(i=1;i<=size(Wa);i++){h=h*Wa[i];} |
---|
748 | } |
---|
749 | ideal Na; |
---|
750 | intvec save_opt=option(get); |
---|
751 | if (size(N)!=0 and (size(LNa)>0)) |
---|
752 | { |
---|
753 | option(returnSB); |
---|
754 | Na=intersect(LNa[1..size(LNa)]); |
---|
755 | option(redSB); |
---|
756 | Na=groebner(Na); // T_ is needed? |
---|
757 | option(set,save_opt); |
---|
758 | } |
---|
759 | attrib(Na,"isSB",1); |
---|
760 | if (reduce(h,Na,1)==0) |
---|
761 | { |
---|
762 | setring(RR); |
---|
763 | Wc=imap(@P,Wa); |
---|
764 | return(list (ideal(1),Wc,list(ideal(1)))); |
---|
765 | } |
---|
766 | i=1; |
---|
767 | while(i<=size(LNa)) |
---|
768 | { |
---|
769 | if (reduce(h,LNa[i],1)==0){LNa=delete(LNa,i);} |
---|
770 | else{ i++;} |
---|
771 | } |
---|
772 | if (size(LNa)==0) |
---|
773 | { |
---|
774 | setring(RR); |
---|
775 | return(list(ideal(1),ideal(0),list(ideal(1)))); |
---|
776 | } |
---|
777 | option(returnSB); |
---|
778 | ideal Nb=intersect(LNa[1..size(LNa)]); |
---|
779 | option(redSB); |
---|
780 | Nb=groebner(Nb); // T_ is needed? |
---|
781 | option(set,save_opt); |
---|
782 | if (size(Wa)==0) |
---|
783 | { |
---|
784 | setring(RR); |
---|
785 | Nc=imap(@P,Nb); |
---|
786 | Wc=imap(@P,Wa); |
---|
787 | LNb=imap(@P,LNa); |
---|
788 | return(list(Nc,Wc,LNb)); |
---|
789 | } |
---|
790 | Wb=ideal(0); |
---|
791 | attrib(Nb,"isSB",1); |
---|
792 | for (i=1;i<=size(Wa);i++){Wb[i]=reduce(Wa[i],Nb);} |
---|
793 | Wb=facvar(Wb); |
---|
794 | if (size(LNa)!=0) |
---|
795 | { |
---|
796 | setring(RR); |
---|
797 | Nc=imap(@P,Nb); |
---|
798 | Wc=imap(@P,Wb); |
---|
799 | LNb=imap(@P,LNa); |
---|
800 | return(list(Nc,Wc,LNb)) |
---|
801 | } |
---|
802 | else |
---|
803 | { |
---|
804 | setring(RR); |
---|
805 | Nd=imap(@P,Nb); |
---|
806 | Wc=imap(@P,Wb); |
---|
807 | kill LNb; |
---|
808 | list LNb; |
---|
809 | return(list(Nd,Wc,LNb)) |
---|
810 | } |
---|
811 | } |
---|
812 | example |
---|
813 | { "EXAMPLE:"; echo = 2; |
---|
814 | ring r=(0,a,b,c),(x,y),dp; |
---|
815 | setglobalrings(); |
---|
816 | ideal N=(ab-c)*(a-b),(a-bc)*(a-b); |
---|
817 | ideal W=a^2-b^2,bc; |
---|
818 | redspec(N,W); |
---|
819 | } |
---|
820 | |
---|
821 | // lesspol: compare two polynomials by its leading power products |
---|
822 | // input: two polynomials f,g in the ring @R |
---|
823 | // output: 0 if f<g, 1 if f>=g |
---|
824 | proc lesspol(poly f, poly g) |
---|
825 | { |
---|
826 | if (leadmonom(f)<leadmonom(g)){return(1);} |
---|
827 | else |
---|
828 | { |
---|
829 | if (leadmonom(g)<leadmonom(f)){return(0);} |
---|
830 | else |
---|
831 | { |
---|
832 | if (leadcoef(f)<leadcoef(g)){return(1);} |
---|
833 | else {return(0);} |
---|
834 | } |
---|
835 | } |
---|
836 | } |
---|
837 | |
---|
838 | // delfromideal: deletes the i-th polynomial from the ideal F |
---|
839 | proc delfromideal(ideal F, int i) |
---|
840 | { |
---|
841 | int j; |
---|
842 | ideal G; |
---|
843 | if (size(F)<i){ERROR("delfromideal was called incorrect arguments");} |
---|
844 | if (size(F)<=1){return(ideal(0));} |
---|
845 | if (i==0){return(F)} |
---|
846 | for (j=1;j<=size(F);j++) |
---|
847 | { |
---|
848 | if (j!=i){G[size(G)+1]=F[j];} |
---|
849 | } |
---|
850 | return(G); |
---|
851 | } |
---|
852 | |
---|
853 | // delidfromid: deletes the polynomials in J that are in I |
---|
854 | // input: ideals I,J |
---|
855 | // output: the ideal J without the polynomials in I |
---|
856 | proc delidfromid(ideal I, ideal J) |
---|
857 | { |
---|
858 | int i; list r; |
---|
859 | ideal JJ=J; |
---|
860 | for (i=1;i<=size(I);i++) |
---|
861 | { |
---|
862 | r=memberpos(I[i],JJ); |
---|
863 | if (r[1]) |
---|
864 | { |
---|
865 | JJ=delfromideal(JJ,r[2]); |
---|
866 | } |
---|
867 | } |
---|
868 | return(JJ); |
---|
869 | } |
---|
870 | |
---|
871 | // sortideal: sorts the polynomials in an ideal by lm in ascending order |
---|
872 | proc sortideal(ideal Fi) |
---|
873 | { |
---|
874 | def RR=basering; |
---|
875 | setring(@RP); |
---|
876 | def F=imap(RR,Fi); |
---|
877 | def H=F; |
---|
878 | ideal G; |
---|
879 | int i; |
---|
880 | int j; |
---|
881 | poly p; |
---|
882 | while (size(H)!=0) |
---|
883 | { |
---|
884 | j=1; |
---|
885 | p=H[1]; |
---|
886 | for (i=1;i<=size(H);i++) |
---|
887 | { |
---|
888 | if(lesspol(H[i],p)){j=i;p=H[j];} |
---|
889 | } |
---|
890 | G[size(G)+1]=p; |
---|
891 | H=delfromideal(H,j); |
---|
892 | } |
---|
893 | setring(RR); |
---|
894 | def GG=imap(@RP,G); |
---|
895 | return(GG); |
---|
896 | } |
---|
897 | |
---|
898 | // mingb: given a basis (gb reducing) it |
---|
899 | // order the polynomials is ascending order and |
---|
900 | // eliminate the polynomials whose lpp is divisible by some |
---|
901 | // smaller one |
---|
902 | proc mingb(ideal F) |
---|
903 | { |
---|
904 | int t; int i; int j; |
---|
905 | def H=sortideal(F); |
---|
906 | ideal G; |
---|
907 | if (ncols(H)<=1){return(H);} |
---|
908 | G=H[1]; |
---|
909 | for (i=2; i<=ncols(H); i++) |
---|
910 | { |
---|
911 | t=1; |
---|
912 | j=1; |
---|
913 | while (t and (j<i)) |
---|
914 | { |
---|
915 | if((leadmonom(H[i])/leadmonom(H[j]))!=0) {t=0;} |
---|
916 | j++; |
---|
917 | } |
---|
918 | if (t) {G[size(G)+1]=H[i];} |
---|
919 | } |
---|
920 | return(G); |
---|
921 | } |
---|
922 | |
---|
923 | |
---|
924 | // redgb: given a minimal bases (gb reducing) it |
---|
925 | // reduces each polynomial w.r.t. to the others |
---|
926 | proc redgb(ideal F, ideal N, ideal W) |
---|
927 | { |
---|
928 | ideal G; |
---|
929 | ideal H; |
---|
930 | int i; |
---|
931 | if (size(F)==0){return(ideal(0));} |
---|
932 | for (i=1;i<=size(F);i++) |
---|
933 | { |
---|
934 | H=delfromideal(F,i); |
---|
935 | G[i]=pnormalform(pdivi2(F[i],H)[1],N,W); |
---|
936 | } |
---|
937 | return(G); |
---|
938 | } |
---|
939 | |
---|
940 | |
---|
941 | //********************Main routines for buildtree****************** |
---|
942 | |
---|
943 | |
---|
944 | // splitspec: a new leading coefficient f is given to a red-spec |
---|
945 | // then splitspec computes the two new red-spec by |
---|
946 | // considering it null, and non null. |
---|
947 | // in ring @P |
---|
948 | // given f, and the red-spec (N,W) |
---|
949 | // it outputs the null and the non-null red-spec adding f. |
---|
950 | // if some of the output specifications has N=1 then |
---|
951 | // there must be no split and buildtree must continue on |
---|
952 | // the compatible red-spec |
---|
953 | // input: poly f coefficient to split if needed |
---|
954 | // list r=(N,W,LN) redspec |
---|
955 | // output: list L = list(ideal N0, ideal W0), list(ideal N1, ideal W1), cond |
---|
956 | proc splitspec(poly fi, list ri) |
---|
957 | { |
---|
958 | def RR=basering; |
---|
959 | def Ni=ri[1]; |
---|
960 | def Wi=ri[2]; |
---|
961 | setring(@P); |
---|
962 | def f=imap(RR,fi); |
---|
963 | def N=imap(RR,Ni); |
---|
964 | def W=imap(RR,Wi); |
---|
965 | f=Wred(f,W); |
---|
966 | def N0=N; |
---|
967 | def W1=W; |
---|
968 | N0[size(N0)+1]=f; |
---|
969 | def r0=redspec(N0,W); |
---|
970 | W1[size(W1)+1]=f; |
---|
971 | def r1=redspec(N,W1); |
---|
972 | setring(RR); |
---|
973 | def ra0=imap(@P,r0); |
---|
974 | def ra1=imap(@P,r1); |
---|
975 | def cond=imap(@P,f); |
---|
976 | return (list(ra0,ra1,cond)); |
---|
977 | } |
---|
978 | |
---|
979 | // discusspolys: given a basis B and a red-spec (N,W), it analyzes the |
---|
980 | // leadcoef of the polynomials in B until it finds |
---|
981 | // that one of them can be either null or non null. |
---|
982 | // If at the end only the non null option is compatible |
---|
983 | // then the reduced B has all the leadcoef non null. |
---|
984 | // Else recbuildtree must split. |
---|
985 | // ring @R |
---|
986 | // input: ideal B |
---|
987 | // ideal N |
---|
988 | // ideal W (a reduced-specification) |
---|
989 | // output: list of ((N0,W0,LN0),(N1,W1,LN1),Br,cond) |
---|
990 | // cond is the condition to branch |
---|
991 | proc discusspolys(ideal B, list r) |
---|
992 | { |
---|
993 | poly f; poly f1; poly f2; |
---|
994 | poly cond; |
---|
995 | def N=r[1]; def W=r[2]; def LN=r[3]; |
---|
996 | def Ba=B; def F=B; |
---|
997 | ideal N0=1; def W0=W; list LN0=ideal(1); |
---|
998 | def N1=N; def W1=W; def LN1=LN; |
---|
999 | list L; |
---|
1000 | list M; list M0; list M1; |
---|
1001 | list rr; |
---|
1002 | if (size(B)==0) |
---|
1003 | { |
---|
1004 | M0=N0,W0,LN0; // incompatible |
---|
1005 | M1=N1,W1,LN1; |
---|
1006 | M=M0,M1,B,poly(1); |
---|
1007 | return(M); |
---|
1008 | } |
---|
1009 | while ((size(F)!=0) and ((N0[1]==1) or (N1[1]==1))) |
---|
1010 | { |
---|
1011 | f=F[1]; |
---|
1012 | F=delfromideal(F,1); |
---|
1013 | f1=pnormalform(f,N,W); |
---|
1014 | rr=memberpos(f,Ba); |
---|
1015 | if (f1!=0) |
---|
1016 | { |
---|
1017 | Ba[rr[2]]=f1; |
---|
1018 | if (pardeg(leadcoef(f1))!=0) |
---|
1019 | { |
---|
1020 | f2=Wred(leadcoef(f1),W); |
---|
1021 | L=splitspec(f2,list(N,W,LN)); |
---|
1022 | N0=L[1][1]; W0=L[1][2]; LN0=L[1][3]; N1=L[2][1]; W1=L[2][2]; LN1=L[2][3]; |
---|
1023 | cond=L[3]; |
---|
1024 | } |
---|
1025 | } |
---|
1026 | else |
---|
1027 | { |
---|
1028 | Ba=delfromideal(Ba,rr[2]); |
---|
1029 | N0=ideal(1); //F=ideal(0); |
---|
1030 | } |
---|
1031 | } |
---|
1032 | M0=N0,W0,LN0; |
---|
1033 | M1=N1,W1,LN1; |
---|
1034 | M=M0,M1,Ba,cond; |
---|
1035 | return(M); |
---|
1036 | } |
---|
1037 | |
---|
1038 | |
---|
1039 | // lcmlmonoms: computes the lcm of the leading monomials |
---|
1040 | // of the polynomils f and g |
---|
1041 | // ring @R |
---|
1042 | proc lcmlmonoms(poly f,poly g) |
---|
1043 | { |
---|
1044 | def lf=leadmonom(f); |
---|
1045 | def lg=leadmonom(g); |
---|
1046 | def gls=gcd(lf,lg); |
---|
1047 | return((lf*lg)/gls); |
---|
1048 | } |
---|
1049 | |
---|
1050 | // placepairinlist |
---|
1051 | // input: given a new pair of the form (i,j,lmij) |
---|
1052 | // and a list of pairs of the same form |
---|
1053 | // ring @R |
---|
1054 | // output: it inserts the new pair in ascending order of lmij |
---|
1055 | proc placepairinlist(list pair,list P) |
---|
1056 | { |
---|
1057 | list Pr; |
---|
1058 | if (size(P)==0){Pr=insert(P,pair); return(Pr);} |
---|
1059 | if (pair[3]<P[1][3]){Pr=insert(P,pair); return(Pr);} |
---|
1060 | if (pair[3]>=P[size(P)][3]){Pr=insert(P,pair,size(P)); return(Pr);} |
---|
1061 | kill Pr; |
---|
1062 | list Pr; |
---|
1063 | int j; |
---|
1064 | int i=1; |
---|
1065 | int loc=0; |
---|
1066 | while((i<=size(P)) and (loc==0)) |
---|
1067 | { |
---|
1068 | if (pair[3]>=P[i][3]){j=i; i++;} |
---|
1069 | else{loc=1; j=i-1;} |
---|
1070 | } |
---|
1071 | Pr=insert(P,pair,j); |
---|
1072 | return(Pr); |
---|
1073 | } |
---|
1074 | |
---|
1075 | // orderingpairs: |
---|
1076 | // input: ideal F |
---|
1077 | // output: list of ordered pairs (i,j,lcmij) of F in ascending order of lcmij |
---|
1078 | // if a pair verifies Buchberger 1st criterion it is not stored |
---|
1079 | // ring @R |
---|
1080 | proc orderingpairs(ideal F) |
---|
1081 | { |
---|
1082 | int i; |
---|
1083 | int j; |
---|
1084 | poly lm; |
---|
1085 | poly lpf; |
---|
1086 | poly lpg; |
---|
1087 | list P; |
---|
1088 | list pair; |
---|
1089 | if (size(F)<=1){return(P);} |
---|
1090 | for (i=1;i<=size(F)-1;i++) |
---|
1091 | { |
---|
1092 | for (j=i+1;j<=size(F);j++) |
---|
1093 | { |
---|
1094 | lm=lcmlmonoms(F[i],F[j]); |
---|
1095 | // Buchberger 1st criterion |
---|
1096 | lpf=leadmonom(F[i]); |
---|
1097 | lpg=leadmonom(F[j]); |
---|
1098 | if (lpf*lpg!=lm) |
---|
1099 | { |
---|
1100 | pair=(i,j,lm); |
---|
1101 | P=placepairinlist(pair,P); |
---|
1102 | } |
---|
1103 | } |
---|
1104 | } |
---|
1105 | return(P); |
---|
1106 | } |
---|
1107 | |
---|
1108 | // Buchberger 2nd criterion |
---|
1109 | // input: integers i,j |
---|
1110 | // list P of pairs of the form (i,j) not yet verified |
---|
1111 | // ring @R |
---|
1112 | proc criterion(int i, int j, list P, ideal B) |
---|
1113 | { |
---|
1114 | def lcmij=lcmlmonoms(B[i],B[j]); |
---|
1115 | int crit=0; |
---|
1116 | int k=1; |
---|
1117 | list ik; list jk; |
---|
1118 | while ((k<=size(B)) and (crit==0)) |
---|
1119 | { |
---|
1120 | if ((k!=i) and (k!=j)) |
---|
1121 | { |
---|
1122 | if (i<k){ik=i,k;} else{ik=k,i;} |
---|
1123 | if (j<k){jk=i,k;} else{jk=k,j;} |
---|
1124 | if (not((memberpos(ik,P)[1]) or (memberpos(jk,P)[1]))) |
---|
1125 | { |
---|
1126 | if ((lcmij)/leadmonom(B[k])!=0){crit=1;} |
---|
1127 | } |
---|
1128 | } |
---|
1129 | k++; |
---|
1130 | } |
---|
1131 | return(crit); |
---|
1132 | } |
---|
1133 | |
---|
1134 | // discussSpolys: given a basis B and a red-spec (N,W), it analyzes the |
---|
1135 | // leadcoef of the polynomials in B until it finds |
---|
1136 | // that one of them can be either null or non null. |
---|
1137 | // If at the end only the non null option is compatible |
---|
1138 | // then the reduced B has all the leadcoef non null. |
---|
1139 | // Else recbuildtree must split. |
---|
1140 | // ring @R |
---|
1141 | // input: ideal B |
---|
1142 | // ideal N |
---|
1143 | // ideal W (a reduced-specification) |
---|
1144 | // list P current set of pairs of polynomials from B to be tested. |
---|
1145 | // output: list of (N0,W0,LN0),(N1,W1,LN1),Br,Pr,cond] |
---|
1146 | // list Pr the not checked list of pairs. |
---|
1147 | proc discussSpolys(ideal B, list r, list P) |
---|
1148 | { |
---|
1149 | int i; int j; int k; |
---|
1150 | int npols; int nSpols; int tt; |
---|
1151 | poly cond=1; |
---|
1152 | poly lm; poly lpf; poly lpg; |
---|
1153 | def F=B; def Pa=P; list Pa0; |
---|
1154 | def N=r[1]; def W=r[2]; def LN=r[3]; |
---|
1155 | ideal N0=1; def W0=W; list LN0=ideal(1); |
---|
1156 | def N1=N; def W1=W; def LN1=LN; |
---|
1157 | ideal Bw; |
---|
1158 | poly S; |
---|
1159 | list L; list L0; list L1; |
---|
1160 | list M; list M0; list M1; |
---|
1161 | list pair; |
---|
1162 | list KK; int loc; |
---|
1163 | int crit; |
---|
1164 | poly h; |
---|
1165 | if (size(B)==0) |
---|
1166 | { |
---|
1167 | M0=N0,W0,LN0; |
---|
1168 | M1=N1,W1,LN1; |
---|
1169 | M=M0,M1,ideal(0),Pa,cond; |
---|
1170 | return(M); |
---|
1171 | } |
---|
1172 | tt=1; |
---|
1173 | i=1; |
---|
1174 | while ((tt) and (i<=size(B))) |
---|
1175 | { |
---|
1176 | h=B[i]; |
---|
1177 | for (j=1;j<=npars(@R);j++) |
---|
1178 | { |
---|
1179 | h=subst(h,par(j),0); |
---|
1180 | } |
---|
1181 | if (h!=B[i]){tt=0;} |
---|
1182 | i++; |
---|
1183 | } |
---|
1184 | if (tt) |
---|
1185 | { |
---|
1186 | //"T_ a non parametric system occurred"; |
---|
1187 | def RR=basering; |
---|
1188 | def RL=ringlist(RR); |
---|
1189 | RL[1]=0; |
---|
1190 | def LRR=ring(RL); |
---|
1191 | setring(LRR); |
---|
1192 | def BP=imap(RR,B); |
---|
1193 | option(redSB); |
---|
1194 | BP=groebner(BP); |
---|
1195 | setring(RR); |
---|
1196 | B=imap(LRR,BP); |
---|
1197 | M0=ideal(1),W0,LN0; |
---|
1198 | M1=N1,W1,LN1; |
---|
1199 | M=M0,M1,B,list(),cond; |
