[6ba162] | 1 | USAGE: toric_ideal [options] matrix_file |
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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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