1 | % $Id: usercard.tex,v 1.7 2005-09-29 13:10:32 Singular Exp $ |
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2 | |
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3 | % |
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4 | % usercard.tex - Singular user quick reference card. |
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5 | % |
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6 | |
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7 | \input singcard.tex |
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8 | |
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9 | \centerline{\hbf SINGULAR Quick Reference} |
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10 | |
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11 | \centerline{\srm {\ssc Singular} Version 3-0} |
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12 | |
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13 | \bigskip |
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14 | |
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15 | Do not forget to terminate all commands with a {\tt ;} (semicolon)! |
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16 | |
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17 | In particular if \Singular\ prints the continuation prompt {\tt .} |
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18 | (peri\-od) instead of the regular command prompt {\tt >}, then it |
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19 | waits for a command to be terminated by a {\tt ;}. If that does |
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20 | not help, try one or more {\tt "} or {\tt \char`}} to close an |
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21 | opened string or block. |
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22 | |
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23 | Comments start with {\tt //} and extend to end of line. |
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24 | |
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25 | \smallskip |
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26 | Some of the topics concerning interactive use are system dependent. |
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27 | |
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28 | \sec Starting SINGULAR(2.5cm) |
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29 | Singular& start \Singular\cr |
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30 | Singular {\it file} \rep& read {\it files\/} and prompt for further commands\cr |
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31 | Singular --help& print help on command line options and exit\cr |
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32 | \endsec |
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33 | |
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34 | \sec Stopping SINGULAR() |
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35 | quit;& exit \Singular; also {\tt exit;} or {\tt \$}\cr |
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36 | \ctl c& interrupt \Singular\cr |
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37 | \endsec |
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38 | |
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39 | \sec Getting help() |
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40 | help;& enter online help system\cr |
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41 | help {\it topic\/};& describe {\it topic\/}; also {\tt? {\it topic\/};}\cr |
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42 | \subsec{Inside the info help system:} |
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43 | \ctl h& get help on help system\cr |
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44 | q& exit from help system\cr |
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45 | n\rmslash p\rmslash u& go to next/previous/upper node\cr |
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46 | m& pick menu item by name\cr |
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47 | l& go to last visited node/exit from help on help\cr |
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48 | SPC\rmslash DEL& scroll forward/backward one page\cr |
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49 | \endsec |
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50 | |
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51 | \sec Commandline editing() |
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52 | \sectext |
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53 | Commandline editing is similar to that of, e.g., {\tt bash} or {\tt tcsh}:\cr |
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54 | BS\rmslash\ctl d& remove character on the left/right of cursor\cr |
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55 | \ctl p\rmslash\ctl n& get previous/next line from history\cr |
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56 | \ctl b\rmslash\ctl f& move cursor left/right\cr |
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57 | \ctl a\rmslash\ctl e& go to beginning/end of line\cr |
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58 | \ctl u\rmslash\ctl k& delete to beginning/end of line\cr |
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59 | \endsec |
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60 | |
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61 | \sec Names and objects() |
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62 | \sectext |
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63 | Names (= identifiers) have to be declared before they are used:\cr |
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64 | \entryskip |
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65 | \longentry {\it type\/} {\it name\/} \opt{= {\it expression\/}};& |
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66 | declare variable {\it name}\cr |
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67 | kill({\it name\/})& delete variable {\it name}\cr |
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68 | \entryskip |
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69 | \sectext |
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70 | \parskip=\verysmallskipamount |
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71 | Names of type {\tt number}, {\tt poly}, {\tt ideal}, {\tt vector}, {\tt |
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72 | module}, {\tt matrix}, {\tt map}, and {\tt resolution} may be declared |
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73 | only inside a ring. They are local to that ring. The same holds for a {\tt |
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74 | list} if it contains an object of the above types. All other types may be |
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75 | declared at any time. They are globally visible. |
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76 | |
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77 | Names may consist of alphanumeric characters including {\tt \_} |