---|
1200 | return(M); |
---|
1201 | } |
---|
1202 | if (size(Pa)==0){npols=size(B); Pa=orderingpairs(F); nSpols=size(Pa);} |
---|
1203 | while ((size(Pa)!=0) and (N0[1]==1) or (N1[1]==1)) |
---|
1204 | { |
---|
1205 | pair=Pa[1]; |
---|
1206 | i=pair[1]; |
---|
1207 | j=pair[2]; |
---|
1208 | Pa=delete(Pa,1); |
---|
1209 | // Buchberger 1st criterion (not needed here, it is already eliminated |
---|
1210 | // when creating the list of pairs |
---|
1211 | //T_ lpf=leadmonom(F[i]); |
---|
1212 | //T_ lpg=leadmonom(F[j]); |
---|
1213 | //T_ if (lpf*lpg!=pair[3]) |
---|
1214 | //T_ { |
---|
1215 | for (k=1;k<=size(Pa);k++){Pa0[k]=delete(Pa[k],3);} |
---|
1216 | //crit=criterion(i,j,Pa0,F); // produces errors? |
---|
1217 | crit=0; |
---|
1218 | if (not(crit)) |
---|
1219 | { |
---|
1220 | S=pspol(F[i],F[j]); |
---|
1221 | KK=pdivi2(S,F); |
---|
1222 | S=KK[1]; |
---|
1223 | if (S!=0) |
---|
1224 | { |
---|
1225 | S=pnormalform(S,N,W); |
---|
1226 | if (S!=0) |
---|
1227 | { |
---|
1228 | L=discusspolys(ideal(S),list(N,W,LN)); |
---|
1229 | N0=L[1][1]; |
---|
1230 | W0=L[1][2]; |
---|
1231 | LN0=L[1][3]; |
---|
1232 | N1=L[2][1]; |
---|
1233 | W1=L[2][2]; |
---|
1234 | LN1=L[2][3]; |
---|
1235 | S=L[3][1]; |
---|
1236 | cond=L[4]; |
---|
1237 | if (S==1) |
---|
1238 | { |
---|
1239 | M0=ideal(1),W0,list(ideal(1)); |
---|
1240 | M1=N1,W1,LN1; |
---|
1241 | M=M0,M1,ideal(1),list(),cond; |
---|
1242 | return(M); |
---|
1243 | } |
---|
1244 | if (S!=0) |
---|
1245 | { |
---|
1246 | F[size(F)+1]=S; |
---|
1247 | npols=size(F); |
---|
1248 | //"T_ number of polynoms in the basis="; npols; |
---|
1249 | for (k=1;k<size(F);k++) |
---|
1250 | { |
---|
1251 | lm=lcmlmonoms(F[k],S); |
---|
1252 | // Buchberger 1st criterion |
---|
1253 | lpf=leadmonom(F[k]); |
---|
1254 | lpg=leadmonom(S); |
---|
1255 | if (lpf*lpg!=lm) |
---|
1256 | { |
---|
1257 | pair=k,size(F),lm; |
---|
1258 | Pa=placepairinlist(pair,Pa); |
---|
1259 | nSpols=size(Pa); |
---|
1260 | //"T_ number of S-polynoms to test="; nSpols; |
---|
1261 | } |
---|
1262 | } |
---|
1263 | if (N0[1]==1){N=N1; W=W1; LN=LN1;} |
---|
1264 | } |
---|
1265 | } |
---|
1266 | } |
---|
1267 | } |
---|
1268 | //T_ } |
---|
1269 | } |
---|
1270 | M0=N0,W0,LN0; |
---|
1271 | M1=N1,W1,LN1; |
---|
1272 | M=M0,M1,F,Pa,cond; |
---|
1273 | return(M); |
---|
1274 | } |
---|
1275 | |
---|
1276 | |
---|
1277 | // buildtree: Basic routine generating a first reduced CGS |
---|
1278 | // it will define the rings @R, @P and @RP as global rings |
---|
1279 | // and the list @T a global list that will be killed at the output |
---|
1280 | // input: ideal F in ring K[a][x]; |
---|
1281 | // output: list T of lists whose list elements are of the form |
---|
1282 | // T[i]=list(list lab, boolean terminal, ideal B, ideal N, ideal W, list of ideals decomp of N, |
---|
1283 | // ideal of monomials lpp); |
---|
1284 | // all the ideals are in the ring K[a][x]; |
---|
1285 | proc buildtree(ideal F, list #) |
---|
1286 | "USAGE: buildtree(F); |
---|
1287 | F: ideal in K[a][x] (parameters and variables) to be discussed |
---|
1288 | RETURN: Returns a list T describing a dichotomic discussion tree, whose |
---|
1289 | content is the first discussion of the ideal F of K[a][x]. |
---|
1290 | The first element of the list is the root, and contains |
---|
1291 | [1] label: intvec(-1) |
---|
1292 | [2] number of children : int |
---|
1293 | [3] the ideal F |
---|
1294 | [4], [5], [6] the red-spec of the null and non-null conditions |
---|
1295 | given (as option). ideal (0), ideal (0), list(ideal(0)) if |
---|
1296 | no optional conditions are given. |
---|
1297 | [7] the set of lpp of ideal F |
---|
1298 | [8] condition that was taken to reach the vertex |
---|
1299 | (poly 1, for the root). |
---|
1300 | The remaining elements of the list represent vertices of the tree: |
---|
1301 | with the same structure: |
---|
1302 | [1] label: intvec (1,0,0,1,...) gives its position in the tree: |
---|
1303 | first branch condition is taken non-null, second null,... |
---|
1304 | [2] number of children (0 if it is a terminal vertex) |
---|
1305 | [3] the specialized ideal with the previous assumed conditions |
---|
1306 | to reach the vertex |
---|
1307 | [4],[5],[6] the red-spec of the previous assumed conditions |
---|
1308 | to reach the vertex |
---|
1309 | [7] the set of lpp of the specialized ideal at this stage |
---|
1310 | [8] condition that was taken to reach the vertex from the |
---|
1311 | father's vertex (that was taken non-null if the last |
---|
1312 | integer in the label is 1, and null if it is 0) |
---|
1313 | The terminal vertices form a disjoint partition of the parameter space |
---|
1314 | whose bases specialize to the reduced Groebner basis of the |
---|
1315 | specialized ideal on each point of the segment and preserve |
---|
1316 | the lpp. So they form a disjoint reduced CGS. |
---|
1317 | NOTE: The basering R, must be of the form K[a][x], a=parameters, |
---|
1318 | x=variables, and should be defined previously. The ideal must |
---|
1319 | be defined on R. |
---|
1320 | The disjoint and reduced CGS built by buildtree can be obtained |
---|
1321 | from the output of buildtree by calling finalcases(T); this |
---|
1322 | selects the terminal vertices. |
---|
1323 | The content of buildtree can be written in a file that is readable |
---|
1324 | by Maple in order to plot its content using buildtreetoMaple; |
---|
1325 | The file written by buildtreetoMaple when read in a Maple |
---|
1326 | worksheet can be plotted using the dbgb routine tplot; |
---|
1327 | |
---|
1328 | KEYWORDS: CGS, disjoint, reduced, comprehensive Groebner system |
---|
1329 | EXAMPLE: buildtree; shows an example" |
---|
1330 | { |
---|
1331 | list @T; |
---|
1332 | exportto(Top,@T); |
---|
1333 | def @R=basering; |
---|
1334 | setglobalrings(); |
---|
1335 | int i; |
---|
1336 | int j; |
---|
1337 | ideal B; |
---|
1338 | poly f; |
---|
1339 | poly cond=1; |
---|
1340 | def N=ideal(0); |
---|
1341 | def W=ideal(0); |
---|
1342 | list LN; |
---|
1343 | LN[1]=ideal(0); |
---|
1344 | if (size(#)==2) |
---|
1345 | { |
---|
1346 | N=#[1]; |
---|
1347 | W=#[2]; |
---|
1348 | def LL=redspec(N,W); |
---|
1349 | N=LL[1]; |
---|
1350 | W=LL[2]; |
---|
1351 | LN=LL[3]; |
---|
1352 | j=1; |
---|
1353 | for (i=1;i<=size(F);i++) |
---|
1354 | { |
---|
1355 | f=pnormalform(F[i],N,W); |
---|
1356 | if (f!=0){B[j]=f;j++;} |
---|
1357 | } |
---|
1358 | } |
---|
1359 | else {B=F;} |
---|
1360 | def lpp=ideal(0); |
---|
1361 | if (size(B)==0){lpp=ideal(0);} |
---|
1362 | else |
---|
1363 | { |
---|
1364 | for (i=1;i<=size(B);i++){lpp[i]=leadmonom(B[i]);} |
---|
1365 | // lpp=ideal of lead power product of the polys in B |
---|
1366 | } |
---|
1367 | intvec lab=-1; |
---|
1368 | int term=0; |
---|
1369 | list root; |
---|
1370 | root[1]=lab; |
---|
1371 | root[2]=term; |
---|
1372 | root[3]=B; |
---|
1373 | root[4]=N; |
---|
1374 | root[5]=W; |
---|
1375 | root[6]=LN; |
---|
1376 | root[7]=lpp; |
---|
1377 | root[8]=cond; |
---|
1378 | @T[1]=root; |
---|
1379 | list P; |
---|
1380 | recbuildtree(root,P); |
---|
1381 | def T=@T; |
---|
1382 | kill @T; |
---|
1383 | return(T) |
---|
1384 | } |
---|
1385 | example |
---|
1386 | { "EXAMPLE:"; echo = 2; |
---|
1387 | ring R=(0,a1,a2,a3,a4),(x1,x2,x3,x4),dp; |
---|
1388 | ideal F=x4-a4+a2, |
---|
1389 | x1+x2+x3+x4-a1-a3-a4, |
---|
1390 | x1*x3*x4-a1*a3*a4, |
---|
1391 | x1*x3+x1*x4+x2*x3+x3*x4-a1*a4-a1*a3-a3*a4; |
---|
1392 | def T=buildtree(F); |
---|
1393 | finalcases(T); |
---|
1394 | buildtreetoMaple(T,"Tb","Tb.txt"); |
---|
1395 | } |
---|
1396 | |
---|
1397 | // recbuildtree: auxiiary recursive routine called by buildtree |
---|
1398 | proc recbuildtree(list v, list P) |
---|
1399 | { |
---|
1400 | def vertex=v; |
---|
1401 | int i; |
---|
1402 | int j; |
---|
1403 | int pos; |
---|
1404 | list P0; |
---|
1405 | list P1; |
---|
1406 | poly f; |
---|
1407 | def lab=vertex[1]; |
---|
1408 | if ((size(lab)>1) and (lab[1]==-1)) |
---|
1409 | {lab=lab[2..size(lab)];} |
---|
1410 | def term=vertex[2]; |
---|
1411 | def B=vertex[3]; |
---|
1412 | def N=vertex[4]; |
---|
1413 | def W=vertex[5]; |
---|
1414 | def LN=vertex[6]; |
---|
1415 | def lpp=vertex[7]; |
---|
1416 | def cond=vertex[8]; |
---|
1417 | def lab0=lab; |
---|
1418 | def lab1=lab; |
---|
1419 | if ((size(lab)==1) and (lab[1]==-1)) |
---|
1420 | { |
---|
1421 | lab0=0; |
---|
1422 | lab1=1; |
---|
1423 | } |
---|
1424 | else |
---|
1425 | { |
---|
1426 | lab0[size(lab)+1]=0; |
---|
1427 | lab1[size(lab)+1]=1; |
---|
1428 | } |
---|
1429 | list vertex0; |
---|
1430 | list vertex1; |
---|
1431 | ideal B0; |
---|
1432 | ideal lpp0; |
---|
1433 | ideal lpp1; |
---|
1434 | ideal N0=1; |
---|
1435 | def W0=ideal(0); |
---|
1436 | list LN0=ideal(1); |
---|
1437 | def B1=B; |
---|
1438 | def N1=N; |
---|
1439 | def W1=W; |
---|
1440 | list LN1=LN; |
---|
1441 | list L; |
---|
1442 | if (size(P)==0) |
---|
1443 | { |
---|
1444 | L=discusspolys(B,list(N,W,LN)); |
---|
1445 | N0=L[1][1]; |
---|
1446 | W0=L[1][2]; |
---|
1447 | LN0=L[1][3]; |
---|
1448 | N1=L[2][1]; |
---|
1449 | W1=L[2][2]; |
---|
1450 | LN1=L[2][3]; |
---|
1451 | B1=L[3]; |
---|
1452 | cond=L[4]; |
---|
1453 | } |
---|
1454 | if ((size(B1)!=0) and (N0[1]==1)) |
---|
1455 | { |
---|
1456 | L=discussSpolys(B1,list(N1,W1,LN1),P); |
---|
1457 | N0=L[1][1]; |
---|
1458 | W0=L[1][2]; |
---|
1459 | LN0=L[1][3]; |
---|
1460 | N1=L[2][1]; |
---|
1461 | W1=L[2][2]; |
---|
1462 | LN1=L[2][3]; |
---|
1463 | B1=L[3]; |
---|
1464 | P1=L[4]; |
---|
1465 | cond=L[5]; |
---|
1466 | lpp=ideal(0); |
---|
1467 | for (i=1;i<=size(B1);i++){lpp[i]=leadmonom(B1[i]);} |
---|
1468 | } |
---|
1469 | vertex[3]=B1; |
---|
1470 | vertex[4]=N1; // unnecessary |
---|
1471 | vertex[5]=W1; // unnecessary |
---|
1472 | vertex[6]=LN1;// unnecessary |
---|
1473 | vertex[7]=lpp; |
---|
1474 | vertex[8]=cond; |
---|
1475 | if (size(@T)>0) |
---|
1476 | { |
---|
1477 | pos=size(@T)+1; |
---|
1478 | @T[pos]=vertex; |
---|
1479 | } |
---|
1480 | if ((N0[1]!=1) and (N1[1]!=1)) |
---|
1481 | { |
---|
1482 | vertex1[1]=lab1; |
---|
1483 | vertex1[2]=0; |
---|
1484 | vertex1[3]=B1; |
---|
1485 | vertex1[4]=N1; |
---|
1486 | vertex1[5]=W1; |
---|
1487 | vertex1[6]=LN1; |
---|
1488 | vertex1[7]=lpp1; |
---|
1489 | vertex1[8]=cond; |
---|
1490 | if (size(B1)==0){B0=ideal(0); lpp0=ideal(0);} |
---|
1491 | else |
---|
1492 | { |
---|
1493 | j=1; |
---|
1494 | lpp0=ideal(0); |
---|
1495 | for (i=1;i<=size(B1);i++) |
---|
1496 | { |
---|
1497 | f=pnormalform(B1[i],N0,W0); |
---|
1498 | if (f!=0){B0[j]=f; lpp0[j]=leadmonom(f);j++;} |
---|
1499 | } |
---|
1500 | } |
---|
1501 | vertex0[1]=lab0; |
---|
1502 | vertex0[2]=0; |
---|
1503 | vertex0[3]=B0; |
---|
1504 | vertex0[4]=N0; |
---|
1505 | vertex0[5]=W0; |
---|
1506 | vertex0[6]=LN0; |
---|
1507 | vertex0[7]=lpp0; |
---|
1508 | vertex0[8]=cond; |
---|
1509 | recbuildtree(vertex0,P0); |
---|
1510 | recbuildtree(vertex1,P1); |
---|
1511 | } |
---|
1512 | else |
---|
1513 | { |
---|
1514 | vertex[2]=1; |
---|
1515 | B1=mingb(B1); |
---|
1516 | vertex[3]=redgb(B1,N1,W1); |
---|
1517 | vertex[4]=N1; |
---|
1518 | vertex[5]=W1; |
---|
1519 | vertex[6]=LN1; |
---|
1520 | lpp=ideal(0); |
---|
1521 | for (i=1;i<=size(vertex[3]);i++){lpp[i]=leadmonom(vertex[3][i]);} |
---|
1522 | vertex[7]=lpp; |
---|
1523 | vertex[8]=cond; |
---|
1524 | @T[pos]=vertex; |
---|
1525 | } |
---|
1526 | } |
---|
1527 | |
---|
1528 | //****************End of BuildTree************************************* |
---|
1529 | |
---|
1530 | //****************Begin BuildTree To Maple***************************** |
---|
1531 | |
---|
1532 | // buildtreetoMaple: writes the list provided by buildtree to a file |
---|
1533 | // containing the table representing it in Maple |
---|
1534 | |
---|
1535 | // writes the list L=buildtree(F) to a file "writefile" that |
---|
1536 | // is readable by Maple with name T |
---|
1537 | // input: |
---|
1538 | // L: the list output by buildtree |
---|
1539 | // T: the name (string) of the output table in Maple |
---|
1540 | // writefile: the name of the datafile where the output is to be stored |
---|
1541 | // output: |
---|
1542 | // the result is written on the datafile "writefile" containing |
---|
1543 | // the assignment to the table with name "T" |
---|
1544 | proc buildtreetoMaple(list L, string T, string writefile) |
---|
1545 | "USAGE: buildtreetoMaple(T, TM, writefile); |
---|
1546 | T: is the list provided by buildtree, |
---|
1547 | TM: is the name (string) of the table variable in Maple that will represent |
---|
1548 | the output of buildtree, |
---|
1549 | writefile: is the name (string) of the file where to write the content. |
---|
1550 | RETURN: writes the list provided by buildtree to a file |
---|
1551 | containing the table representing it in Maple. |
---|
1552 | KEYWORDS: buildtree, Maple |
---|
1553 | EXAMPLE: buildtreetoMaple; shows an example" |
---|
1554 | { |
---|
1555 | short=0; |
---|
1556 | poly cond; |
---|
1557 | int i; |
---|
1558 | link LLw=":w "+writefile; |
---|
1559 | string La=string("table(",T,");"); |
---|
1560 | write(LLw, La); |
---|
1561 | close(LLw); |
---|
1562 | link LLa=":a "+writefile; |
---|
1563 | def RL=ringlist(@R); |
---|
1564 | list p=RL[1][2]; |
---|
1565 | string param=string(p[1]); |
---|
1566 | if (size(p)>1) |
---|
1567 | { |
---|
1568 | for(i=2;i<=size(p);i++){param=string(param,",",p[i]);} |
---|
1569 | } |
---|
1570 | list v=RL[2]; |
---|
1571 | string vars=string(v[1]); |
---|
1572 | if (size(v)>1) |
---|
1573 | { |
---|
1574 | for(i=2;i<=size(v);i++){vars=string(vars,",",v[i]);} |
---|
1575 | } |
---|
1576 | list xord; |
---|
1577 | list pord; |
---|
1578 | if (RL[1][3][1][1]=="dp"){pord=string("tdeg(",param);} |
---|
1579 | if (RL[1][3][1][1]=="lp"){pord=string("plex(",param);} |
---|
1580 | if (RL[3][1][1]=="dp"){xord=string("tdeg(",vars);} |
---|
1581 | if (RL[3][1][1]=="lp"){xord=string("plex(",vars);} |
---|
1582 | write(LLa,string(T,"[[9]]:=",xord,");")); |
---|
1583 | write(LLa,string(T,"[[10]]:=",pord,");")); |
---|
1584 | write(LLa,string(T,"[[11]]:=true; ")); |
---|
1585 | list S; |
---|
1586 | for (i=1;i<=size(L);i++) |
---|
1587 | { |
---|
1588 | if (L[i][2]==0) |
---|
1589 | { |
---|
1590 | cond=L[i][8]; |
---|
1591 | S=btcond(T,L[i],cond); |
---|
1592 | write(LLa,S[1]); |
---|
1593 | write(LLa,S[2]); |
---|
1594 | } |
---|
1595 | S=btbasis(T,L[i]); |
---|
1596 | write(LLa,S); |
---|
1597 | S=btN(T,L[i]); |
---|
1598 | write(LLa,S); |
---|
1599 | S=btW(T,L[i]); |
---|
1600 | write(LLa,S); |
---|
1601 | if (L[i][2]==1) {S=btterminal(T,L[i]); write(LLa,S);} |
---|
1602 | S=btlpp(T,L[i]); |
---|
1603 | write(LLa,S); |
---|
1604 | } |
---|
1605 | close(LLa); |
---|
1606 | } |
---|
1607 | example |
---|
1608 | { "EXAMPLE:"; echo = 2; |
---|
