1 | USAGE: toric_ideal [options] matrix_file |
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2 | |
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3 | |
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4 | |
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5 | DESCRIPTION: |
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6 | |
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7 | toric_ideal is a program for computing the toric ideal of a matrix A. |
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8 | This ideal is always given by the reduced Groebner basis with respect |
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9 | to a given term ordering which is computed via BuchbergerŽs |
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10 | algorithm. |
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11 | For this purpose, we can use six different algorithms: |
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12 | The algorithm of Conti/Traverso (ct) computes the toric ideal of an |
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13 | extended matrix. This ideal can be used later for solving integer |
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14 | programming problems for given right hand vectors. By eliminating |
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15 | auxiliary variables, we can get the toric ideal of the original matrix |
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16 | from it. The same is true for the "positive" variant of the |
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17 | Conti-Traverso-algorithm (pct) which can only be applied if A has |
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18 | nonnegative entries, but should be faster in that case (it computes |
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19 | the toric ideal of a smaller extended matrix). |
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20 | All other algorithms compute the toric ideal of A itself. Except from |
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21 | the elimination version of the Conti-Traverso algorithm (ect), they |
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22 | should be more efficient than the algorithm mentionned before. These |
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23 | are the algorithms of Pottier (pt), Bigatti/La Scala/Robbiano (blr), |
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24 | Hosten/Sturmfels (hs) and Di Biase/Urbanke (du). The last two seem to |
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25 | be the fastest in the actual implementation. |
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26 | |
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27 | |
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28 | |
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29 | FILE FORMAT: |
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30 | |
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31 | The input file has to be a MATRIX file. toric_ideal produces a |
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32 | GROEBNER file named like the MATRIX file with extensions replaced by |
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33 | |
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34 | .GB.<alg> |
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35 | |
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36 | where GB stands for GROEBNER and <alg> is the abbreviation for the |
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37 | used algorithm as above. |
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38 | |
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39 | A MATRIX file should look as follows: |
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40 | |
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41 | |
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42 | MATRIX |
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43 | |
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44 | columns: |
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45 | <number of columns> |
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46 | |
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47 | cost vector: |
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48 | <coefficients of the cost vector> |
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49 | |
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50 | rows: |
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51 | <number of rows> |
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52 | |
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53 | matrix: |
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54 | <matrix coefficients> |
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55 | |
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56 | positive row space vector: |
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57 | <coefficients of row space vector> |
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58 | |
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59 | |
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60 | The last two lines are only needed when toric_ideal is called with the |
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61 | algorithms of Hosten/Sturmfels or Bigatti/La Scala/Robbiano, i.e. the |
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62 | options |
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63 | |
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64 | -alg hs |
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65 | or |
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66 | -alg blr |
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67 | |
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68 | The other algorithms ignore these lines. |
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69 | |
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70 | Example: |
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71 | |
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72 | |
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73 | MATRIX |
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74 | |
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75 | columns: |
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76 | 3 |
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77 | |
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78 | cost vector: |
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79 | 1 1 1 |
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80 | |
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81 | rows: |
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82 | 2 |
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83 | |
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84 | matrix: |
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85 | 1 2 3 |
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86 | 4 5 6 |
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87 | |
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88 | positive row space vector: |
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89 | 1 2 3 |
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90 | |
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91 | |
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92 | A GROEBNER file looks as follows: |
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93 | |
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94 | |
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95 | GROEBNER |
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96 | |
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97 | computed with algorithm: |
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98 | <algorithm name abbreviation> (* abbreviations as above *) |
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99 | from file(s): (* the following four lines are |
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100 | <name of respective MATRIX file> optional *) |
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101 | computation time: |
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102 | <computation time in seconds> |
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103 | |
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104 | term ordering: |
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105 | elimination block |
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106 | <number of elimination variables> |
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107 | <LEX / DEG_LEX (* only if number of elimination |
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108 | / DEG_REV_LEX> variables >0 *) |
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109 | weighted block |
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110 | <number of weighted variables> |
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111 | <W_LEX / W_REV_LEX (* number of weighted variables |
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112 | / W_DEG_LEX / W_DEG_REV_LEX> should always be >0 *) |
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113 | <weight_vector> |
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114 | |
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115 | size: |
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116 | <number of elements> |
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117 | |
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118 | Groebner basis: |
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119 | <basis elements> |
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120 | |
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121 | <settings for the Buchberger |
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122 | algorithm and compiler settings> (* optional *) |
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123 | |
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124 | |
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125 | The Groebner basis consists always of binomials of the form x^a - x^b |
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126 | where x^a and x^b are relatively prime. Such a binomial can be |
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127 | represented by the vector a-b. The basis elements in the GROEBNER file |
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128 | are given by the coefficients of this vector representation. |
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129 | The settings for BuchbergerŽs algorithm and the compiler flags are |
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130 | produced when the GROEBNER file is generated by a call of toric_ideal |
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131 | with the verbose output option |
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132 | |
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133 | -v, --verbose |
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134 | |
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135 | Example (not belonging to the example above): |
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136 | |
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137 | |
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138 | GROEBNER |
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139 | |
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140 | computed with algorithm: |
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141 | du |
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142 | |
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143 | term ordering: |
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144 | elimination block: |
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145 | 0 |
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146 | weighted block: |
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147 | 3 |
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148 | W_LEX |
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149 | 1 2 3 |
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150 | |
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151 | size: |
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152 | 1 |
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153 | |
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154 | Groebner basis: |
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155 | 2 3 -2 (* x^2 * y^3 - z^2 *) |
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156 | |
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157 | |
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158 | |
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159 | OPTIONS: |
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160 | |
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161 | -alg <alg>, |
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162 | --algorithm <alg> algorithm to use for computing the toric |
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163 | ideal; <alg> may be |
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164 | ct for Conti/Traverso, |
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165 | pct for the positive Conti/Traverso, |
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166 | ect for Conti/Traverso with elimination, |
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167 | pt for Pottier, |
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168 | hs for Hosten/Sturmfels, |
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169 | du for Di Biase/Urbanke, |
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170 | blr for Bigatti-LaScal-Robbiano. |
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171 | |
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172 | -p <number> percentage of new generators to cause an |
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173 | autoreduction during BuchbergerŽs algorithm; |
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174 | <number> may be an arbitrary float, a |
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175 | negative value allows no intermediate |
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176 | autoreductions |
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177 | default is |
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178 | -p 12.0 |
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179 | |
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180 | -S [RP] [M] [B] [M] [2] criteria to use in BuchbergerŽs algorithm |
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181 | for discarding unneccessary S-pairs |
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182 | RP relatively prime leading terms |
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183 | M Gebauer-Moeller criterion M |
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184 | F Gebauer-Moeller criterion F |
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185 | B Gebauer-Moeller criterion B |
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186 | 2 BuchbergerŽs second criterion |
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187 | default is |
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188 | -S RP M B |
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189 | |
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190 | -v, |
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191 | --verbose verbose output mode; writes the settings for |
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192 | BuchbergerŽs algorithm and the compiler |
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193 | flags into the output GROEBNER file |
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194 | |
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195 | -V <number>, |
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196 | --version <number> version of BuchbergerŽs algorithm to use; |
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197 | <number> may be 1, 1a, 2 or 3 |
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198 | default is |
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199 | -V 1 |
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200 | |
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201 | |
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