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78 | (underscore) and have to start with a letter. Capital and small letters |
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79 | are distinguished. Names may be followed by an integer expression in |
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80 | parentheses, resulting in so-called {\it indexed names}.\cr |
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81 | \entryskip |
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82 | {\it name\/}({\it n}..{\it m\/})& |
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83 | shortcut for {\tt {\it name\/}({\it |
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84 | n\/}), $\ldots$, {\it name\/}({\it |
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85 | m\/})}\hfil\break |
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86 | (\eg {\tt ring r = 0, x(1..3), dp;})\cr |
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87 | \_ {\rm (underscore)}& refers to the value of the last expression |
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88 | printed\cr |
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89 | \endsec |
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90 | |
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91 | \vfill |
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92 | \centerline{\srm \copyright 1998-2005 \qquad Permissions on back} |
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93 | |
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94 | \eject |
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95 | |
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96 | \sec Ring declaration() |
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97 | \longentry ring {\it name\/} = {\it basefield}, ({\it ringvars\/}), {\it ordering\/};& |
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98 | \hyphenpenalty=50 \tolerance=200 declare ring |
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99 | {\it name\/} and make it the new base\-ring. |
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100 | {\it ringvars\/} has to be a list of names, the |
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101 | other items are described below. Example:\par |
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102 | {\tt ring r = 32003, (x, y, z), dp;}\cr |
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103 | \longentry qring {\it name\/} = {\it ideal\/};& |
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104 | declare quotient ring {\it name\/} of the |
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105 | current base\-ring with respect to {\it |
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106 | ideal\/}. {\it ideal\/} has to be a standard |
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107 | basis. Make {\it name\/} the new basering.\cr |
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108 | \subsec{Examples of available {\bit basefields\/}:} |
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109 | 0& the rational numbers\cr |
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110 | \it p& the finite field $Z_p$ with {\it p\/} |
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111 | elements,\par |
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112 | $2 \le p \le 2147483629$ a prime\cr |
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113 | ({\it p\/}\^{}{\it n}, {\it gen\/})& |
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114 | the finite field with $p^n$ elements, {\it p\/} |
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115 | a prime and\par $4 \le p^n \le 32671$. The name |
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116 | {\it gen\/} refers to some generator of the |
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117 | cyclic group of unities.\cr |
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118 | ({\it p}, {\it alpha\/})& algebraic extension of $Q$ or $Z_p$ ($p = |
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119 | 0$ or as above) by {\it alpha}. The minpoly |
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120 | $\mu_{\hbox{\tit alpha}}$ for {\it alpha\/} has |
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121 | to be specified with an assignment to {\tt |
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122 | minpoly} (\eg {\tt minpoly=a\^{}2+1;}, for |
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123 | $\hbox{\it alpha}=\hbox{\tt a}$). {\it |
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124 | alpha\/} has to be a name.\cr |
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125 | ({\it p}, $t_1$, $\ldots$)& transcendental extension of $Q$ or $Z_p$ |
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126 | ($p = 0$ or as above) by~$t_i$. The $t_i$ have |
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127 | to be names.\cr |
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128 | real,{\it len}& the real numbers represented by long floating |
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129 | point numbers of length{\it len}\cr |
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130 | \endsec |
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131 | |
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132 | \sec Term orderings() |
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133 | \sectext |
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134 | An {\it ordering\/} as referred to in the ring declaration may either be |
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135 | a global, local, or matrix ordering or a list of these resulting in a |
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136 | pro\-duct ordering. The list may include extra weight vectors and may be |
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137 | preceded or followed by a module ordering specification.\cr |
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138 | \subsec{Global orderings} |
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139 | lp& lexicographical ordering\cr |
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140 | dp& degree reverse lexicographical ordering\cr |
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141 | Dp& degree lexicographical ordering\cr |
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142 | wp($w_1$, $\ldots$)& weighted reverse lexicographical ordering\cr |
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143 | Wp($w_1$, $\ldots$)& weighted lexicographical ordering\cr |
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144 | & The $w_i$ have to be positive integers.\cr |
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145 | \subsec{Local orderings} |
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146 | ls& negative lexicographical ordering\cr |
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147 | ds& negative degree reverse lexicographical ordering\cr |
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148 | Ds& negative degree lexicographical ordering\cr |
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149 | ws($w_1$, $\ldots$)& general weighted reverse lexicographical ordering\cr |