1609 | ring R=(0,a1,a2,a3,a4),(x1,x2,x3,x4),dp; |
---|
1610 | ideal F=x4-a4+a2, |
---|
1611 | x1+x2+x3+x4-a1-a3-a4, |
---|
1612 | x1*x3*x4-a1*a3*a4, |
---|
1613 | x1*x3+x1*x4+x2*x3+x3*x4-a1*a4-a1*a3-a3*a4; |
---|
1614 | def T=buildtree(F); |
---|
1615 | finalcases(T); |
---|
1616 | buildtreetoMaple(T,"Tb","Tb.txt"); |
---|
1617 | } |
---|
1618 | |
---|
1619 | // auxiliary routine called by buildtreetoMaple |
---|
1620 | // input: |
---|
1621 | // list L: element i of the list of buildtree(F) |
---|
1622 | // output: |
---|
1623 | // the string of T[[lab,1]]:=label; in Maple |
---|
1624 | proc btterminal(string T, list L) |
---|
1625 | { |
---|
1626 | int i; |
---|
1627 | string Li; |
---|
1628 | string term; |
---|
1629 | string coma=","; |
---|
1630 | if (L[2]==0){term="false";} else {term="true";} |
---|
1631 | def lab=L[1]; |
---|
1632 | string slab; |
---|
1633 | if ((size(lab)==1) and lab[1]==-1) |
---|
1634 | {slab="";coma="";} //if (size(lab)==0) |
---|
1635 | else |
---|
1636 | { |
---|
1637 | slab=string(lab[1]); |
---|
1638 | if (size(lab)>=1) |
---|
1639 | { |
---|
1640 | for (i=2;i<=size(lab);i++){slab=string(slab,",",lab[i]);} |
---|
1641 | } |
---|
1642 | } |
---|
1643 | Li=string(T,"[[",slab,coma,"6]]:=",term,"; "); |
---|
1644 | return(Li); |
---|
1645 | } |
---|
1646 | |
---|
1647 | // auxiliary routine called by buildtreetoMaple |
---|
1648 | // input: |
---|
1649 | // list L: element i of the list of buildtree(F) |
---|
1650 | // output: |
---|
1651 | // the string of T[[lab,3]] (basis); in Maple |
---|
1652 | proc btbasis(string T, list L) |
---|
1653 | { |
---|
1654 | int i; |
---|
1655 | string Li; |
---|
1656 | string coma=","; |
---|
1657 | def lab=L[1]; |
---|
1658 | string slab; |
---|
1659 | if ((size(lab)==1) and lab[1]==-1) |
---|
1660 | {slab="";coma="";} //if (size(lab)==0) |
---|
1661 | else |
---|
1662 | { |
---|
1663 | slab=string(lab[1]); |
---|
1664 | if (size(lab)>=1) |
---|
1665 | { |
---|
1666 | for (i=2;i<=size(lab);i++){slab=string(slab,",",lab[i]);} |
---|
1667 | } |
---|
1668 | } |
---|
1669 | Li=string(T,"[[",slab,coma,"3]]:=[",L[3],"]; "); |
---|
1670 | return(Li); |
---|
1671 | } |
---|
1672 | |
---|
1673 | // auxiliary routine called by buildtreetoMaple |
---|
1674 | // input: |
---|
1675 | // list L: element i of the list of buildtree(F) |
---|
1676 | // output: |
---|
1677 | // the string of T[[lab,4]] (null conditions ideal); in Maple |
---|
1678 | proc btN(string T, list L) |
---|
1679 | { |
---|
1680 | int i; |
---|
1681 | string Li; |
---|
1682 | string coma=","; |
---|
1683 | def lab=L[1]; |
---|
1684 | string slab; |
---|
1685 | if ((size(lab)==1) and lab[1]==-1) |
---|
1686 | {slab=""; coma="";} |
---|
1687 | else |
---|
1688 | { |
---|
1689 | slab=string(lab[1]); |
---|
1690 | if (size(lab)>=1) |
---|
1691 | { |
---|
1692 | for (i=2;i<=size(lab);i++){slab=string(slab,",",lab[i]);} |
---|
1693 | } |
---|
1694 | } |
---|
1695 | if ((size(lab)==1) and lab[1]==-1) |
---|
1696 | {Li=string(T,"[[",slab,coma,"4]]:=[ ]; ");} |
---|
1697 | else |
---|
1698 | {Li=string(T,"[[",slab,coma,"4]]:=[",L[4],"]; ");} |
---|
1699 | return(Li); |
---|
1700 | } |
---|
1701 | |
---|
1702 | // auxiliary routine called by buildtreetoMaple |
---|
1703 | // input: |
---|
1704 | // list L: element i of the list of buildtree(F) |
---|
1705 | // output: |
---|
1706 | // the string of T[[lab,5]] (null conditions ideal); in Maple |
---|
1707 | proc btW(string T, list L) |
---|
1708 | { |
---|
1709 | int i; |
---|
1710 | string Li; |
---|
1711 | string coma=","; |
---|
1712 | def lab=L[1]; |
---|
1713 | string slab; |
---|
1714 | if ((size(lab)==1) and lab[1]==-1) |
---|
1715 | {slab=""; coma="";} |
---|
1716 | else |
---|
1717 | { |
---|
1718 | slab=string(lab[1]); |
---|
1719 | if (size(lab)>=1) |
---|
1720 | { |
---|
1721 | for (i=2;i<=size(lab);i++){slab=string(slab,",",lab[i]);} |
---|
1722 | } |
---|
1723 | } |
---|
1724 | if (size(L[5])==0) |
---|
1725 | {Li=string(T,"[[",slab,coma,"5]]:={ }; ");} |
---|
1726 | else |
---|
1727 | {Li=string(T,"[[",slab,coma,"5]]:={",L[5],"}; ");} |
---|
1728 | return(Li); |
---|
1729 | } |
---|
1730 | |
---|
1731 | // auxiliary routine called by buildtreetoMaple |
---|
1732 | // input: |
---|
1733 | // list L: element i of the list of buildtree(F) |
---|
1734 | // output: |
---|
1735 | // the string of T[[lab,12]] (lpp); in Maple |
---|
1736 | proc btlpp(string T, list L) |
---|
1737 | { |
---|
1738 | int i; |
---|
1739 | string Li; |
---|
1740 | string coma=",";; |
---|
1741 | def lab=L[1]; |
---|
1742 | string slab; |
---|
1743 | if ((size(lab)==1) and lab[1]==-1) |
---|
1744 | {slab=""; coma="";} |
---|
1745 | else |
---|
1746 | { |
---|
1747 | slab=string(lab[1]); |
---|
1748 | if (size(lab)>=1) |
---|
1749 | { |
---|
1750 | for (i=2;i<=size(lab);i++){slab=string(slab,",",lab[i]);} |
---|
1751 | } |
---|
1752 | } |
---|
1753 | if (size(L[7])==0) |
---|
1754 | { |
---|
1755 | Li=string(T,"[[",slab,coma,"12]]:=[ ]; "); |
---|
1756 | } |
---|
1757 | else |
---|
1758 | { |
---|
1759 | Li=string(T,"[[",slab,coma,"12]]:=[",L[7],"]; "); |
---|
1760 | } |
---|
1761 | return(Li); |
---|
1762 | } |
---|
1763 | |
---|
1764 | // auxiliary routine called by buildtreetoMaple |
---|
1765 | // input: |
---|
1766 | // list L: element i of the list of buildtree(F) |
---|
1767 | // output: |
---|
1768 | // the list of strings of (T[[lab,0]]=0,T[[lab,1]]<>0); in Maple |
---|
1769 | proc btcond(string T, list L, poly cond) |
---|
1770 | { |
---|
1771 | int i; |
---|
1772 | string Li1; |
---|
1773 | string Li2; |
---|
1774 | def lab=L[1]; |
---|
1775 | string slab; |
---|
1776 | string coma=",";; |
---|
1777 | if ((size(lab)==1) and lab[1]==-1) |
---|
1778 | {slab=""; coma="";} |
---|
1779 | else |
---|
1780 | { |
---|
1781 | slab=string(lab[1]); |
---|
1782 | if (size(lab)>=1) |
---|
1783 | { |
---|
1784 | for (i=2;i<=size(lab);i++){slab=string(slab,",",lab[i]);} |
---|
1785 | } |
---|
1786 | } |
---|
1787 | Li1=string(T,"[[",slab+coma,"0]]:=",L[8],"=0; "); |
---|
1788 | Li2=string(T,"[[",slab+coma,"1]]:=",L[8],"<>0; "); |
---|
1789 | return(list(Li1,Li2)); |
---|
1790 | } |
---|
1791 | |
---|
1792 | //*****************End of BuildtreetoMaple********************* |
---|
1793 | |
---|
1794 | //*****************Begin of Selectcases************************ |
---|
1795 | |
---|
1796 | // given an intvec with sum=n |
---|
1797 | // it returns the list of intvect with the sum=n+1 |
---|
1798 | proc comp1(intvec l) |
---|
1799 | { |
---|
1800 | list L; |
---|
1801 | int p=size(l); |
---|
1802 | int i; |
---|
1803 | if (p==0){return(l);} |
---|
1804 | if (p==1){return(list(intvec(l[1]+1)));} |
---|
1805 | L[1]=intvec((l[1]+1),l[2..p]); |
---|
1806 | L[p]=intvec(l[1..p-1],(l[p]+1)); |
---|
1807 | for (i=2;i<p;i++) |
---|
1808 | { |
---|
1809 | L[i]=intvec(l[1..(i-1)],(l[i]+1),l[(i+1)..p]); |
---|
1810 | } |
---|
1811 | return(L); |
---|
1812 | } |
---|
1813 | |
---|
1814 | // comp: p-compositions of n |
---|
1815 | // input |
---|
1816 | // int n; |
---|
1817 | // int p; |
---|
1818 | // return |
---|
1819 | // the list of all intvec (p-composition of n) |
---|
1820 | proc comp(int n,int p) |
---|
1821 | { |
---|
1822 | if (n<0){ERROR("comp was called with negative argument");} |
---|
1823 | if (n==0){return(list(0:p));} |
---|
1824 | int i; |
---|
1825 | int k; |
---|
1826 | list L1=comp(n-1,p); |
---|
1827 | list L=comp1(L1[1]); |
---|
1828 | list l; |
---|
1829 | list la; |
---|
1830 | for (i=2; i<=size(L1);i++) |
---|
1831 | { |
---|
1832 | l=comp1(L1[i]); |
---|
1833 | for (k=1;k<=size(l);k++) |
---|
1834 | { |
---|
1835 | if(not(memberpos(l[k],L)[1])) |
---|
1836 | {L[size(L)+1]=l[k];} |
---|
1837 | } |
---|
1838 | } |
---|
1839 | return(L); |
---|
1840 | } |
---|
1841 | |
---|
1842 | // given the matrices of coefficients and monomials m amd m1 of |
---|
1843 | // two polynomials (the first one contains all the terms of f |
---|
1844 | // and the second only those of f |
---|
1845 | // it returns the list with the common monomials and the list of coefficients |
---|
1846 | // of the polynomial f with zeroes if necessary. |
---|
1847 | proc adaptcoef(matrix m, matrix m1) |
---|
1848 | { |
---|
1849 | int i; |
---|
1850 | int j; |
---|
1851 | int ncm=ncols(m); |
---|
1852 | int ncm1=ncols(m1); |
---|
1853 | ideal T; |
---|
1854 | for (i=1;i<=ncm;i++){T[i]=m[1,i];} |
---|
1855 | ideal C; |
---|
1856 | for (i=1;i<=ncm;i++){C[i]=0;} |
---|
1857 | for (i=1;i<=ncm;i++) |
---|
1858 | { |
---|
1859 | j=1; |
---|
1860 | while((j<ncm1) and (m1[1,j]>m[1,i])){j++;} |
---|
1861 | if (m1[1,j]==m[1,i]){C[i]=m1[2,j];} |
---|
1862 | } |
---|
1863 | return(list(T,C)); |
---|
1864 | } |
---|
1865 | |
---|
1866 | // given the ideal of non-null conditions and an intvec lambda |
---|
1867 | // with the exponents of each w in W |
---|
1868 | // it returns the polynomial prod (w_i)^(lambda_i). |
---|
1869 | proc WW(ideal W, intvec lambda) |
---|
1870 | { |
---|
1871 | if (size(W)==0){return(poly(1));} |
---|
1872 | poly w=1; |
---|
1873 | int i; |
---|
1874 | for (i=1;i<=ncols(W);i++) |
---|
1875 | { |
---|
1876 | w=w*(W[i])^(lambda[i]); |
---|
1877 | } |
---|
1878 | return(w); |
---|
1879 | } |
---|
1880 | |
---|
1881 | // given a polynomial f and the non-null conditions W |
---|
1882 | // WPred eliminates the factors in f that are in W |
---|
1883 | // ring @PAB |
---|
1884 | // input: |
---|
1885 | // poly f: |
---|
1886 | // ideal W of non-null conditions (already supposed that it is facvar) |
---|
1887 | // output: |
---|
1888 | // poly f2 where the non-null conditions in W have been dropped from f |
---|
1889 | proc WPred(poly f, ideal W) |
---|
1890 | { |
---|
1891 | if (f==0){return(f);} |
---|
1892 | def l=factorize(f,2); |
---|
1893 | int i; |
---|
1894 | poly f1=1; |
---|
1895 | for(i=1;i<=size(l[1]);i++) |
---|
1896 | { |
---|
1897 | if (memberpos(l[1][i],W)[1]){;} |
---|
1898 | else{f1=f1*((l[1][i])^(l[2][i]));} |
---|
1899 | } |
---|
1900 | return(f1); |
---|
1901 | } |
---|
1902 | |
---|
1903 | //genimage |
---|
1904 | // ring @R |
---|
1905 | //input: |
---|
1906 | // poly f1, idel N1,ideal W1,poly f2, ideal N2, ideal W2 |
---|
1907 | // corresponding to two polynomials having the same lpp |
---|
1908 | // f1 in the redspec given by N1,W1, f2 in the redspec given by N2,W2 |
---|
1909 | //output: |
---|
1910 | // the list of (ideal GG, list(list r1, list r2)) |
---|
1911 | // where g an ideal whose elements have the same lpp as f1 and f2 |
---|
1912 | // that specialize well to f1 in N1,W1 and to f2 in N2,W2. |
---|
1913 | // If it doesn't exist a genimage, then g=ideal(0). |
---|
1914 | proc genimage(poly f1, ideal N1, ideal W1, poly f2, ideal N2, ideal W2) |
---|
1915 | { |
---|
1916 | int i; ideal W12; poly ff1; poly g1=0; ideal GG; |
---|
1917 | int tt=1; |
---|
1918 | // detect whether f1 reduces to 0 on segment 2 |
---|
1919 | ff1=pnormalform(f1,N2,W2); |
---|
1920 | if (ff1==0) |
---|
1921 | { |
---|
1922 | // detect whether N1 is included in N2 |
---|
1923 | def RR=basering; |
---|
1924 | setring @P; |
---|
1925 | def NP1=imap(RR,N1); |
---|
1926 | def NP2=imap(RR,N2); |
---|
1927 | poly nr; |
---|
1928 | i=1; |
---|
1929 | while ((tt) and (i<=size(NP1))) |
---|
1930 | { |
---|
1931 | nr=reduce(NP1[i],NP2); |
---|
1932 | if (nr!=0){tt=0;} |
---|
1933 | i++; |
---|
1934 | } |
---|
1935 | setring(RR); |
---|
1936 | } |
---|
1937 | else{tt=0;} |
---|
1938 | if (tt==1) |
---|
1939 | { |
---|
1940 | // detect whether W1 intersect W2 is non-empty |
---|
1941 | for (i=1;i<=size(W1);i++) |
---|
1942 | { |
---|
1943 | if (memberpos(W1[i],W2)[1]) |
---|
1944 | { |
---|
1945 | W12[size(W12)+1]=W1[i]; |
---|
1946 | } |
---|
1947 | else |
---|
1948 | { |
---|
1949 | if (nonnull(W1[i],N2,W2)) |
---|
1950 | { |
---|
1951 | W12[size(W12)+1]=W1[i]; |
---|
1952 | } |
---|
1953 | } |
---|
1954 | } |
---|
1955 | for (i=1;i<=size(W2);i++) |
---|
1956 | { |
---|
1957 | if (not(memberpos(W2[i],W12)[1])) |
---|
1958 | { |
---|
1959 | W12[size(W12)+1]=W2[i]; |
---|
1960 | } |
---|
1961 | } |
---|
1962 | } |
---|
1963 | if (tt==1){g1=extendpoly(f1,N1,W12);} |
---|
1964 | if (g1!=0) |
---|
1965 | { |
---|
1966 | //T_ "genimage has found a more generic basis (method 1)"; |
---|
1967 | //T_ "f1:"; f1; "N1:"; N1; "W1:"; W1; |
---|
1968 | //T_ "f2:"; f2; "N2:"; N2; "W2:"; W2; |
---|
1969 | //T_ "g1:"; g1; |
---|
1970 | if (pnormalform(g1,N1,W1)==0) |
---|
1971 | { |
---|
1972 | GG=f1,g1; |
---|
1973 | //T_ "A sheaf has been found (method 2)"; |
---|
1974 | } |
---|
1975 | else |
---|
1976 | { |
---|
1977 | GG=g1; |
---|
1978 | } |
---|
1979 | return(GG); |
---|
1980 | } |
---|
1981 | |
---|
1982 | // begins the second step; |
---|
1983 | int bound=6; |
---|
1984 | // in ring @R |
---|
1985 | int j; int g=0; int alpha; int r1; int s1=1; int s2=1; |
---|
1986 | poly G; |
---|
1987 | matrix qT; |
---|
1988 | matrix T; |
---|
1989 | ideal N10; |
---|
1990 | poly GT; |
---|
1991 | ideal N12=N1,N2; |
---|
1992 | def varx=maxideal(1); |
---|
1993 | int nx=size(varx); |
---|
1994 | poly pvarx=1; |
---|
1995 | for (i=1;i<=nx;i++){pvarx=pvarx*varx[i];} |
---|
1996 | def m=coef(43*f1+157*f2,pvarx); |
---|
1997 | def m1=coef(f1,pvarx); |
---|
1998 | def m2=coef(f2,pvarx); |
---|
1999 | list L1=adaptcoef(m,m1); |
---|
2000 | list L2=adaptcoef(m,m2); |
---|
2001 | ideal Tm=L1[1]; |
---|
2002 | ideal c1=L1[2]; |
---|
2003 | ideal c2=L2[2]; |
---|
2004 | poly ww1; |
---|
2005 | poly ww2; |
---|
2006 | poly cA1; |
---|
2007 | poly cB1; |
---|
2008 | matrix TT; |
---|
2009 | poly H; |
---|
2010 | list r; |
---|
2011 | ideal q; |
---|
2012 | poly mu; |
---|
2013 | ideal N; |
---|
2014 | |
---|
2015 | // in ring @PAB |
---|
2016 | list Px=ringlist(@P); |
---|
2017 | list v="@A","@B"; |
---|
2018 | Px[2]=Px[2]+v; |
---|
2019 | def npx=size(Px[3][1][2]); |
---|
2020 | Px[3][1][2]=1:(npx+size(v)); |
---|
2021 | def @PAB=ring(Px); |
---|
2022 | setring(@PAB); |
---|
2023 | |
---|
2024 | poly PH; |
---|
2025 | ideal NP; |
---|
2026 | list rP; |
---|
2027 | def PN1=imap(@R,N1); |
---|
2028 | def PW1=imap(@R,W1); |
---|
2029 | def PN2=imap(@R,N2); |
---|
2030 | def PW2=imap(@R,W2); |
---|
2031 | def a1=imap(@R,c1); |
---|
2032 | def a2=imap(@R,c2); |
---|
2033 | matrix PT; |
---|
2034 | ideal PN; |
---|
2035 | ideal PN12=PN1,PN2; |
---|
2036 | PN=liftstd(PN12,PT); |
---|
2037 | list compos1; |
---|
2038 | list compos2; |
---|
2039 | list compos0; |
---|
2040 | intvec comp0; |
---|
2041 | poly w1=0; |
---|
2042 | poly w2=0; |
---|
2043 | poly h; |
---|
2044 | poly cA=0; |
---|
2045 | poly cB=0; |
---|
2046 | int t=0; |
---|
2047 | list l; |
---|
2048 | poly h1; |
---|
2049 | g=0; |
---|
2050 | while ((g<=bound) and not(t)) |
---|
2051 | { |
---|
2052 | compos0=comp(g,2); |
---|
2053 | r1=1; |
---|
2054 | while ((r1<=size(compos0)) and not(t)) |
---|
2055 | { |
---|
2056 | comp0=compos0[r1]; |
---|
2057 | if (comp0[1]<=bound/2) |
---|
2058 | { |
---|
2059 | compos1=comp(comp0[1],ncols(PW1)); |
---|
2060 | s1=1; |
---|
2061 | while ((s1<=size(compos1)) and not(t)) |
---|
2062 | { |
---|
2063 | if (comp0[2]<=bound/2) |
---|
2064 | { |
---|
2065 | compos2=comp(comp0[2],ncols(PW2)); |
---|
2066 | s2=1; |
---|
2067 | while ((s2<=size(compos2)) and not(t)) |
---|
2068 | { |
---|
2069 | w1=WW(PW1,compos1[s1]); |
---|
2070 | w2=WW(PW2,compos2[s2]); |
---|
2071 | h=@A*w1*a1[1]-@B*w2*a2[1]; |
---|
2072 | h=reduce(h,PN); |
---|
2073 | if (h==0){cA=1;cB=-1;} |
---|
2074 | else |
---|
2075 | { |
---|
2076 | l=factorize(h,2); |
---|
2077 | h1=1; |
---|
2078 | for(i=1;i<=size(l[1]);i++) |
---|
2079 | { |
---|
2080 | if ((memberpos(@A,variables(l[1][i]))[1]) or (memberpos(@B,variables(l[1][i]))[1])) |
---|
2081 | {h1=h1*l[1][i];} |
---|
2082 | } |
---|
2083 | cA=diff(h1,@B); |
---|
2084 | cB=diff(h1,@A); |
---|
2085 | } |
---|
2086 | if ((cA!=0) and (cB!=0) and (jet(cA,0)==cA) and (jet(cB,0)==cB)) |
---|
2087 | { |
---|
2088 | t=1; |
---|
2089 | alpha=1; |
---|
2090 | while((t) and (alpha<=ncols(a1))) |
---|
2091 | { |
---|
2092 | h=cA*w1*a1[alpha]+cB*w2*a2[alpha]; |
---|
2093 | if (not(reduce(h,PN,1)==0)){t=0;} |
---|
2094 | alpha++; |
---|
2095 | } |
---|
2096 | } |
---|
2097 | else{t=0;} |
---|
2098 | s2++; |
---|
2099 | } |
---|
2100 | } |
---|
2101 | s1++; |
---|
2102 | } |
---|
2103 | } |
---|
2104 | r1++; |
---|
2105 | } |
---|
2106 | g++; |
---|
2107 | } |
---|
2108 | setring(@R); |
---|
2109 | ww1=imap(@PAB,w1); |
---|
2110 | ww2=imap(@PAB,w2); |
---|
2111 | T=imap(@PAB,PT); |
---|
2112 | N=imap(@PAB,PN); |
---|
2113 | cA1=imap(@PAB,cA); |
---|
2114 | cB1=imap(@PAB,cB); |
---|
2115 | if (t) |
---|
2116 | { |
---|
2117 | G=0; |
---|
2118 | for (alpha=1;alpha<=ncols(Tm);alpha++) |
---|
2119 | { |
---|
2120 | H=cA1*ww1*c1[alpha]+cB1*ww2*c2[alpha]; |
---|
2121 | setring(@PAB); |
---|
2122 | PH=imap(@R,H); |
---|
2123 | PN=imap(@R,N); |
---|
2124 | rP=division(PH,PN); |
---|
2125 | setring(@R); |
---|
2126 | r=imap(@PAB,rP); |
---|
2127 | if (r[2][1]!=0){ERROR("the division is not null and it should be");} |
---|
2128 | q=r[1]; |
---|
2129 | qT=transpose(matrix(q)); |
---|
2130 | N10=N12; |
---|
2131 | for (i=size(N1)+1;i<=size(N1)+size(N2);i++){N10[i]=0;} |
---|
2132 | G=G+(cA1*ww1*c1[alpha]-(matrix(N10)*T*qT)[1,1])*Tm[alpha]; |
---|
2133 | } |
---|
2134 | //T_ "genimage has found a more generic basis (method 2)"; |
---|
2135 | //T_ "f1:"; f1; "N1:"; N1; "W1:"; W1; |
---|
2136 | //T_ "f2:"; f2; "N2:"; N2; "W2:"; W2; |
---|
2137 | //T_ "G:"; G; |
---|
2138 | GG=ideal(G); |
---|
2139 | } |
---|
2140 | else{GG=ideal(0);} |
---|
2141 | return(GG); |
---|
2142 | } |
---|
2143 | |
---|
2144 | // purpose: given a polynomial f (in the reduced basis) |
---|
2145 | // the null-conditions ideal N in the segment |
---|
2146 | // end the set of non-null polynomials common to the segment and |
---|
2147 | // a new segment, |
---|
2148 | // to obtain an equivalent polynomial with a leading coefficient |
---|
2149 | // that is non-null in the second segment. |
---|
2150 | // input: |
---|
2151 | // poly f: a polynomials of the reduced basis in the segment (N,W) |
---|
2152 | // ideal N: the null-conditions ideal in the segment |
---|
2153 | // ideal W12: the set of non-null polynomials common to the segment and |
---|
2154 | // a second segment |
---|
2155 | proc extendpoly(poly f, ideal N, ideal W12) |
---|
2156 | { |
---|
2157 | int bound=4; |
---|
2158 | ideal cfs; |
---|
2159 | ideal cfsn; |
---|
2160 | ideal ppfs; |
---|
2161 | poly p=f; |
---|
2162 | poly fn; |
---|
2163 | poly lm; poly lc; |
---|
2164 | int tt=0; |
---|
2165 | int i; |
---|
2166 | while (p!=0) |
---|
2167 | { |
---|
2168 | lm=leadmonom(p); |
---|
2169 | lc=leadcoef(p); |
---|
2170 | cfs[size(cfs)+1]=lc; |
---|
2171 | ppfs[size(ppfs)+1]=lm; |
---|
2172 | p=p-lc*lm; |
---|
2173 | } |
---|
2174 | def lcf=cfs[1]; |
---|
2175 | int r1=0; int s1; |
---|
2176 | def RR=basering; |
---|
2177 | setring @P; |
---|
2178 | list compos1; |
---|
2179 | poly w1; |
---|
2180 | ideal q; |
---|
2181 | def lcfp=imap(RR,lcf); |
---|
2182 | def W=imap(RR,W12); |
---|
2183 | def Np=imap(RR,N); |
---|
2184 | def cfsp=imap(RR,cfs); |
---|
2185 | ideal cfspn; |
---|
2186 | matrix T; |
---|
2187 | ideal H=lcfp,Np; |
---|
2188 | def G=liftstd(H,T); |
---|
2189 | list r; |
---|
2190 | while ((r1<=bound) and not(tt)) |
---|
2191 | { |
---|
2192 | compos1=comp(r1,ncols(W)); |
---|
2193 | s1=1; |
---|
2194 | while ((s1<=size(compos1)) and not(tt)) |
---|
2195 | { |
---|
2196 | w1=WW(W,compos1[s1]); |
---|
2197 | cfspn=ideal(0); |
---|
2198 | cfspn[1]=w1; |
---|
2199 | tt=1; |
---|
2200 | i=2; |
---|
2201 | while ((i<=size(cfsp)) and (tt)) |
---|
2202 | { |
---|
2203 | r=division(w1*cfsp[i],G); |
---|
2204 | if (r[2][1]!=0){tt=0;} |
---|
2205 | else |
---|
2206 | { |
---|
2207 | q=r[1]; |
---|
2208 | cfspn[i]=(T*transpose(matrix(q)))[1,1]; |
---|
2209 | } |
---|
2210 | i++; |
---|
2211 | } |
---|
2212 | s1++; |
---|
2213 | } |
---|
2214 | r1++; |
---|
2215 | } |
---|
2216 | setring RR; |
---|
2217 | if (tt) |
---|
2218 | { |
---|
2219 | cfsn=imap(@P,cfspn); |
---|
2220 | fn=0; |
---|
2221 | for (i=1;i<=size(ppfs);i++) |
---|
2222 | { |
---|
2223 | fn=fn+cfsn[i]*ppfs[i]; |
---|
2224 | } |
---|
2225 | } |
---|
2226 | else{fn=0;} |
---|
2227 | return(fn); |
---|
2228 | } |
---|
2229 | |
---|
2230 | // nonnull |
---|
2231 | // ring @P (or @R) |
---|
2232 | // input: |
---|
2233 | // poly f |
---|
2234 | // ideal N |
---|
2235 | // ideal W |
---|
2236 | // output: |
---|
2237 | // 1 if f is nonnull in the segment (N,W) |
---|
2238 | // 0 if it can be zero |
---|
2239 | proc nonnull(poly f, ideal N, ideal W) |
---|
2240 | { |
---|
2241 | int tt; |
---|
2242 | ideal N0=N; |
---|
2243 | N0[size(N0)+1]=f; |
---|
2244 | poly h=1; |
---|
2245 | int i; |
---|
2246 | for (i=1;i<=size(W);i++){h=h*W[i];} |
---|
2247 | def RR=basering; |
---|
2248 | setring(@P); |
---|
2249 | list Px=ringlist(@P); |
---|
2250 | list v="@C"; |
---|
2251 | Px[2]=Px[2]+v; |
---|
2252 | def npx=size(Px[3][1][2]); |
---|
2253 | Px[3][1][1]="dp"; |
---|
2254 | Px[3][1][2]=1:(npx+size(v)); |
---|
2255 | def @PC=ring(Px); |
---|
2256 | setring(@PC); |
---|
2257 | def N1=imap(RR,N0); |
---|
2258 | def h1=imap(RR,h); |
---|
2259 | ideal G=1-@C*h1; |
---|
2260 | G=G+N1; |
---|
2261 | option(redSB); |
---|
2262 | ideal G1=groebner(G); |
---|
2263 | if (G1[1]==1){tt=1;} else{tt=0;} |
---|
2264 | setring(RR); |
---|
2265 | return(tt); |
---|
2266 | } |
---|
2267 | |
---|
2268 | // decide |
---|
2269 | // input: |
---|
2270 | // given two corresponding polynomials g1 and g2 with the same lpp |
---|
2271 | // g1 belonging to the basis in the segment N1,W1 |
---|
2272 | // g2 belonging to the basis in the segment N2,W2 |
---|
2273 | // output: |
---|
2274 | // an ideal (with a single polynomial of more if a sheaf is needed) |
---|
2275 | // that specializes well on both segments to g1 and g2 respectivelly. |
---|
2276 | // If ideal(0) is output, then no such polynomial nor sheaf exists. |
---|
2277 | proc decide(poly g1, ideal N1, ideal W1, poly g2, ideal N2, ideal W2) |
---|
2278 | { |
---|
2279 | poly S; |
---|
2280 | poly S1; |
---|
2281 | poly S2; |
---|
2282 | S=leadcoef(g2)*g1-leadcoef(g1)*g2; |
---|
2283 | def RR=basering; |
---|
2284 | setring(@RP); |
---|
2285 | def SR=imap(RR,S); |
---|
2286 | def N1R=imap(RR,N1); |
---|
2287 | def N2R=imap(RR,N2); |
---|
2288 | attrib(N1R,"isSB",1); |
---|
2289 | attrib(N2R,"isSB",1); |
---|
2290 | poly S1R=reduce(SR,N1R); |
---|
2291 | poly S2R=reduce(SR,N2R); |
---|
2292 | setring(RR); |
---|
2293 | S1=imap(@RP,S1R); |
---|
2294 | S2=imap(@RP,S2R); |
---|
2295 | if ((S2==0) and (nonnull(leadcoef(g1),N2,W2))){return(ideal(g1));} |
---|
2296 | if ((S1==0) and (nonnull(leadcoef(g2),N1,W1))){return(ideal(g2));} |
---|
2297 | if ((S1==0) and (S2==0)) |
---|
2298 | { |
---|
2299 | //T_ "A sheaf has been found (method 1)"; |
---|
2300 | return(ideal(g1,g2)); |
---|
2301 | } |
---|
2302 | return(ideal(genimage(g1,N1,W1,g2,N2,W2))); |
---|
2303 | } |
---|
2304 | |
---|
2305 | // input: the tree (list) from buildtree output |
---|
2306 | // output: the list of terminal vertices. |
---|
2307 | proc finalcases(list T) |
---|
2308 | "USAGE: finalcases(T); |
---|
2309 | T is the list provided by buildtree |
---|
2310 | RETURN: A list with the CGS determined by buildtree. |
---|
2311 | Each element of the list represents one segment |
---|
2312 | of the buildtree CGS. |
---|
2313 | The list elements have the following structure: |
---|
2314 | [1]: label (an intvec(1,0,..)) that indicates the position |
---|
2315 | in the buildtree but that is irrelevant for the CGS |
---|
2316 | [2]: 1 (integer) it is also irrelevant and indicates |
---|
2317 | that this was a terminal vertex in buildtree. |
---|
2318 | [3]: the reduced basis of the segment. |
---|
2319 | [4], [5], [6]: the red-spec of the null and non-null conditions |
---|
2320 | of the segment. |
---|
2321 | [4] is the null-conditions radical ideal N, |
---|
2322 | [5] is the non-null polynomials set (ideal) W, |
---|
2323 | [6] is the set of prime components (ideals) of N. |
---|
2324 | [7]: is the set of lpp |
---|
2325 | [8]: poly 1 (irrelevant) is the condition to branch (but no |
---|
2326 | more branch is necessary in the discussion, so 1 is the result. |
---|
2327 | NOTE: It can be called having as argument the list output by buildtree |
---|
2328 | KEYWORDS: buildtree, buildtreetoMaple, CGS |
---|
2329 | EXAMPLE: finalcases; shows an example" |
---|
2330 | { |
---|
2331 | int i; |
---|
2332 | list L; |
---|
2333 | for (i=1;i<=size(T);i++) |
---|
2334 | { |
---|
2335 | if (T[i][2]) |
---|
2336 | {L[size(L)+1]=T[i];} |
---|
2337 | } |
---|
2338 | return(L); |
---|
2339 | } |
---|
2340 | example |
---|
2341 | { "EXAMPLE:"; echo = 2; |
---|
2342 | ring R=(0,a1,a2,a3,a4),(x1,x2,x3,x4),dp; |
---|
2343 | ideal F=x4-a4+a2, x1+x2+x3+x4-a1-a3-a4, x1*x3*x4-a1*a3*a4, x1*x3+x1*x4+x2*x3+x3*x4-a1*a4-a1*a3-a3*a4; |
---|
2344 | def T=buildtree(F); |
---|
2345 | finalcases(T); |
---|
2346 | } |
---|
2347 | |
---|
2348 | // input: the list of terminal vertices of buildtree (output of finalcases) |
---|
2349 | // output: the same terminal vertices grouped by lpp |
---|
2350 | proc groupsegments(list T) |
---|
2351 | { |
---|
2352 | int i; |
---|
2353 | list L; |
---|
2354 | list lpp; |
---|
2355 | list lp; |
---|
2356 | list ls; |
---|
2357 | int n=size(T); |
---|
2358 | lpp[1]=T[n][7]; |
---|
2359 | L[1]=list(lpp[1],list(list(T[n][1],T[n][3],T[n][4],T[n][5],T[n][6]))); |
---|
2360 | if (n>1) |
---|
2361 | { |
---|
2362 | for (i=1;i<=size(T)-1;i++) |
---|
2363 | { |
---|
2364 | lp=memberpos(T[n-i][7],lpp); |
---|
2365 | if(lp[1]==1) |
---|
2366 | { |
---|
2367 | ls=L[lp[2]][2]; |
---|
2368 | ls[size(ls)+1]=list(T[n-i][1],T[n-i][3],T[n-i][4],T[n-i][5],T[n-i][6]); |
---|
2369 | L[lp[2]][2]=ls; |
---|
2370 | } |
---|
2371 | else |
---|
2372 | { |
---|
2373 | lpp[size(lpp)+1]=T[n-i][7]; |
---|
2374 | L[size(L)+1]=list(T[n-i][7],list(list(T[n-i][1],T[n-i][3],T[n-i][4],T[n-i][5],T[n-i][6]))); |
---|
2375 | } |
---|
2376 | } |
---|
2377 | } |
---|
2378 | return(L); |
---|
2379 | } |
---|
2380 | |
---|
2381 | // eliminates repeated elements form an ideal |
---|
2382 | proc elimrepeated(ideal F) |
---|
2383 | { |
---|
2384 | int i; |
---|
2385 | int j; |
---|
2386 | ideal FF; |
---|
2387 | FF[1]=F[1]; |
---|
2388 | for (i=2;i<=ncols(F);i++) |
---|
2389 | { |
---|
2390 | if (not(memberpos(F[i],FF)[1])) |
---|
2391 | { |
---|
2392 | FF[size(FF)+1]=F[i]; |
---|
2393 | } |
---|
2394 | } |
---|
2395 | return(FF); |
---|
2396 | } |
---|
2397 | |
---|
2398 | |
---|
2399 | // decide F is the same as decide but allows as first element a sheaf F |
---|
2400 | proc decideF(ideal F,ideal N,ideal W, poly f2, ideal N2, ideal W2) |
---|
2401 | { |
---|
2402 | int i; |
---|
2403 | ideal G=F; |
---|
2404 | ideal g; |
---|
2405 | if (ncols(F)==1) {return(decide(F[1],N,W,f2,N2,W2));} |
---|
2406 | for (i=1;i<=ncols(F);i++) |
---|
2407 | { |
---|
2408 | G=G+decide(F[i],N,W,f2,N2,W2); |
---|
2409 | } |
---|
2410 | return(elimrepeated(G)); |
---|
2411 | } |
---|
2412 | |
---|
2413 | // newredspec |
---|
2414 | // input: two redspec in the form of N,W and Nj,Wj |
---|
2415 | // output: a redspec representing the minimal redspec segment that contains |
---|
2416 | // both input segments. |
---|
2417 | proc newredspec(ideal N,ideal W, ideal Nj, ideal Wj) |
---|
2418 | { |
---|
2419 | ideal nN; |
---|
2420 | ideal nW; |
---|
2421 | int u; |
---|
2422 | def RR=basering; |
---|
2423 | setring(@P); |
---|
2424 | list r; |
---|
2425 | def Np=imap(RR,N); |
---|
2426 | def Wp=imap(RR,W); |
---|
2427 | def Njp=imap(RR,Nj); |
---|
2428 | def Wjp=imap(RR,Wj); |
---|
2429 | Np=intersect(Np,Njp); |
---|
2430 | ideal WR; |
---|
2431 | for(u=1;u<=size(Wjp);u++) |
---|
2432 | { |
---|
2433 | if(nonnull(Wjp[u],Np,Wp)){WR[size(WR)+1]=Wjp[u];} |
---|
2434 | } |
---|
2435 | for(u=1;u<=size(Wp);u++) |
---|
2436 | { |
---|
2437 | if((not(memberpos(Wp[u],WR)[1])) and (nonnull(Wp[u],Njp,Wjp))) |
---|
2438 | { |
---|
2439 | WR[size(WR)+1]=Wp[u]; |
---|
2440 | } |
---|
2441 | } |
---|
2442 | r=redspec(Np,WR); |
---|
2443 | option(redSB); |
---|
2444 | Np=groebner(r[1]); |
---|
2445 | Wp=r[2]; |
---|
2446 | setring(RR); |
---|
2447 | nN=imap(@P,Np); |
---|
2448 | nW=imap(@P,Wp); |
---|
2449 | return(list(nN,nW)); |
---|
2450 | } |
---|
2451 | |
---|
2452 | // selectcases |
---|
2453 | // input: |
---|
2454 | // list bT: the list output by buildtree. |
---|
2455 | // output: |
---|
2456 | // list L it contins the list of segments allowing a common |
---|
2457 | // reduced basis. The elements of L are of the form |
---|
2458 | // list (lpp,B,list(list(N,W,L),..list(N,W,L)) ) |
---|
2459 | proc selectcases(list bT) |
---|
2460 | { |
---|
2461 | list T=groupsegments(finalcases(bT)); |
---|
2462 | list T0=bT[1]; |
---|
2463 | // first element of the list of buildtree |
---|
2464 | list TT0; |
---|
2465 | TT0[1]=list(T0[7],T0[3],list(list(T0[4],T0[5],T0[6]))); |
---|
2466 | // first element of the output of selectcases |
---|
2467 | list T1=T; // the initial list; it is only actualized (split) |
---|
2468 | // when a segment is completely revised (all split are |
---|
2469 | // already be considered); |
---|
2470 | // ( (lpp, ((lab,B,N,W,L),.. ()) ), .. (..) ) |
---|
2471 | list TT; // the output list ( (lpp,B,((N,W,L),..()) ),.. (..) ) |
---|
2472 | // case i |
---|
2473 | list S1; // the segments in case i T1[i][2]; ( (lab,B,N,W,L),..() ) |
---|
2474 | list S2; // the segments in case i that are being summarized in |
---|
2475 | // actual segment ( (N,W,L),..() ) |
---|
2476 | list S3; // the segments in case i that cannot be summarized in |
---|
2477 | // the actual case. When the case is finished a new case |