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150 | Ws($w_1$, $\ldots$)& general weighted lexicographical ordering\cr |
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151 | & $w_1$ has to be a non-zero integer, every other |
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152 | $w_i$ may be any integer\cr |
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153 | \subsec{Matrix orderings} |
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154 | \longentry M($m_{11}$, $m_{12}$, $\ldots$, $m_{nn}$)& |
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155 | {\it m\/} has to be an invertible matrix with |
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156 | integer coeffi\-cients. Coefficients have to be |
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157 | specified row-wise.\cr |
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158 | \subsec{Product orderings} |
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159 | \longentry($o_1$\opt{($k_1$)}, $o_2$\opt{($k_2$)}, $\ldots$, $o_n$\opt{($k_n$)})& |
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160 | the $o_i$ have to be any of the above orderings. |
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161 | {\tt lp}, {\tt dp}, {\tt Dp}, {\tt ls}, {\tt |
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162 | ds}, {\tt Ds} may be followed by an integer |
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163 | expression $k_i$ in parentheses specifying the |
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164 | number of variables $o_i$ refers to (\eg {\tt |
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165 | (lp(3), dp(2))}).\cr |
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166 | \subsec{Extra weight vector} |
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167 | a($w_1$, $\ldots$)& any of the above degree orderings may be |
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168 | preceded by an extra weight vector\cr |
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169 | \noalign{\eject} |
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170 | \subsec{Module orderings} |
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171 | ({\it c}, $o_1$, $\ldots$)& sort by components first\cr |
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172 | ($o_1$, $\ldots$, {\it c\/})& sort by variables first\cr |
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173 | & $o_i$ may be any of the above orderings or an |
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174 | extra weight vector, {\it c\/} may be one of |
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175 | {\tt C} or {\tt c}:\cr |
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176 | C& sort generators in ascending order (\ie {\tt |
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177 | gen({\it i\/})} $<$ {\tt gen({\it j\/})} iff $i < |
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178 | j$)\cr |
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179 | c& sort generators in descending order\cr |
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180 | \endsec |
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181 | |
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182 | \sec Data types(1.5cm) |
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183 | \sectext |
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184 | Examples of ring-independent types:\cr |
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185 | \entryskip |
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186 | \longitem |
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187 | int i1 = 101; int i2 = 13 div 3;\cr |
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188 | \entryskip |
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189 | \longitem |
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190 | intvec iv = 13 div 3, -4, i1;\cr |
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191 | \entryskip |
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192 | \longentry |
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193 | intmat im[2][2] = 13 div 3, -4, i1;& |
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194 | a $2\times 2$ matrix. Entries are filled |
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195 | row-wise, missing entries are set to zero, extra |
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196 | entries are ignored. vector/matrix elements are |
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197 | accessed using the {\tt[$\ldots$]} operator, |
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198 | where the first element has index one (\eg {\tt |
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199 | iv[3]; im[1, 2];}).\cr |
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200 | \entryskip |
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201 | \longitem |
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202 | string s1 = "a quote \char"5C " and a backslash \char"5C \char"5C";\par |
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203 | string s2 = "con" + "catenation";\cr |
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204 | \entryskip |
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205 | \sectext |
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206 | Basering in the following is {\tt ring r = 0, (x, y, z, mu, nu), dp;}\cr |
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207 | \entryskip |
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208 | \longitem |
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209 | number n = 5/3;\cr |
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210 | \entryskip |
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211 | \longentry |
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212 | poly p(1) = 3/4x3yz4+2xy2;\par |
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213 | poly p(2) = (5/3)*mu\^{}2*nu\^{}3+n*yz2;& |
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214 | {\tt p(1)} equals $3/4x^3yz^4+2xy^2$. Short |
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215 | format of mono\-mials is valid for one-character |
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216 | ring variables only.\cr |
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217 | \entryskip |
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218 | \longentry |
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219 | ideal i = p(1..2), x+y;& note the use of indexed names\cr |
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220 | \entryskip |
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221 | \longentry |
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222 | vector v = [p(1), p(2), x+y];\par |
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223 | vector w = 2*p(1)*gen(6)+n*nu*gen(1);& |
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224 | vectors may be written in brackets ({\tt |
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225 | [$\ldots$]}) or expressed as linear |