---|
2478 | // is created with them ( (lab,B,N,W,L),..() ) |
---|
2479 | list s3; // list of integers s whose segment cannot be summarized |
---|
2480 | // in the actual case |
---|
2481 | ideal lpp; // the summarized lpp (can contain repetitions) |
---|
2482 | ideal lppi;// in proecess of sumarizing lpp (can contain repetitions) |
---|
2483 | ideal B; // the summarized B (can contain polynomials with |
---|
2484 | // the same lpp (sheaves)) |
---|
2485 | ideal Bi; // in process of summarizing B (can contain polynomials with |
---|
2486 | // the same lpp (sheaves)) |
---|
2487 | ideal N; // the summarized N |
---|
2488 | ideal W; // the summarized W |
---|
2489 | ideal F; // the summarized polynomial j (can contain a sheaf instead of |
---|
2490 | // a single poly) |
---|
2491 | ideal FF; // the same as F but it can be ideal(0) |
---|
2492 | poly lpj; |
---|
2493 | poly fj; |
---|
2494 | ideal Nj; |
---|
2495 | ideal Wj; |
---|
2496 | ideal G; |
---|
2497 | int i; // the index of the case i in T1; |
---|
2498 | int j; // the index of the polynomial j of the basis |
---|
2499 | int s; // the index of the segment s in S1; |
---|
2500 | int u; |
---|
2501 | int tests; // true if al the polynomial in segment s have been generalized; |
---|
2502 | list r; |
---|
2503 | // initializing the new list |
---|
2504 | i=1; |
---|
2505 | while(i<=size(T1)) |
---|
2506 | { |
---|
2507 | S1=T1[i][2]; // ((lab,B,N,W,L)..) of the segments in case i |
---|
2508 | if (size(S1)==1) |
---|
2509 | { |
---|
2510 | TT[i]=list(T1[i][1],S1[1][2],list(list(S1[1][3],S1[1][4],S1[1][5]))); |
---|
2511 | } |
---|
2512 | else |
---|
2513 | { |
---|
2514 | S2=list(); |
---|
2515 | S3=list(); // ((lab,B,N,W,L)..) of the segments in case i to |
---|
2516 | // create another segment i+1 |
---|
2517 | s3=list(); |
---|
2518 | B=S1[1][2]; |
---|
2519 | Bi=ideal(0); |
---|
2520 | lpp=T1[i][1]; |
---|
2521 | j=1; |
---|
2522 | tests=1; |
---|
2523 | while (j<=size(S1[1][2])) |
---|
2524 | { // j desings the new j-th polynomial |
---|
2525 | N=S1[1][3]; |
---|
2526 | W=S1[1][4]; |
---|
2527 | F=ideal(S1[1][2][j]); |
---|
2528 | s=2; |
---|
2529 | while (s<=size(S1) and not(memberpos(s,s3)[1])) |
---|
2530 | { // s desings the new segment s |
---|
2531 | fj=S1[s][2][j]; |
---|
2532 | Nj=S1[s][3]; |
---|
2533 | Wj=S1[s][4]; |
---|
2534 | FF=decideF(F,N,W,fj,Nj,Wj); |
---|
2535 | if (FF[1]==0) |
---|
2536 | { |
---|
2537 | if (@ish) |
---|
2538 | { |
---|
2539 | "Warning: Dealing with an homogeneous ideal"; |
---|
2540 | "mrcgs was not able to summarize all lpp cases into a single segment"; |
---|
2541 | "Please send a mail with your Problem to antonio.montes@upc.edu"; |
---|
2542 | "You found a counterexample of the complete success of the actual mrcgs algorithm"; |
---|
2543 | //"T_"; "f1:"; F; "N1:"; N; "W1:"; W; "f2:"; fj; "N2:"; Nj; "W2:"; Wj; |
---|
2544 | } |
---|
2545 | S3[size(S3)+1]=S1[s]; |
---|
2546 | s3[size(s3)+1]=s; |
---|
2547 | tests=0; |
---|
2548 | } |
---|
2549 | else |
---|
2550 | { |
---|
2551 | F=FF; |
---|
2552 | lpj=leadmonom(fj); |
---|
2553 | r=newredspec(N,W,Nj,Wj); |
---|
2554 | N=r[1]; |
---|
2555 | W=r[2]; |
---|
2556 | } |
---|
2557 | s++; |
---|
2558 | } |
---|
2559 | if (Bi[1]==0){Bi=FF;} |
---|
2560 | else |
---|
2561 | { |
---|
2562 | Bi=Bi+FF; |
---|
2563 | } |
---|
2564 | j++; |
---|
2565 | } |
---|
2566 | if (tests) |
---|
2567 | { |
---|
2568 | B=Bi; |
---|
2569 | lpp=ideal(0); |
---|
2570 | for (u=1;u<=size(B);u++){lpp[u]=leadmonom(B[u]);} |
---|
2571 | } |
---|
2572 | for (s=1;s<=size(T1[i][2]);s++) |
---|
2573 | { |
---|
2574 | if (not(memberpos(s,s3)[1])) |
---|
2575 | { |
---|
2576 | S2[size(S2)+1]=list(S1[s][3],S1[s][4],S1[s][5]); |
---|
2577 | } |
---|
2578 | } |
---|
2579 | TT[i]=list(lpp,B,S2); |
---|
2580 | // for (s=1;s<=size(s3);s++){S1=delete(S1,s);} |
---|
2581 | T1[i][2]=S2; |
---|
2582 | if (size(S3)>0){T1=insert(T1,list(T1[i][1],S3),i);} |
---|
2583 | } |
---|
2584 | i++; |
---|
2585 | } |
---|
2586 | for (i=1;i<=size(TT);i++){TT0[i+1]=TT[i];} |
---|
2587 | return(TT0); |
---|
2588 | } |
---|
2589 | |
---|
2590 | //*****************End of Selectcases************************** |
---|
2591 | |
---|
2592 | //*****************Begin of CanTree**************************** |
---|
2593 | |
---|
2594 | // equalideals |
---|
2595 | // input: 2 ideals F and G; |
---|
2596 | // output: 1 if they are identical (the same polynomials in the same order) |
---|
2597 | // 0 else |
---|
2598 | proc equalideals(ideal F, ideal G) |
---|
2599 | { |
---|
2600 | int i=1; int t=1; |
---|
2601 | if (size(F)!=size(G)){return(0);} |
---|
2602 | while ((i<=size(F)) and (t)) |
---|
2603 | { |
---|
2604 | if (F[i]!=G[i]){t=0;} |
---|
2605 | i++; |
---|
2606 | } |
---|
2607 | return(t); |
---|
2608 | } |
---|
2609 | |
---|
2610 | // delintvec |
---|
2611 | // input: intvec V |
---|
2612 | // int i |
---|
2613 | // output: |
---|
2614 | // intvec W (equal to V but the coordinate i is deleted |
---|
2615 | proc delintvec(intvec V, int i) |
---|
2616 | { |
---|
2617 | int j; |
---|
2618 | intvec W; |
---|
2619 | for (j=1;j<i;j++){W[j]=V[j];} |
---|
2620 | for (j=i+1;j<=size(V);j++){W[j-1]=V[j];} |
---|
2621 | return(W); |
---|
2622 | } |
---|
2623 | |
---|
2624 | // redtocanspec |
---|
2625 | // Computes the canonical specification of a redspec (N,W,L). |
---|
2626 | // input: |
---|
2627 | // ideal N (null conditions, must be radical) |
---|
2628 | // ideal W (non-null conditions ideal) |
---|
2629 | // list L must contain the radical decomposition of N. |
---|
2630 | // output: |
---|
2631 | // the list of elements of the (ideal N1,list(ideal M11,..,ideal M1k)) |
---|
2632 | // determining the canonical specification of the difference of |
---|
2633 | // V(N) \ V(h), where h=prod(w in W). |
---|
2634 | proc redtocanspec(intvec lab, int child, list rs) |
---|
2635 | { |
---|
2636 | ideal N=rs[1]; ideal W=rs[2]; list L=rs[3]; |
---|
2637 | intvec labi; intvec labij; |
---|
2638 | int childi; |
---|
2639 | int i; int j; list L0; |
---|
2640 | L0[1]=list(lab,size(L)); |
---|
2641 | if (W[1]==0) |
---|
2642 | { |
---|
2643 | for (i=1;i<=size(L);i++) |
---|
2644 | { |
---|
2645 | labi=lab,child+i; |
---|
2646 | L0[size(L0)+1]=list(labi,1,L[i]); |
---|
2647 | labij=labi,1; |
---|
2648 | L0[size(L0)+1]=list(labij,0,ideal(1)); |
---|
2649 | } |
---|
2650 | return(L0); |
---|
2651 | } |
---|
2652 | if (N[1]==1) |
---|
2653 | { |
---|
2654 | L0[1]=list(lab,1); |
---|
2655 | labi=lab,child+1; |
---|
2656 | L0[size(L0)+1]=list(labi,1,ideal(1)); |
---|
2657 | labij=labi,1; |
---|
2658 | L0[size(L0)+1]=list(labij,0,ideal(1)); |
---|
2659 | } |
---|
2660 | def RR=basering; |
---|
2661 | setring(@P); |
---|
2662 | ideal Np=imap(RR,N); |
---|
2663 | ideal Wp=imap(RR,W); |
---|
2664 | poly h=1; |
---|
2665 | for (i=1;i<=size(Wp);i++){h=h*Wp[i];} |
---|
2666 | list Lp=imap(RR,L); |
---|
2667 | list r; list Ti; list LL; |
---|
2668 | LL[1]=list(lab,size(Lp)); |
---|
2669 | for (i=1;i<=size(Lp);i++) |
---|
2670 | { |
---|
2671 | Ti=minAssGTZ(Lp[i]+h); |
---|
2672 | for(j=1;j<=size(Ti);j++) |
---|
2673 | { |
---|
2674 | option(redSB); |
---|
2675 | Ti[j]=groebner(Ti[j]); |
---|
2676 | } |
---|
2677 | labi=lab,child+i; |
---|
2678 | childi=size(Ti); |
---|
2679 | LL[size(LL)+1]=list(labi,childi,Lp[i]); |
---|
2680 | for (j=1;j<=childi;j++) |
---|
2681 | { |
---|
2682 | labij=labi,j; |
---|
2683 | LL[size(LL)+1]=list(labij,0,Ti[j]); |
---|
2684 | } |
---|
2685 | } |
---|
2686 | LL[1]=list(lab,size(Lp)); |
---|
2687 | setring(RR); |
---|
2688 | return(imap(@P,LL)); |
---|
2689 | } |
---|
2690 | |
---|
2691 | // difftocanspec |
---|
2692 | // Computes the canonical specification of a diffspec V(N) \ V(M) |
---|
2693 | // input: |
---|
2694 | // intvec lab: label where to hang the canspec |
---|
2695 | // list N ideal of null conditions. |
---|
2696 | // ideal M ideal of the variety to be substacted |
---|
2697 | // output: |
---|
2698 | // the list of elements determining the canonical specification of |
---|
2699 | // the difference V(N) \ V(M): |
---|
2700 | // ( (intvec(i),children), ...(lab, children, prime ideal),...) |
---|
2701 | proc difftocanspec(intvec lab, int child, ideal N, ideal M) |
---|
2702 | { |
---|
2703 | int i; int j; list LLL; |
---|
2704 | def RR=basering; |
---|
2705 | setring(@P); |
---|
2706 | ideal Np=imap(RR,N); |
---|
2707 | ideal Mp=imap(RR,M); |
---|
2708 | def L=minAssGTZ(Np); |
---|
2709 | for(j=1;j<=size(L);j++) |
---|
2710 | { |
---|
2711 | option(redSB); |
---|
2712 | L[j]=groebner(L[j]); |
---|
2713 | } |
---|
2714 | intvec labi; intvec labij; |
---|
2715 | int childi; |
---|
2716 | list LL; |
---|
2717 | if ((Mp[1]==0) or ((size(L)==1) and (L[1][1]==1))) |
---|
2718 | { |
---|
2719 | //LL[1]=list(lab,1); |
---|
2720 | //labi=lab,1; |
---|
2721 | //LL[2]=list(labi,1,ideal(1)); |
---|
2722 | //labij=labi,1; |
---|
2723 | //LL[3]=list(labij,0,ideal(1)); |
---|
2724 | setring(RR); |
---|
2725 | return(LLL); |
---|
2726 | } |
---|
2727 | list r; list Ti; |
---|
2728 | def k=0; |
---|
2729 | LL[1]=list(lab,0); |
---|
2730 | for (i=1;i<=size(L);i++) |
---|
2731 | { |
---|
2732 | Ti=minAssGTZ(L[i]+Mp); |
---|
2733 | for(j=1;j<=size(Ti);j++) |
---|
2734 | { |
---|
2735 | option(redSB); |
---|
2736 | Ti[j]=groebner(Ti[j]); |
---|
2737 | } |
---|
2738 | if (not((size(Ti)==1) and (equalideals(L[i],Ti[1])))) |
---|
2739 | { |
---|
2740 | k++; |
---|
2741 | labi=lab,child+k; |
---|
2742 | childi=size(Ti); |
---|
2743 | LL[size(LL)+1]=list(labi,childi,L[i]); |
---|
2744 | for (j=1;j<=childi;j++) |
---|
2745 | { |
---|
2746 | labij=labi,j; |
---|
2747 | LL[size(LL)+1]=list(labij,0,Ti[j]); |
---|
2748 | } |
---|
2749 | } |
---|
2750 | else{setring(RR); return(LLL);} |
---|
2751 | } |
---|
2752 | if (size(LL)>0) |
---|
2753 | { |
---|
2754 | LL[1]=list(lab,k); |
---|
2755 | setring(RR); |
---|
2756 | return(imap(@P,LL)); |
---|
2757 | } |
---|
2758 | else {setring(RR); return(LLL);} |
---|
2759 | } |
---|
2760 | |
---|
2761 | // tree |
---|
2762 | // purpose: given a label and the list L of vertices of the tree, |
---|
2763 | // whose content |
---|
2764 | // are of the form list(intvec lab, int children, ideal P) |
---|
2765 | // to obtain the vertex and its position |
---|
2766 | // input: |
---|
2767 | // intvec lab: label of the vertex |
---|
2768 | // list: L the list containing the vertices |
---|
2769 | // output: |
---|
2770 | // list V the vertex list(lab, children, P) |
---|
2771 | proc tree(intvec lab,list L) |
---|
2772 | { |
---|
2773 | int i=0; int tt=1; list V; intvec labi; |
---|
2774 | while ((i<size(L)) and (tt)) |
---|
2775 | { |
---|
2776 | i++; |
---|
2777 | labi=L[i][1]; |
---|
2778 | if (labi==lab) |
---|
2779 | { |
---|
2780 | V=list(L[i],i); |
---|
2781 | tt=0; |
---|
2782 | } |
---|
2783 | } |
---|
2784 | if (tt==0){return(V);} |
---|
2785 | else{return(list(list(intvec(0)),0));} |
---|
2786 | } |
---|
2787 | |
---|
2788 | // GCS (generalized canonical specification) |
---|
2789 | // new structure of a GCS |
---|
2790 | |
---|
2791 | // L is a list of vertices V of the GCS. |
---|
2792 | // first vertex=list(intvec lab, int children, ideal lpp, ideal B) |
---|
2793 | // other vertices=list(intvec lab, int children, ideal P) |
---|
2794 | // the individual vertices can be accessed with the function tree |
---|
2795 | // by the call V=tree(lab,L), that outputs the vertex if it exists |
---|
2796 | // and its position in L, or nothing if it does not exist. |
---|
2797 | // The first element of the list must be the root of the tree and has |
---|
2798 | // label lab=i, and other information. |
---|
2799 | |
---|
2800 | // example: |
---|
2801 | // the canonical specification |
---|
2802 | // V(a^2-ac-ba+c-abc) \ (union( V(b,a), V(c,a), V(b,a-c), V(c,a-b))) |
---|
2803 | // is represented by the list |
---|
2804 | // L=((intvec(i),children=1,lpp,B),(intvec(i,1),4,ideal(a^2-ac-ba+c-abc)), |
---|
2805 | // (intvec(i,1,1),0,ideal(b,a)), (intvec(i,1,2),0,ideal(c,a)), |
---|
2806 | // (intvec(i,1,3),0,ideal(b,a-c)), (intvec(i,1,4),0,ideal(c,a-b)) |
---|
2807 | // ) |
---|
2808 | // example: |
---|
2809 | // the canonical specification |
---|
2810 | // (V(a)\(union(V(c,a),V(b+c,a),V(b,a)))) union |
---|
2811 | // (V(b)\(union(V(b,a),V(b,a-c)))) union |
---|
2812 | // (V(c)\(union(V(c,a),V(c,a-b)))) |
---|
2813 | // is represented by the list |
---|
2814 | // L=((i,children=3,lpp,B), |
---|
2815 | // (intvec(i,1),3,ideal(a)), |
---|
2816 | // (intvec(i,1,1),0,(c,a)),(intvec(i,1,2),0,(b+c,a)),(intvec(i,1,3),0,(b,a)), |
---|
2817 | // (intvec(i,2),2,ideal(b)), |
---|
2818 | // (intvec(i,2,1),0,(b,a)),(intvec(i,2,2),0,(b,a-c)), |
---|
2819 | // (intvec(i,3),2,ideal(c)), |
---|
2820 | // (intvec(i,3,1),0,(c,a)),(intvec(i,3,2),0,(c,a-b)) |
---|
2821 | // ) |
---|
2822 | // If L is the list in the last example, the call |
---|
2823 | // tree(intvec(i,2,1),L) will output ((intvec(i,2,1),0,(b,a)),7) |
---|
2824 | |
---|
2825 | // GCS |
---|
2826 | // input: list T is supposed to be an element L[i] of selectcases: |
---|
2827 | // T= list( ideal lpp, ideal B, list(N,W,L),.., list(N,W,L)) |
---|
2828 | // output: the list L of vertices being the GCS of the addition of |
---|
2829 | // all the segments in T. |
---|
2830 | // list(list(intvec lab, int children, ideal lpp, ideal B), |
---|
2831 | // list(intvec lab, int children, ideal P),.. |
---|
2832 | // ) |
---|
2833 | proc GCS(intvec lab, list case) |
---|
2834 | { |
---|
2835 | int i; int ii; int t; |
---|
2836 | list @L; |
---|
2837 | @L[1]=list(lab,0,case[1],case[2]); |
---|
2838 | exportto(Top,@L); |
---|
2839 | int j; |
---|
2840 | list u; intvec labu; int childu; |
---|
2841 | list v; intvec labv; int childv; |
---|
2842 | list T=case[3]; |
---|
2843 | for (j=1;j<=size(T);j++) |
---|
2844 | { |
---|
2845 | t=addcase(lab,T[j]); |
---|
2846 | deletebrotherscontaining(lab); |
---|
2847 | } |
---|
2848 | relabelingindices(lab,lab); |
---|
2849 | list L=@L; |
---|
2850 | kill @L; |
---|
2851 | return(L); |
---|
2852 | } |
---|
2853 | |
---|
2854 | // sorbylab: |
---|
2855 | // pupose: given the list of mrcgs to order is by increasing label |
---|
2856 | proc sortbylab(list L) |
---|
2857 | { |
---|
2858 | int n=L[1][2]; |
---|
2859 | int i; int j; |
---|
2860 | list H=L; |
---|
2861 | list LL; |
---|
2862 | list L1; |
---|
2863 | //LL[1]=L[1]; |
---|
2864 | //H=delete(H,1); |
---|
2865 | while (size(H)!=0) |
---|
2866 | { |
---|
2867 | j=1; |
---|
2868 | L1=H[1]; |
---|
2869 | for (i=1;i<=size(H);i++) |
---|
2870 | { |
---|
2871 | if(lesslab(H[i],L1)){j=i;L1=H[j];} |
---|
2872 | } |
---|
2873 | LL[size(LL)+1]=L1; |
---|
2874 | H=delete(H,j); |
---|
2875 | } |
---|
2876 | return(LL); |
---|
2877 | } |
---|
2878 | |
---|
2879 | // lesslab |
---|
2880 | // purpose: given two elements of the list of mrcgs it |
---|