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226 | combinations of the canonical generators {\tt |
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227 | gen({\it i\/})}\cr |
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228 | \entryskip |
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229 | \longitem |
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230 | module mo = v, w, x+y*gen(1);\cr |
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231 | \entryskip |
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232 | \longitem |
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233 | resolution r = sres(std(mo), 0);\cr |
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234 | \entryskip |
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235 | \longentry |
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236 | matrix ma[2][2] = 5/3, p(1), 101;& |
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237 | the rules for declaring, filling, and accessing |
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238 | integer matrices apply to types {\tt matrix} |
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239 | and {\tt vector}, too\cr |
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240 | \entryskip |
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241 | \longentry |
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242 | list l = iv, v, p(1..2), mo;& |
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243 | lists may collect objects of any type. They are |
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244 | ring-dependent iff one of the entries is.\cr |
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245 | \entryskip |
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246 | \longentry |
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247 | def d = read("MPfile:r example.mp");& |
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248 | a name of type {\tt def} inherits the type of |
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249 | the object assigned first to it. Useful if the |
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250 | actual type of an object is unknown.\cr |
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251 | \endsec |
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252 | |
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253 | \sec Monitoring and debugging tools() |
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254 | timer = 1;& print time used for commands to execute\cr |
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255 | \longentry int t = timer; {\it command\/}; \rep; timer-t;& |
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256 | print time used for {\it commands\/} to execute\cr |
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257 | memory(1);& print number of bytes allocated from system\cr |
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258 | option(prot);& show algorithm protocol\cr |
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259 | option(mem);& show algorithm memory usage\cr |
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260 | \entryskip |
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261 | TRACE = 1;& print protocol on execution of procedures\cr |
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262 | listvar(all);& list all (user-)defined names\cr |
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263 | \longentry listvar({\it ringname\/});& |
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264 | list all names belonging to {\it ringname}\cr |
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265 | \endsec |
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266 | |
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267 | \eject |
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268 | |
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269 | \sec Options() |
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270 | option();& show current option settings\cr |
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271 | \longentry option($option_1$, no$option_2$, $\ldots$);& |
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272 | switch $option_1$ on and $option_2$ off, resp.\cr |
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273 | option(none);& reset all options to default values\cr |
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274 | \sectext |
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275 | Type {\tt help option;} for a list of all options.\cr |
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276 | \subsec{Monitoring} |
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277 | debugLib& show loading of procedures from libraries\cr |
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278 | mem& show algorithm memory usage\cr |
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279 | prot& show algorithm protocol\cr |
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280 | \subsec{Standard bases} |
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281 | fastHC& try to find highest corner as fast as possible\cr |
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282 | intStrategy& avoid divisions\cr |
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283 | morePairs& create additional pairs\cr |
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284 | notSugar& disable sugar strategy\cr |
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285 | redSB& compute reduced standard bases\cr |
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286 | redTail& reduce tails\cr |
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287 | sugarCrit& use sugar criteria\cr |
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288 | weightM& automatically compute weights\cr |
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289 | \subsec{Resolutions} |
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290 | minRes& do additional minimizing\cr |
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291 | notRegularity& disable regularity bound\cr |
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292 | \subsec{Miscellany} |
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293 | returnSB& let some functions return standard bases\cr |
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294 | \endsec |
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295 | |
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296 | \sec System variables() |
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297 | \sectext |
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298 | Type {\tt help System variables;} for a list of all system variables.\cr |
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299 | \subsec{Standard bases} |
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300 | degBound& stop if (weighted) total degree exceeds {\tt |
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301 | degBound}\cr |
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302 | multBound& stop if multiplicity gets smaller than {\tt |
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303 | multBound}\cr |