2881 | // returns 1 if the label of the first is less than that of the second |
---|
2882 | proc lesslab(list l1, list l2) |
---|
2883 | { |
---|
2884 | intvec lab1=l1[1]; |
---|
2885 | intvec lab2=l2[1]; |
---|
2886 | int n1=size(lab1); |
---|
2887 | int n2=size(lab2); |
---|
2888 | int n=n1; |
---|
2889 | if (n2<n1){n=n2;} |
---|
2890 | int tt=0; |
---|
2891 | int j=1; |
---|
2892 | while ((lab1[j]==lab2[j]) and (j<n)){j++;} |
---|
2893 | if (lab1[j]<lab2[j]){tt=1;} |
---|
2894 | if ((j==n) and (lab1[j]==lab2[j]) and (n2>n1)){tt=1;} |
---|
2895 | return(tt); |
---|
2896 | } |
---|
2897 | |
---|
2898 | // cantree |
---|
2899 | // input: the list provided by selectcases |
---|
2900 | // output: the list providing the canonicaltree |
---|
2901 | proc cantree(list S) |
---|
2902 | { |
---|
2903 | string method=" "; |
---|
2904 | list T0=S[1]; |
---|
2905 | // first element of the list of selectcases |
---|
2906 | int i; int j; |
---|
2907 | list L; |
---|
2908 | list T; |
---|
2909 | L[1]=list(intvec(0),size(S)-1,T0[1],T0[2],T0[3][1],method); |
---|
2910 | for (i=2;i<=size(S);i++) |
---|
2911 | { |
---|
2912 | T=GCS(intvec(i-1),S[i]); |
---|
2913 | T=sortbylab(T); |
---|
2914 | for (j=1;j<=size(T);j++) |
---|
2915 | {L[size(L)+1]=T[j];} |
---|
2916 | } |
---|
2917 | return(L); |
---|
2918 | } |
---|
2919 | |
---|
2920 | // addcase |
---|
2921 | // recursive routine that adds to the list @L, (an already GCS) |
---|
2922 | // a new redspec rs=(N,W,L); |
---|
2923 | // and returns the test t whose value is |
---|
2924 | // 0 if the new canspec is not to be hung to the fathers vertex, |
---|
2925 | // 1 if yes. |
---|
2926 | proc addcase(intvec labu, list rs) |
---|
2927 | { |
---|
2928 | int i; int j; int childu; ideal Pu; |
---|
2929 | list T; int nchildu; |
---|
2930 | def N=rs[1]; def W=rs[2]; def PN=rs[3]; |
---|
2931 | ideal NN; ideal MM; |
---|
2932 | int tt=1; |
---|
2933 | poly h=1; for (i=1;i<=size(W);i++){h=h*W[i];} |
---|
2934 | list u=tree(labu,@L); childu=u[1][2]; |
---|
2935 | list v; intvec labv; int childv; list w; intvec labw; |
---|
2936 | if (childu>0) |
---|
2937 | { |
---|
2938 | v=firstchild(u[1][1]); |
---|
2939 | while(v[2][1]!=0) |
---|
2940 | { |
---|
2941 | labv=v[1][1]; |
---|
2942 | w=firstchild(labv); |
---|
2943 | while(w[2][1]!=0) |
---|
2944 | { |
---|
2945 | labw=w[1][1]; |
---|
2946 | if(addcase(labw,rs)==0) |
---|
2947 | {tt=0;} |
---|
2948 | w=nextbrother(labw); |
---|
2949 | } |
---|
2950 | u=tree(labu,@L); |
---|
2951 | childu=u[1][2]; |
---|
2952 | v=nextbrother(v[1][1]); |
---|
2953 | } |
---|
2954 | deletebrotherscontaining(labu); |
---|
2955 | relabelingindices(labu,labu); |
---|
2956 | } |
---|
2957 | if (tt==1) |
---|
2958 | { |
---|
2959 | u=tree(labu,@L); |
---|
2960 | nchildu=lastchildrenindex(labu); |
---|
2961 | if (size(labu)==1) |
---|
2962 | { |
---|
2963 | T=redtocanspec(labu,nchildu,rs); |
---|
2964 | tt=0; |
---|
2965 | } |
---|
2966 | else |
---|
2967 | { |
---|
2968 | NN=N; |
---|
2969 | if (containedP(u[1][3],N)){tt=0;} |
---|
2970 | for (i=1;i<=size(u[1][3]);i++) |
---|
2971 | { |
---|
2972 | NN[size(NN)+1]=u[1][3][i]; |
---|
2973 | } |
---|
2974 | MM=NN; |
---|
2975 | MM[size(MM)+1]=h; |
---|
2976 | T=difftocanspec(labu,nchildu,NN,MM); |
---|
2977 | } |
---|
2978 | if (size(T)>0) |
---|
2979 | { |
---|
2980 | @L[u[2]][2]=@L[u[2]][2]+T[1][2]; |
---|
2981 | for (i=2;i<=size(T);i++){@L[size(@L)+1]=T[i];} |
---|
2982 | if (size(labu)>1) |
---|
2983 | { |
---|
2984 | simplifynewadded(labu); |
---|
2985 | } |
---|
2986 | } |
---|
2987 | else{tt=1;} |
---|
2988 | } |
---|
2989 | return(tt); |
---|
2990 | } |
---|
2991 | |
---|
2992 | // reduceR |
---|
2993 | // reduces the polynomial f w.r.t. N, in the ring @P |
---|
2994 | proc reduceR(poly f, ideal N) |
---|
2995 | { |
---|
2996 | def RR=basering; |
---|
2997 | setring(@P); |
---|
2998 | def fP=imap(RR,f); |
---|
2999 | def NP=imap(RR,N); |
---|
3000 | attrib(NP,"isSB",1); |
---|
3001 | def rp=reduce(fP,NP); |
---|
3002 | setring(RR); |
---|
3003 | return(imap(@P,rp)); |
---|
3004 | } |
---|
3005 | |
---|
3006 | // containedP |
---|
3007 | // returns 1 if ideal Pu is contained in ideal Pv |
---|
3008 | // returns 0 if not |
---|
3009 | // in ring @P |
---|
3010 | proc containedP(ideal Pu,ideal Pv) |
---|
3011 | { |
---|
3012 | int t=1; |
---|
3013 | int n=size(Pu); |
---|
3014 | int i=0; |
---|
3015 | poly r=0; |
---|
3016 | while ((t) and (i<n)) |
---|
3017 | { |
---|
3018 | i++; |
---|
3019 | r=reduceR(Pu[i],Pv); |
---|
3020 | if (r!=0){t=0;} |
---|
3021 | } |
---|
3022 | return(t); |
---|
3023 | } |
---|
3024 | |
---|
3025 | // simplifynewadded |
---|
3026 | // auxiliary routine of addcase |
---|
3027 | // when a new redspec is added to a non terminal vertex, |
---|
3028 | // it is applied to simplify the addition. |
---|
3029 | // When Pu==Pv, the children of w are hung from u fathers |
---|
3030 | // and deleted the whole new addition. |
---|
3031 | // Finally, deletebrotherscontaining is applied to u fathers |
---|
3032 | // in order to eliminate branches contained. |
---|
3033 | proc simplifynewadded(intvec labu) |
---|
3034 | { |
---|
3035 | int t; int ii; int k; int kk; int j; |
---|
3036 | intvec labfu=delintvec(labu,size(labu)); list fu; int childfu; |
---|
3037 | list u=tree(labu,@L); int childu=u[1][2]; ideal Pu=u[1][3]; |
---|
3038 | list v; intvec labv; int childv; ideal Pv; |
---|
3039 | list w; intvec labw; intvec nlab; list ww; |
---|
3040 | if (childu>0) |
---|
3041 | { |
---|
3042 | v=firstchild(u[1][1]); labv=v[1][1]; childv=v[1][2]; Pv=v[1][3]; |
---|
3043 | ii=0; |
---|
3044 | t=0; |
---|
3045 | while ((not(t)) and (ii<childu)) |
---|
3046 | { |
---|
3047 | ii++; |
---|
3048 | if (equalideals(Pu,Pv)) |
---|
3049 | { |
---|
3050 | fu=tree(labfu,@L); |
---|
3051 | childfu=fu[1][2]; |
---|
3052 | j=lastchildrenindex(fu[1][1])+1; |
---|
3053 | k=0; |
---|
3054 | w=firstchild(v[1][1]); |
---|
3055 | childv=v[1][2]; |
---|
3056 | for (kk=1;kk<=childv;kk++) |
---|
3057 | { |
---|
3058 | if (kk<childv){ww=nextbrother(w[1][1]);} |
---|
3059 | nlab=labfu,j; |
---|
3060 | @L[w[2]][1]=nlab; |
---|
3061 | j++; |
---|
3062 | if (kk<childv){w=ww;} |
---|
3063 | } |
---|
3064 | childfu=fu[1][2]+childv-1; |
---|
3065 | @L[fu[2]][2]=childfu; |
---|
3066 | @L[v[2]][2]=0; |
---|
3067 | t=1; |
---|
3068 | deleteverts(labu); |
---|
3069 | } |
---|
3070 | } |
---|
3071 | } |
---|
3072 | deletebrotherscontaining(labfu); |
---|
3073 | } |
---|
3074 | |
---|
3075 | // given the label labfu of the vertex fu it returns the last |
---|
3076 | // int of the label of the last existing children. |
---|
3077 | // if no child exists, then it outputs 0. |
---|
3078 | proc lastchildrenindex(intvec labfu) |
---|
3079 | { |
---|
3080 | int i; |
---|
3081 | int lastlabi; intvec labi; intvec labfi; |
---|
3082 | int lastlab=0; |
---|
3083 | for (i=1;i<=size(@L);i++) |
---|
3084 | { |
---|
3085 | labi=@L[i][1]; |
---|
3086 | if (size(labi)>1) |
---|
3087 | { |
---|
3088 | labfi=delintvec(labi,size(labi)); |
---|
3089 | if (labfu==labfi) |
---|
3090 | { |
---|
3091 | lastlabi=labi[size(labi)]; |
---|
3092 | if (lastlab<lastlabi) |
---|
3093 | { |
---|
3094 | lastlab=lastlabi; |
---|
3095 | } |
---|
3096 | } |
---|
3097 | } |
---|
3098 | } |
---|
3099 | return(lastlab); |
---|
3100 | } |
---|
3101 | |
---|
3102 | // given the vertex u it provides the next brother of u. |
---|
3103 | // if it does not exist, then it outputs v=list(list(intvec(0)),0) |
---|
3104 | proc nextbrother(intvec labu) |
---|
3105 | { |
---|
3106 | list L; int i; int j; list next; |
---|
3107 | int lastlabu=labu[size(labu)]; |
---|
3108 | intvec labfu=delintvec(labu,size(labu)); |
---|
3109 | int lastlabi; intvec labi; intvec labfi; |
---|
3110 | for (i=1;i<=size(@L);i++) |
---|
3111 | { |
---|
3112 | labi=@L[i][1]; |
---|
3113 | if (size(labi)>1) |
---|
3114 | { |
---|
3115 | labfi=delintvec(labi,size(labi)); |
---|
3116 | if (labfu==labfi) |
---|
3117 | { |
---|
3118 | lastlabi=labi[size(labi)]; |
---|
3119 | if (lastlabu<lastlabi) |
---|
3120 | {L[size(L)+1]=list(lastlabi,list(@L[i],i));} |
---|
3121 | } |
---|
3122 | } |
---|
3123 | } |
---|
3124 | if (size(L)==0){return(list(intvec(0),0));} |
---|
3125 | next=L[1]; |
---|
3126 | for (i=2;i<=size(L);i++) |
---|
3127 | { |
---|
3128 | if (L[i][1]<next[1]){next=L[i];} |
---|
3129 | } |
---|
3130 | return(next[2]); |
---|
3131 | } |
---|
3132 | |
---|
3133 | // gives the first child of vertex fu |
---|
3134 | proc firstchild(def labfu) |
---|
3135 | { |
---|
3136 | intvec labfu0=labfu; |
---|
3137 | labfu0[size(labfu0)+1]=0; |
---|
3138 | return(nextbrother(labfu0)); |
---|
3139 | } |
---|
3140 | |
---|
3141 | // purpose: eliminate the children vertices of fu and all its descendents |
---|
3142 | // whose prime ideal Pu contains a prime ideal Pv of some brother vertex w. |
---|
3143 | proc deletebrotherscontaining(intvec labfu) |
---|
3144 | { |
---|
3145 | int i; int t; |
---|
3146 | list fu=tree(labfu,@L); |
---|
3147 | int childfu=fu[1][2]; |
---|
3148 | list u; intvec labu; ideal Pu; |
---|
3149 | list v; intvec labv; ideal Pv; |
---|
3150 | u=firstchild(labfu); |
---|
3151 | for (i=1;i<=childfu;i++) |
---|
3152 | { |
---|
3153 | labu=u[1][1]; |
---|
3154 | Pu=u[1][3]; |
---|
3155 | v=firstchild(fu[1][1]); |
---|
3156 | t=1; |
---|
3157 | while ((t) and (v[2]!=0)) |
---|
3158 | { |
---|
3159 | labv=v[1][1]; |
---|
3160 | Pv=v[1][3]; |
---|
3161 | if (labu!=labv) |
---|
3162 | { |
---|
3163 | if (containedP(Pv,Pu)) |
---|
3164 | { |
---|
3165 | deleteverts(labu); |
---|
3166 | fu=tree(labfu,@L); |
---|
3167 | @L[fu[2]][2]=fu[1][2]-1; |
---|
3168 | t=0; |
---|
3169 | } |
---|
3170 | } |
---|
3171 | if (t!=0) |
---|
3172 | { |
---|
3173 | v=nextbrother(v[1][1]); |
---|
3174 | } |
---|
3175 | } |
---|
3176 | if (i<childfu) |
---|
3177 | { |
---|
3178 | u=nextbrother(u[1][1]); |
---|
3179 | } |
---|
3180 | } |
---|
3181 | } |
---|
3182 | |
---|
3183 | // purpose: delete all descendent vertices from u included u |
---|
3184 | // from the list @L. |
---|
3185 | // It must be noted that after the operation, the number of children |
---|
3186 | // in fathers vertex must be decreased in 1 unitity. This operation is not |
---|
3187 | // performed inside this recursive routine. |
---|
3188 | proc deleteverts(intvec labu) |
---|
3189 | { |
---|
3190 | int i; int ii; list v; intvec labv; |
---|
3191 | list u=tree(labu,@L); |
---|
3192 | int childu=u[1][2]; |
---|
3193 | @L=delete(@L,u[2]); |
---|
3194 | if (childu>0) |
---|
3195 | { |
---|
3196 | v=firstchild(labu); |
---|
3197 | labv=v[1][1]; |
---|
3198 | for (ii=1;ii<=childu;ii++) |
---|
3199 | { |
---|
3200 | deleteverts(labv); |
---|
3201 | if (ii<childu) |
---|
3202 | { |
---|
3203 | v=nextbrother(v[1][1]); |
---|
3204 | labv=v[1][1]; |
---|
3205 | } |
---|
3206 | } |
---|
3207 | } |
---|
3208 | } |
---|
3209 | |
---|
3210 | // purpose: starting from vertex olab (initially nlab=olab) |
---|
3211 | // relabels the vertices of @L to be consecutive |
---|
3212 | proc relabelingindices(intvec olab, intvec nlab) |
---|
3213 | { |
---|
3214 | int i; |
---|
3215 | intvec nlabi; intvec labv; |
---|
3216 | list u=tree(olab,@L); |
---|
3217 | int childu=u[1][2]; |
---|
3218 | list v; |
---|
3219 | if (childu==0){@L[u[2]][1]=nlab;} |
---|
3220 | else |
---|
3221 | { |
---|
3222 | v=firstchild(u[1][1]); |
---|
3223 | @L[u[2]][1]=nlab; |
---|
3224 | i=1; |
---|
3225 | while(v[2]!=0) |
---|
3226 | { |
---|
3227 | labv=v[1][1]; |
---|
3228 | nlabi=nlab,i; |
---|
3229 | relabelingindices(labv,nlabi); |
---|
3230 | v=nextbrother(labv); |
---|
3231 | i++; |
---|
3232 | } |
---|
3233 | } |
---|
3234 | } |
---|
3235 | |
---|
3236 | // mrcgs |
---|
3237 | // fundamental routine giving the |
---|
3238 | // "Minimal Reduced Comprehensive Groebner System" |
---|
3239 | // input: F = ideal in ring R=K[a][x] |
---|
3240 | // output: a list L representing the tree of the mrcgs. |
---|
3241 | proc mrcgs(ideal F, list #) |
---|
3242 | "USAGE: mrcgs(F); |
---|
3243 | F is the ideal from which to obtain the Minimal Reduced CGS. |
---|
3244 | Alternatively, as option: |
---|
3245 | mrcgs(F,L); |
---|
3246 | where L is a list of the null conditions ideal N, and W the set of |
---|
3247 | non-null polynomials (ideal). If this option is set, the ideals N and W |
---|
3248 | must depend only on the parameters and the parameter space is |
---|
3249 | reduced to V(N) \ V(h), where h=prod(w), for w in W. |
---|
3250 | A reduced specification of (N,W) will be computed and used to |
---|
3251 | restrict the parameter-space. The output will omit the known restrictions |
---|
3252 | given as option. |
---|
3253 | RETURN: The list representing the Minimal Reduced CGS. |
---|
3254 | The description given here is identical for rcgs and crcgs. |
---|
3255 | The elements of the list T computed by mrcgs are lists representing |
---|
3256 | a rooted tree. |
---|
3257 | Each element has as the two first entries with the following content:@* |
---|
3258 | [1]: The label (intvec) representing the position in the rooted |
---|
3259 | tree: 0 for the root (and this is a special element) |
---|
3260 | i for the root of the segment i |
---|
3261 | (i,...) for the children of the segment i |
---|
3262 | [2]: the number of children (int) of the vertex. |
---|
3263 | There thus three kind of vertices: |
---|
3264 | 1) the root (first element labelled 0), |
---|
3265 | 2) the vertices labelled with a single integer i, |
---|
3266 | 3) the rest of vertices labelled with more indices. |
---|
3267 | Description of the root. Vertex type 1) |
---|
3268 | There is a special vertex (the first one) whose content is |
---|
3269 | the following: |
---|
3270 | [3] lpp of the given ideal |
---|
3271 | [4] the given ideal |
---|
3272 | [5] the red-spec of the (optional) given null and non-null conditions |
---|
3273 | (see redspec for the description) |
---|
3274 | [6] MRCGS (to remember which algorithm has been used). If the |
---|
3275 | algorithm used is rcgs of crcgs then this will be stated |
---|
3276 | at this vertex (RCGS or CRCGS). |
---|
3277 | Description of vertices type 2). These are the vertices that |
---|
3278 | initiate a segment, and are labelled with a single integer. |
---|
3279 | [3] lpp (ideal) of the reduced basis. If they are repeated lpp's this |
---|
3280 | will correspond to a sheaf. |
---|
3281 | [4] the reduced basis (ideal) of the segment. |
---|
3282 | Description of vertices type 3). These vertices have as first |
---|
3283 | label i and descend form vertex i in the position of the label |
---|
3284 | (i,...). They contain moreover a unique prime ideal in the parameters |
---|
3285 | and form ascending chains of ideals. |
---|
3286 | How is to be read the mrcgs tree? The vertices with an even number of |
---|
3287 | integers in the label are to be considered as additive and those |
---|
3288 | with an odd number of integers in the label are to be considered as |
---|
3289 | subtraction. As an example consider the following vertices: |
---|
3290 | v1=((i),2,lpp,B), |
---|
3291 | v2=((i,1),2,P_{(i,1)}), |
---|
3292 | v3=((i,1,1),2,P_{(i,1,1)}, |
---|
3293 | v4=((i,1,1,1),1,P_{(i,1,1,1)}, |
---|
3294 | v5=((i,1,1,1,1),0,P_{(i,1,1,1,1)}, |
---|
3295 | v6=((i,1,1,2),1,P_{(i,1,1,2)}, |
---|
3296 | v7=((i,1,1,2,1),0,P_{(i,1,1,2,1)}, |
---|
3297 | v8=((i,1,2),0,P_{(i,1,2)}, |
---|
3298 | v9=((i,2),1,P_{(i,2)}, |
---|
3299 | v10=((i,2,1),0,P_{(i,2,1)}, |
---|
3300 | They represent the segment: |
---|
3301 | (V(i,1)\(((V(i,1,1) \ ((V(i,1,1,1) \ V(i,1,1,1,1)) u (V(i,1,1,2) \ V(i,1,1,2,1))))) |
---|
3302 | u V(i,1,2)) u (V(i,2) \ V(i,2,1)) |
---|
3303 | and can also be represented by |
---|
3304 | (V(i,1) \ (V(i,1,1) u V(i,1,2))) u |
---|
3305 | (V(i,1,1,1) \ V(i,1,1,1)) u |
---|
3306 | (V(i,1,1,2) \ V(i,1,1,2,1)) u |
---|
3307 | (V(i,2) \ V(i,2,1)) |
---|
3308 | where V(i,j,..) = V(P_{(i,j,..)} |
---|
3309 | NOTE: There are three fundamental routines in the library: mrcgs, rcgs and crcgs. |
---|
3310 | mrcgs (Minimal Reduced CGS) is an algorithm that packs so much as it |
---|
3311 | is able to do (using algorithms adhoc) the segments with the same lpp, |
---|
3312 | obtaining the minimal number of segments. The hypothesis is that this |
---|
3313 | is also canonical, but for the moment there is no proof of the uniqueness |
---|
3314 | of that minimal packing. Moreover, the segments that are obtained are not |
---|
3315 | locally closed, i.e. there are not always the difference of two varieties, |
---|
3316 | but can be a union of differences. |
---|
3317 | The output can be visualized using cantreetoMaple, that will |
---|
3318 | write a file with the content of mrcgs that can be read in Maple |
---|
3319 | and plotted using the Maple plotcantree routine of the Monte's dpgb library |
---|
3320 | You can also try the routine cantodiffcgs when the segments are all |
---|
3321 | difference of two varieties to have a simpler view of the output. |
---|
3322 | But it will give an error if the output is not locally closed. |
---|
3323 | KEYWORDS: rcgs, crcgs, buildtree, cantreetoMaple, cantodiffcgs |
---|
3324 | EXAMPLE: mrcgs; shows an example" |
---|
3325 | { |
---|
3326 | int i=1; |
---|
3327 | int @ish=1; |
---|
3328 | exportto(Top,@ish); |
---|
3329 | while((@ish) and (i<=size(F))) |
---|
3330 | { |
---|
3331 | @ish=ishomog(F[i]); |
---|
3332 | i++; |
---|
3333 | } |
---|
3334 | list L=buildtree(F, #); |
---|
3335 | list S=selectcases(L); |
---|
3336 | list T=cantree(S); |
---|
3337 | T[1][6]="MRCGS"; |
---|
3338 | T[1][4]=F; |
---|
3339 | for (i=1;i<=size(F);i++) |
---|
3340 | { |
---|
3341 | T[1][3][i]=leadmonom(F[i]); |
---|
3342 | } |
---|
3343 | if (size(#)>0) |
---|
3344 | { |
---|
3345 | ideal N=#[1]; |
---|
3346 | ideal W=#[2]; |
---|
3347 | T=reduceconds(T,N,W); |
---|
3348 | } |
---|
3349 | kill @ish; |
---|
3350 | return(T); |
---|
3351 | } |
---|
3352 | example |
---|
3353 | { "EXAMPLE:"; echo = 2; |
---|
3354 | ring R=(0,b,c,d,e,f),(x,y),dp; |
---|
3355 | ideal F=x^2+b*y^2+2*c*x*y+2*d*x+2*e*y+f, 2*x+2*c*y+2*d, 2*b*y+2*c*x+2*e; |
---|
3356 | def T=mrcgs(F); |
---|
3357 | T; |
---|
3358 | cantreetoMaple(T,"Tm","Tm.txt"); |
---|
3359 | //cantodiffcgs(T); // has non locally closed segments |
---|
3360 | ring R=(0,a1,a2,a3,a4),(x1,x2,x3,x4),dp; |
---|
3361 | ideal F2=x4-a4+a2, x1+x2+x3+x4-a1-a3-a4, x1*x3*x4-a1*a3*a4, x1*x3+x1*x4+x2*x3+x3*x4-a1*a4-a1*a3-a3*a4; |
---|
3362 | def T2=mrcgs(F2); |
---|
3363 | T2; |
---|
3364 | cantreetoMaple(T2,"T2m","T2m.txt"); |
---|
3365 | cantodiffcgs(T2); |
---|
3366 | } |
---|
3367 | |
---|
3368 | // reduceconds: when null and nonnull conditions are specified it |
---|
3369 | // takes the output of cantree and reduces the tree |
---|
3370 | // assuming the null and nonnull conditions |
---|
3371 | // input: list T (the output of cantree computed with null and nonull conditions |
---|
3372 | // ideal N: null conditions |
---|
3373 | // ideal W: non-null conditions |
---|
3374 | // output: the list T assuming the null and non-null conditions |
---|
3375 | proc reduceconds(list T,ideal N,ideal W) |
---|
3376 | { |
---|
3377 | int i; intvec lab; intvec labfu; list fu; int j; int t; |
---|
3378 | list @L=T; |
---|
3379 | exportto(Top,@L); |
---|
3380 | int n=size(W); |
---|
3381 | for (i=2;i<=size(@L);i++) |
---|
3382 | { |
---|
3383 | t=0; j=0; |
---|
3384 | while ((not(t)) and (j<n)) |
---|
3385 | { |
---|
3386 | j++; |
---|
3387 | if (size(@L[i][1])>1) |
---|
3388 | { |
---|
3389 | if (memberpos(W[j],@L[i][3])[1]) |
---|
3390 | { |
---|
3391 | t=1; |
---|
3392 | @L[i][3]=ideal(1); |
---|
3393 | } |
---|
3394 | } |
---|
3395 | } |
---|
3396 | } |
---|
3397 | for (i=2;i<=size(@L);i++) |
---|
3398 | { |
---|
3399 | if (size(@L[i][1])>1) |
---|
3400 | { |
---|
3401 | @L[i][3]=delidfromid(N,@L[i][3]); |
---|
3402 | } |
---|
3403 | } |
---|
3404 | for (i=2;i<=size(@L);i++) |
---|
3405 | { |
---|
3406 | if ((size(@L[i][1])>1) and (size(@L[i][1]) mod 2==1) and (equalideals(@L[i][3],ideal(0)))) |
---|
3407 | { |
---|
3408 | lab=@L[i][1]; |
---|
3409 | labfu=delintvec(lab,size(lab)); |
---|
3410 | fu=tree(labfu,@L); |
---|
3411 | @L[fu[2]][2]=@L[fu[2]][2]-1; |
---|
3412 | deleteverts(lab); |
---|
3413 | } |
---|
3414 | } |
---|
3415 | for (j=2; j<=size(@L); j++) |
---|
3416 | { |
---|
3417 | if (@L[j][2]>0) |
---|
3418 | { |
---|
3419 | deletebrotherscontaining(@L[j][1]); |
---|
3420 | } |
---|
3421 | } |
---|
3422 | for (i=1;i<=@L[1][2];i++) |
---|
3423 | { |
---|
3424 | relabelingindices(intvec(i),intvec(i)); |
---|
3425 | } |
---|
3426 | list TT=@L; |
---|
3427 | kill @L; |
---|
3428 | return(TT); |
---|
3429 | } |
---|
3430 | |
---|
3431 | //**************End of cantree****************************** |
---|
3432 | |
---|
3433 | //**************Begin of CanTreeTo Maple******************** |
---|
3434 | |
---|
3435 | // cantreetoMaple |
---|
3436 | // input: list L: the output of cantree |
---|
3437 | // string T: the name of the table of Maple that represents L |
---|
3438 | // in Maple |
---|
3439 | // string writefile: the name of the file where the table T |
---|
3440 | // is written |
---|
3441 | proc cantreetoMaple(list L, string T, string writefile) |
---|
3442 | "USAGE: cantreetoMaple(T, TM, writefile); |
---|
3443 | T: is the list provided by mrcgs or crcgs or crcgs, |
---|
3444 | TM: is the name (string) of the table variable in Maple that will represent |
---|
3445 | the output of the fundamental routines, |
---|
3446 | writefile: is the name (string) of the file where to write the content. |
---|
3447 | RETURN: writes the list provided by mrcgs or crcgs or crcgs to a file |
---|
3448 | containing the table representing it in Maple. |
---|
3449 | NOTE: It can be called from the output of mrcgs or rcgs of crcgs |
---|
3450 | KEYWORDS: mrcgs, rcgs, crcgs, Maple |
---|
3451 | EXAMPLE: cantreetoMaple; shows an example" |
---|
3452 | { |
---|
3453 | short=0; |
---|
3454 | int i; |
---|
3455 | list L0=L[1]; |
---|
3456 | int numcases=L0[2]; |
---|
3457 | link LLw=":w "+writefile; |
---|
3458 | string La=string("table(",T,");"); |
---|
3459 | write(LLw, La); |
---|
3460 | close(LLw); |
---|
3461 | link LLa=":a "+writefile; |
---|
3462 | def RL=ringlist(@R); |
---|
3463 | list p=RL[1][2]; |
---|
3464 | string param=string(p[1]); |
---|
3465 | if (size(p)>1) |
---|
3466 | { |
---|
3467 | for(i=2;i<=size(p);i++){param=string(param,",",p[i]);} |
---|
3468 | } |
---|
3469 | list v=RL[2]; |
---|
3470 | string vars=string(v[1]); |
---|
3471 | if (size(v)>1) |
---|
3472 | { |
---|
3473 | for(i=2;i<=size(v);i++){vars=string(vars,",",v[i]);} |
---|
3474 | } |
---|
3475 | list xord; |
---|
3476 | list pord; |
---|
3477 | if (RL[1][3][1][1]=="dp"){pord=string("tdeg(",param);} |
---|
3478 | else |
---|
3479 | { |
---|
3480 | if (RL[1][3][1][1]=="lp"){pord=string("plex(",param);} |
---|
3481 | } |
---|
3482 | if (RL[3][1][1]=="dp"){xord=string("tdeg(",vars);} |
---|
3483 | else |
---|
3484 | { |
---|
3485 | if (RL[3][1][1]=="lp"){xord=string("plex(",vars);} |
---|
3486 | } |
---|
3487 | write(LLa,string(T,"[[___xord]]:=",xord,");")); |
---|
3488 | write(LLa,string(T,"[[___pord]]:=",pord,");")); |
---|
3489 | //write(LLa,string(T,"[[11]]:=true; ")); |
---|
3490 | list S; |
---|
3491 | S=string(T,"[[0]]:=",numcases,";"); |
---|
3492 | write(LLa,S); |
---|
3493 | S=string(T,"[[___method]]:=",L[1][6],";"); |
---|
3494 | // Method L[1][6]; |
---|
3495 | write(LLa,S); |
---|
3496 | S=string(T,"[[___basis]]:=[",L0[4],"];"); |
---|
3497 | write(LLa,S); |
---|
3498 | S=string(T,"[[___nullcond]]:=[",L0[5][1],"];"); |
---|
3499 | write(LLa,S); |
---|
3500 | S=string(T,"[[___notnullcond]]:={",L0[5][2],"};"); |
---|
3501 | write(LLa,S); |
---|
3502 | for (i=1;i<=numcases;i++) |
---|
3503 | { |
---|
3504 | S=ctlppbasis(T,L,intvec(i)); |
---|
3505 | write(LLa,S[1]); |
---|
3506 | write(LLa,S[2]); |
---|
3507 | write(LLa,S[3]); |
---|
3508 | //write(LLa,S[4]); |
---|
3509 | ctrecwrite(LLa, L, T, intvec(i),S[4]); |
---|
3510 | } |
---|
3511 | close(LLa); |
---|
3512 | } |
---|
3513 | example |
---|
3514 | { "EXAMPLE:"; echo = 2; |
---|
3515 | ring R=(0,b,c,d,e,f),(x,y),dp; |
---|
3516 | ideal F=x^2+b*y^2+2*c*x*y+2*d*x+2*e*y+f, 2*x+2*c*y+2*d, 2*b*y+2*c*x+2*e; |
---|
3517 | def T=mrcgs(F); |
---|
3518 | T; |
---|
3519 | cantreetoMaple(T,"Tm","Tm.txt"); |
---|
3520 | } |
---|
3521 | |
---|
3522 | // ctlppbasis: auxiliary cantreetoMaple routine |
---|
3523 | // input: |
---|
3524 | // string T: the name of the table in Maple |
---|
3525 | // intvec lab: the label of the case |
---|
3526 | // ideal B: the basis of the case |
---|
3527 | // output: |
---|
3528 | // the string of T[[lab]] (basis); in Maple |
---|
3529 | proc ctlppbasis(string T, list L, intvec lab) |
---|
3530 | { |
---|
3531 | list u; |
---|
3532 | intvec lab0=lab,0; |
---|
3533 | u=tree(lab,L); |
---|
3534 | list Li; |
---|
3535 | Li[1]=string(T,"[[",lab,",___lpp]]:=[",u[1][3],"]; "); |
---|
3536 | Li[2]=string(T,"[[",lab,"]]:=[",u[1][4],"]; "); |
---|
3537 | Li[3]=string(T,"[[",lab0,"]]:=",u[1][2],"; "); |
---|
3538 | Li[4]=u[1][2]; |
---|
3539 | return(Li); |
---|
3540 | } |
---|
3541 | |
---|
3542 | // ctlppbasis: auxiliary cantreetoMaple routine |
---|
3543 | // recursive routine to write all elements |
---|
3544 | proc ctrecwrite(LLa, list L, string T, intvec lab, int n) |
---|
3545 | { |
---|
3546 | int i; |
---|
3547 | intvec labi; intvec labi0; |
---|
3548 | string S; |
---|
3549 | list u; |
---|
3550 | for (i=1;i<=n;i++) |
---|
3551 | { |
---|
3552 | labi=lab,i; |
---|
3553 | u=tree(labi,L); |
---|
3554 | S=string(T,"[[",labi,"]]:=[",u[1][3],"];"); |
---|
3555 | write(LLa,S); |
---|
3556 | labi0=labi,0; |
---|
3557 | S=string(T,"[[",labi0,"]]:=",u[1][2],";"); |
---|
3558 | write(LLa,S); |
---|
3559 | ctrecwrite(LLa, L, T, labi, u[1][2]); |
---|
3560 | } |
---|
3561 | } |
---|
3562 | |
---|
3563 | //**************End of CanTreeTo Maple******************** |
---|
3564 | |
---|
3565 | //**************Begin homogenizing************************ |
---|
3566 | |
---|
3567 | // ishomog: |
---|
3568 | // Purpose: test if a polynomial is homogeneous in the variables or not |
---|
3569 | // input: poly f |
---|
3570 | // output 1 if f is homogeneous, 0 if not |
---|
3571 | proc ishomog(def f) |
---|
3572 | { |
---|
3573 | int i; poly r; int d; int dr; |
---|
3574 | if (f==0){return(1);} |
---|
3575 | d=deg(f); dr=d; r=f; |
---|
3576 | while ((d==dr) and (r!=0)) |
---|
3577 | { |
---|
3578 | r=r-lead(r); |
---|
3579 | dr=deg(r); |
---|
3580 | } |
---|
3581 | if (r==0){return(1);} |
---|
3582 | else{return(0);} |
---|
3583 | } |
---|
3584 | |
---|
3585 | proc rcgs(ideal F, list #) |
---|
3586 | "USAGE: rcgs(F); |
---|
3587 | F is the ideal from which to obtain the Reduced CGS. |
---|
3588 | rcgs(F,L); |
---|
3589 | where L is a list of the null conditions ideal N, and W the set of |
---|
3590 | non-null polynomials (ideal). If this option is set, the ideals N and W |
---|
3591 | must depend only on the parameters and the parameter space is |
---|
3592 | reduced to V(N) \ V(h), where h=prod(w), for w in W. |
---|
3593 | A reduced specification of (N,W) will be computed and used to |
---|
3594 | restrict the parameter-space. The output will omit the known restrictions |
---|
3595 | given as option. |
---|
3596 | RETURN: The list representing the Reduced CGS. |
---|
3597 | The description given here is analogous as for mrcgs and crcgs. |
---|
3598 | The elements of the list T computed by rcgs are lists representing |
---|
3599 | a rooted tree. |
---|
3600 | Each element has as the two first entries with the following content:@* |
---|
3601 | [1]: The label (intvec) representing the position in the rooted |
---|
3602 | tree: 0 for the root (and this is a special element) |
---|
3603 | i for the root of the segment i |
---|
3604 | (i,...) for the children of the segment i |
---|
3605 | [2]: the number of children (int) of the vertex. |
---|
3606 | There thus three kind of vertices: |
---|
3607 | 1) the root (first element labelled 0), |
---|
3608 | 2) the vertices labelled with a single integer i, |
---|
3609 | 3) the rest of vertices labelled with more indices. |
---|
3610 | Description of the root. Vertex type 1) |
---|
3611 | There is a special vertex (the first one) whose content is |
---|
3612 | the following: |
---|
3613 | [3] lpp of the given ideal |
---|
3614 | [4] the given ideal |
---|
3615 | [5] the red-spec of the (optional) given null and non-null conditions |
---|
3616 | (see redspec for the description) |
---|
3617 | [6] RCGS (to remember which algorithm has been used). If the |
---|
3618 | algorithm used is mrcgs or crcgs then this will be stated |
---|
3619 | at this vertex (mrcgs or CRCGS). |
---|
3620 | Description of vertices type 2). These are the vertices that |
---|
3621 | initiate a segment, and are labelled with a single integer. |
---|
3622 | [3] lpp (ideal) of the reduced basis. If they are repeated lpp's this |
---|
3623 | will correspond to a sheaf. |
---|
3624 | [4] the reduced basis (ideal) of the segment. |
---|
3625 | Description of vertices type 3). These vertices have as first |
---|
3626 | label i and descend form vertex i in the position of the label |
---|
3627 | (i,...). They contain moreover a unique prime ideal in the parameters |
---|
3628 | and form ascending chains of ideals. |
---|
3629 | How is to be read the rcgs tree? The vertices with an even number of |
---|
3630 | integers in the label are to be considered as additive and those |
---|
3631 | with an odd number of integers in the label are to be considered as |
---|
3632 | subtraction. As an example consider the following vertices: |
---|
3633 | v1=((i),2,lpp,B), |
---|
3634 | v2=((i,1),2,P_{(i,1)}), |
---|
3635 | v3=((i,1,1),0,P_{(i,1,1)}, v4=((i,1,2),0,P_{(i,1,1)}), |
---|
3636 | v5=((i,2),2,P_{(i,2)}, |
---|
3637 | v6=((i,2,1),0,P_{(i,2,1)}, v7=((i,2,2),0,P_{(i,2,2)} |
---|
3638 | They represent the segment: |
---|
3639 | (V(i,1)\(V(i,1,1) u V(i,1,2))) u |
---|
3640 | (V(i,2)\(V(i,2,1) u V(i,2,2))) |
---|
3641 | where V(i,j,..) = V(P_{(i,j,..)} |
---|
3642 | NOTE: There are three fundamental routines in the library: mrcgs, rcgs and crcgs. |
---|
3643 | rcgs (Reduced CGS) is an algorithm that first homogenizes the |
---|
3644 | basis of the given ideal then applies mrcgs and finally de-homogenizes |
---|
3645 | and reduces the resulting bases. (See the note of mrcgs). |
---|
3646 | As a result of Wibmer's Theorem, the resulting segments are |
---|
3647 | locally closed (i.e. difference of varieties). Nevertheless, the |
---|
3648 | output is not completely canonical as the homogeneous ideal considered |
---|
3649 | is not the homogenized ideal of the given ideal but only the ideal |
---|
3650 | obtained by homogenizing the given basis. |
---|
3651 | |
---|
3652 | The output can be visualized using cantreetoMaple, that will |
---|
3653 | write a file with the content of mrcgs that can be read in Maple |
---|
3654 | and plotted using the Maple plotcantree routine of the Monte's dpgb library |
---|
3655 | You can also use the routine cantodiffcgs as the segments are all |
---|
3656 | difference of two varieties to have a simpler view of the output. |
---|
3657 | KEYWORDS: rcgs, crcgs, buildtree, cantreetoMaple, cantodiffcgs |
---|
3658 | EXAMPLE: rcgs; shows an example" |
---|
3659 | { |
---|
3660 | ideal N; |
---|
3661 | ideal W; |
---|
3662 | int j; int i; |
---|
3663 | poly f; |
---|
3664 | if (size(#)==2) |
---|
3665 | { |
---|
3666 | N=#[1]; |
---|
3667 | W=#[2]; |
---|
3668 | } |
---|
3669 | i=1; int postred=0; |
---|
3670 | int ish=1; |
---|
3671 | while ((ish) and (i<=size(F))) |
---|
3672 | { |
---|
3673 | ish=ishomog(F[i]); |
---|
3674 | i++; |
---|
3675 | } |
---|
3676 | if (ish){return(mrcgs(F, #));} |
---|
3677 | def RR=basering; |
---|
3678 | list RRL=ringlist(RR); |
---|
3679 | if (RRL[3][1][1]!="dp"){ERROR("the order must be dp");} |
---|
3680 | poly @t; |
---|
3681 | ring H=0,@t,dp; |
---|
3682 | def RH=RR+H; |
---|
3683 | setring(RH); |
---|
3684 | setglobalrings(); |
---|
3685 | def FH=imap(RR,F); |
---|
3686 | list u; ideal B; ideal lpp; intvec lab; |
---|
3687 | FH=homog(FH,@t); |
---|
3688 | def Nh=imap(RR,N); |
---|
3689 | def Wh=imap(RR,W); |
---|
3690 | list L; |
---|
3691 | if ((size(Nh)>0) or (size(Wh)>0)) |
---|
3692 | { |
---|
3693 | L=mrcgs(FH,list(Nh,Wh)); |
---|
3694 | } |
---|
3695 | else |
---|
3696 | { |
---|
3697 | L=mrcgs(FH); |
---|
3698 | } |
---|
3699 | L[1][3]=subst(L[1][3],@t,1); |
---|
3700 | L[1][4]=subst(L[1][4],@t,1); |
---|
3701 | for (i=1; i<=L[1][2]; i++) |
---|
3702 | { |
---|
3703 | lab=intvec(i); |
---|
3704 | u=tree(lab,L); |
---|
3705 | postred=difflpp(u[1][3]); |
---|
3706 | B=sortideal(subst(L[u[2]][4],@t,1)); |
---|
3707 | lpp=sortideal(subst(L[u[2]][3],@t,1)); |
---|
3708 | if (memberpos(1,B)[1]){B=ideal(1); lpp=ideal(1);} |
---|
3709 | if (postred) |
---|
3710 | { |
---|
3711 | lpp=ideal(0); |
---|
3712 | B=postredgb(mingb(B)); |
---|
3713 | for (j=1;j<=size(B);j++){lpp[j]=leadmonom(B[j]);} |
---|
3714 | } |
---|
3715 | else{"Sheaves present, not reduced bases in the case with:";lpp;} |
---|
3716 | L[u[2]][4]=B; |
---|
3717 | L[u[2]][3]=lpp; |
---|
3718 | } |
---|
3719 | setring(RR); |
---|
3720 | setglobalrings(); |
---|
3721 | list LL=imap(RH,L); |
---|
3722 | LL[1][6]="RCGS"; |
---|
3723 | return(LL); |
---|
3724 | } |
---|
3725 | example |
---|
3726 | { "EXAMPLE:"; echo = 2; |
---|
3727 | ring R=(0,b,c,d,e,f),(x,y),dp; |
---|
3728 | ideal F=x^2+b*y^2+2*c*x*y+2*d*x+2*e*y+f, 2*x+2*c*y+2*d, 2*b*y+2*c*x+2*e; |
---|
3729 | def T=rcgs(F); |
---|
3730 | T; |
---|
3731 | cantreetoMaple(T,"Tr","Tr.txt"); |
---|
3732 | cantodiffcgs(T); |
---|
3733 | } |
---|
3734 | |
---|
3735 | proc difflpp(ideal lpp) |
---|
3736 | { |
---|
3737 | int t=1; int i; |
---|
3738 | poly lp1=lpp[1]; |
---|
3739 | poly lp; |
---|
3740 | i=2; |
---|
3741 | while ((i<=size(lpp)) and (t)) |
---|
3742 | { |
---|
3743 | lp=lpp[i]; |
---|
3744 | if (lp==lp1){t=0;} |
---|
3745 | lp1=lp; |
---|
3746 | i++; |
---|
3747 | } |
---|
3748 | return(t); |
---|
3749 | } |
---|
3750 | |
---|
3751 | // redgb: given a minimal bases (gb reducing) it |
---|
3752 | // reduces each polynomial w.r.t. to the others |
---|
3753 | proc postredgb(ideal F) |
---|
3754 | { |
---|
3755 | ideal G; |
---|
3756 | ideal H; |
---|
3757 | int i; |
---|
3758 | if (size(F)==0){return(ideal(0));} |
---|
3759 | for (i=1;i<=size(F);i++) |
---|
3760 | { |
---|
3761 | H=delfromideal(F,i); |
---|
3762 | G[i]=pdivi2(F[i],H)[1]; |
---|
3763 | } |
---|
3764 | return(G); |
---|
3765 | } |
---|
3766 | |
---|
3767 | proc crcgs(ideal F, list #) |
---|
3768 | "USAGE: crcgs(F); |
---|
3769 | F is the ideal from which to obtain the Canonical Reduced CGS. |
---|
3770 | crcgs(F,L); |
---|
3771 | where L is a list of the null conditions ideal N, and W the set of |
---|
3772 | non-null polynomials (ideal). If this option is set, the ideals N and W |
---|
3773 | must depend only on the parameters and the parameter space is |
---|
3774 | reduced to V(N) \ V(h), where h=prod(w), for w in W. |
---|
3775 | A reduced specification of (N,W) will be computed and used to |
---|
3776 | restrict the parameter-space. The output will omit the known restrictions |
---|
3777 | given as option. |
---|
3778 | RETURN: The list representing the Canonical Reduced CGS. |
---|
3779 | The description given here is identical for mrcgs and rcgs. |
---|
3780 | The elements of the list T computed by crcgs are lists representing |
---|
3781 | a rooted tree. |
---|
3782 | Each element has as the two first entries with the following content:@* |
---|
3783 | [1]: The label (intvec) representing the position in the rooted |
---|
3784 | tree: 0 for the root (and this is a special element) |
---|
3785 | i for the root of the segment i |
---|
3786 | (i,...) for the children of the segment i |
---|
3787 | [2]: the number of children (int) of the vertex. |
---|
3788 | There thus three kind of vertices: |
---|
3789 | 1) the root (first element labelled 0), |
---|
3790 | 2) the vertices labelled with a single integer i, |
---|
3791 | 3) the rest of vertices labelled with more indices. |
---|
3792 | Description of the root. Vertex type 1) |
---|
3793 | There is a special vertex (the first one) whose content is |
---|
3794 | the following: |
---|
3795 | [3] lpp of the given ideal |
---|
3796 | [4] the given ideal |
---|
3797 | [5] the red-spec of the (optional) given null and non-null conditions |
---|
3798 | (see redspec for the description) |
---|
3799 | [6] mrcgs (to remember which algorithm has been used). If the |
---|
3800 | algorithm used is rcgs of crcgs then this will be stated |
---|
3801 | at this vertex (RCGS or CRCGS). |
---|
3802 | Description of vertices type 2). These are the vertices that |
---|
3803 | initiate a segment, and are labelled with a single integer. |
---|
3804 | [3] lpp (ideal) of the reduced basis. If they are repeated lpp's this |
---|
3805 | will correspond to a sheaf. |
---|
3806 | [4] the reduced basis (ideal) of the segment. |
---|
3807 | Description of vertices type 3). These vertices have as first |
---|
3808 | label i and descend form vertex i in the position of the label |
---|
3809 | (i,...). They contain moreover a unique prime ideal in the parameters |
---|
3810 | and form ascending chains of ideals. |
---|
3811 | How is to be read the mrcgs tree? The vertices with an even number of |
---|
3812 | integers in the label are to be considered as additive and those |
---|
3813 | with an odd number of integers in the label are to be considered as |
---|
3814 | subtraction. As an example consider the following vertices: |
---|
3815 | v1=((i),2,lpp,B), |
---|
3816 | v2=((i,1),2,P_{(i,1)}), |
---|
3817 | v3=((i,1,1),0,P_{(i,1,1)}, v4=((i,1,2),0,P_{(i,1,1)}), |
---|
3818 | v5=((i,2),2,P_{(i,2)}, |
---|
3819 | v6=((i,2,1),0,P_{(i,2,1)}, v7=((i,2,2),0,P_{(i,2,2)} |
---|
3820 | They represent the segment: |
---|
3821 | (V(i,1)\(V(i,1,1) u V(i,1,2))) u |
---|
3822 | (V(i,2)\(V(i,2,1) u V(i,2,2))) |
---|
3823 | where V(i,j,..) = V(P_{(i,j,..)} |
---|
3824 | NOTE: There are three fundamental routines in the library: mrcgs, rcgs and crcgs. |
---|
3825 | crcgs (Canonical Reduced CGS) is an algorithm that first homogenizes the |
---|
3826 | the given ideal then applies mrcgs and finally de-homogenizes |
---|
3827 | and reduces the resulting bases. (See the note of mrcgs). |
---|
3828 | As a result of Wibmer's Theorem, the resulting segments are |
---|
3829 | locally closed (i.e. difference of varieties) and the partition is |
---|
3830 | canonical as the homogenized ideal is uniquely associated to the given |
---|
3831 | ideal not depending of the given basis. |
---|
3832 | |
---|
3833 | Nevertheless the computations to do are usually more time consuming |
---|
3834 | and so it is preferable to compute first the rcgs and only if |
---|
3835 | it success you can try crcgs. |
---|
3836 | |
---|
3837 | The output can be visualized using cantreetoMaple, that will |
---|
3838 | write a file with the content of crcgs that can be read in Maple |
---|
3839 | and plotted using the Maple plotcantree routine of the Monte's dpgb library |
---|
3840 | You can also use the routine cantodiffcgs as the segments are all |
---|
3841 | difference of two varieties to have a simpler view of the output. |
---|
3842 | KEYWORDS: mrcgs, rcgs, buildtree, cantreetoMaple, cantodiffcgs |
---|
3843 | EXAMPLE: mrcgs; shows an example" |
---|
3844 | { |
---|
3845 | int ish=1; int i=1; |
---|
3846 | while ((ish) and (i<=size(F))) |
---|
3847 | { |
---|
3848 | ish=ishomog(F[i]); |
---|
3849 | i++; |
---|
3850 | } |
---|
3851 | if (ish){return(mrcgs(F, #));} |
---|
3852 | list L; |
---|
3853 | def RR=basering; |
---|
3854 | setglobalrings(); |
---|
3855 | setring(@RP); |
---|
3856 | ideal FP=imap(RR,F); |
---|
3857 | option(redSB); |
---|
3858 | def G=groebner(FP); |
---|
3859 | setring(RR); |
---|
3860 | def GR=imap(@RP,G); |
---|
3861 | kill @RP; |
---|
3862 | kill @P; |
---|
3863 | L=rcgs(GR, #); |
---|
3864 | L[1][6]="CRCGS"; |
---|
3865 | return(L); |
---|
3866 | } |
---|
3867 | example |
---|
3868 | { "EXAMPLE:"; echo = 2; |
---|
3869 | ring R=(0,b,c,d,e,f),(x,y),dp; |
---|
3870 | ideal F=x^2+b*y^2+2*c*x*y+2*d*x+2*e*y+f, 2*x+2*c*y+2*d, 2*b*y+2*c*x+2*e; |
---|
3871 | def T=crcgs(F); |
---|
3872 | T; |
---|
3873 | cantreetoMaple(T,"Tc","Tc.txt"); |
---|
3874 | cantodiffcgs(T); |
---|
3875 | } |
---|
3876 | |
---|
3877 | //purpose ideal intersection called in @R and computed in @P |
---|
3878 | proc idintR(ideal N, ideal M) |
---|
3879 | { |
---|
3880 | def RR=basering; |
---|
3881 | setring(@P); |
---|
3882 | def Np=imap(RR,N); |
---|
3883 | def Mp=imap(RR,M); |
---|
3884 | def Jp=idint(Np,Mp); |
---|
3885 | setring(RR); |
---|
3886 | return(imap(@P,Jp)); |
---|
3887 | } |
---|
3888 | |
---|
3889 | //purpose reduced groebner basis called in @R and computed in @P |
---|
3890 | proc gbR(ideal N) |
---|
3891 | { |
---|
3892 | def RR=basering; |
---|
3893 | setring(@P); |
---|
3894 | def Np=imap(RR,N); |
---|
3895 | option(redSB); |
---|
3896 | Np=groebner(Np); |
---|
3897 | setring(RR); |
---|
3898 | return(imap(@P,Np)); |
---|
3899 | } |
---|
3900 | |
---|
3901 | // purpose: given the output of a locally closed CGS (i.e. from rcgs or crcgs) |
---|
3902 | // it returns the segments as difference of varieties. |
---|
3903 | proc cantodiffcgs(list L) |
---|
3904 | "USAGE: canttodiffcgs(T); |
---|
3905 | T: is the list provided by mrcgs or crcgs or crcgs, |
---|
3906 | RETURN: The list transforming the content of these routines to a simpler |
---|
3907 | output where each segment corresponds to a single element of the list |
---|
3908 | that is described as difference of two varieties. |
---|
3909 | |
---|
3910 | The first element of the list is identical to the first element |
---|
3911 | of the list provided by the corresponding cgs algorithm, and |
---|
3912 | contains general information on the call (see mrcgs). |
---|
3913 | The remaining elements are lists of 4 elements, |
---|
3914 | representing segments. These elements are |
---|
3915 | [1]: the lpp of the segment |
---|
3916 | [2]: the basis of the segment |
---|
3917 | [3]; the ideal of the first variety (radical) |
---|
3918 | [4]; the ideal of the second variety (radical) |
---|
3919 | The segment is V([3]) \ V([4]). |
---|
3920 | |
---|
3921 | NOTE: It can be called from the output of mrcgs or rcgs of crcgs |
---|
3922 | KEYWORDS: mrcgs, rcgs, crcgs, Maple |
---|
3923 | EXAMPLE: cantodiffcgs; shows an example" |
---|
3924 | { |
---|
3925 | int i; int j; int k; int depth; list LL; list u; list v; list w; |
---|
3926 | ideal N; ideal Nn; ideal M; ideal Mn; ideal N0; ideal W0; |
---|
3927 | LL[1]=L[1]; |
---|
3928 | N0=L[1][5][1]; |
---|
3929 | W0=L[1][5][2]; |
---|
3930 | def RR=basering; |
---|
3931 | setring(@P); |
---|
3932 | def N0P=imap(RR,N0); |
---|
3933 | def W0P=imap(RR,N0); |
---|
3934 | ideal NP; |
---|
3935 | ideal MP; |
---|
3936 | setring(RR); |
---|
3937 | for (i=2;i<=size(L);i++) |
---|
3938 | { |
---|
3939 | depth=size(L[i][1]); |
---|
3940 | if (depth>3){ERROR("the given CGS has non locally closed segments");} |
---|
3941 | } |
---|
3942 | for (i=1;i<=L[1][2];i++) |
---|
3943 | { |
---|
3944 | N=ideal(1); |
---|
3945 | M=ideal(1); |
---|
3946 | u=tree(intvec(i),L); |
---|
3947 | for (j=1;j<=u[1][2];j++) |
---|
3948 | { |
---|
3949 | v=tree(intvec(i,j),L); |
---|
3950 | Nn=v[1][3]; |
---|
3951 | N=idintR(N,Nn); |
---|
3952 | for (k=1;k<=v[1][2];k++) |
---|
3953 | { |
---|
3954 | w=tree(intvec(i,j,k),L); |
---|
3955 | Mn=w[1][3]; |
---|
3956 | M=idintR(M,Mn); |
---|
3957 | } |
---|
3958 | } |
---|
3959 | setring(@P); |
---|
3960 | def NP=imap(RR,N); |
---|
3961 | def MP=imap(RR,M); |
---|
3962 | MP=MP+N0P; |
---|
3963 | for (j=1;j<=size(W0P);j++){MP=MP+ideal(W0P[j]);} |
---|
3964 | NP=NP+N0P; |
---|
3965 | NP=gbR(NP); |
---|
3966 | MP=gbR(MP); |
---|
3967 | setring(RR); |
---|
3968 | N=imap(@P,NP); |
---|
3969 | M=imap(@P,MP); |
---|
3970 | LL[i+1]=list(u[1][3],u[1][4],N,M); |
---|
3971 | } |
---|
3972 | return(LL); |
---|
3973 | } |
---|
3974 | example |
---|
3975 | { "EXAMPLE:"; echo = 2; |
---|
3976 | ring R=(0,b,c,d,e,f),(x,y),dp; |
---|
3977 | ideal F=x^2+b*y^2+2*c*x*y+2*d*x+2*e*y+f, 2*x+2*c*y+2*d, 2*b*y+2*c*x+2*e; |
---|
3978 | def T=crcgs(F); |
---|
3979 | T; |
---|
3980 | cantreetoMaple(T,"Tc","Tc.txt"); |
---|
3981 | cantodiffcgs(T); |
---|
3982 | } |
---|
3983 | |
---|
3984 | //**************End homogenizing************************ |
---|