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304 | noether& cut off all monomials above monomial {\tt |
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305 | noether}\cr |
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306 | \subsec{Miscellany} |
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307 | basering& current basering\cr |
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308 | minpoly& minimal polynomial for algebraic extensions\cr |
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309 | short& do not print monomials in short format if zero\cr |
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310 | timer& on assignment of a non-zero value show time |
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311 | used for execution of executed commands. On |
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312 | evaluation, return system time in seconds used |
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313 | by \Singular\ since start\cr |
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314 | TRACE& print information on procedures being executed |
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315 | if larger than one\cr |
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316 | \endsec |
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317 | |
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318 | \sec Input and output() |
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319 | < "{\it filename\/}";& load and execute {\it filename\/}\cr |
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320 | \longentry write("{\it filename\/}", {\it expression}, \rep)& |
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321 | write {\it expressions\/} to ASCII file {\it |
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322 | filename}\cr |
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323 | \longentry read("{\it filename\/}");& |
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324 | read ASCII file {\it filename\/} and return |
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325 | content as a string. See also example below.\cr |
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326 | \longentry |
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327 | dump("MPfile: {\it filename\/}");\par |
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328 | getdump("MPfile: {\it filename\/}");& |
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329 | dump current state of {\sc Singular} to {\it |
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330 | filename} and retrieve it, resp.\cr |
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331 | \entryskip |
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332 | \sectext |
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333 | An example how to write one single expression (in this case the |
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334 | ideal {\tt i}) to a file and read it back from there: |
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335 | |
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336 | {\tt |
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337 | write("i.save", i);\par |
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338 | execute("ideal i=" + read("i.save") + ";"); |
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339 | }\cr |
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340 | \endsec |
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341 | |
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342 | \eject |
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343 | |
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344 | \sec Libraries() |
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345 | LIB "{\it library\/}";& load {\it library}\cr |
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346 | help {\it library\/};& show help on {\it library}\cr |
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347 | help all.lib;& show list of all libraries\cr |
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348 | \endsec |
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349 | |
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350 | \sec Mapping(1.5cm) |
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351 | \longentry map {\it name\/} = {\it ringname}, {\it ideal\/};& |
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352 | declare a map {\it name\/} from {\it ringname\/} |
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353 | to current basering. The $i$-th ring variable |
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354 | from {\it ringname\/} is mapped to the $i$-th |
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355 | generator of {\it ideal}.\cr |
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356 | \longentry {\it mapname\/}({\it expression\/})& |
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357 | apply map {\it mapname\/} to {\it expression}\cr |
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358 | \entryskip |
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359 | \sectext |
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360 | Coefficients between rings with different basefields are mapped in the following |
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361 | way (non-canonical maps only):\strut |
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362 | \abovedisplayskip=0pt |
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363 | \belowdisplayskip=0pt |
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364 | $$ |
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365 | \eqalign{Z_p \rightarrow Q& :[i]_p \mapsto i \in [-p/2,p/2] \subset Z\cr |
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366 | Z_p \rightarrow Z_q&:[i]_p \mapsto i \in [-p/2,p/2] \subset Z, i \mapsto [i]_q} |
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367 | $$ |
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368 | \cr |
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369 | \noalign{\vskip -9pt} % dirty trick, gobbles the trailing \strut from \sectext |
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370 | \entryskip |
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371 | \longentry fetch({\it ringname}, {\it name\/})& |
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372 | map from ring {\it ringname\/} to current |
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373 | base\-ring. The rings have to be identical up |
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374 | to names of ring variables\cr |
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375 | \longentry imap({\it ringname}, {\it name\/})& |
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376 | map from subring {\it ringname\/} to current |
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377 | basering\cr |
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378 | \longentry subst({\it expression}, {\it ringvar}, {\it monomial\/})& |
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379 | substitute {\it ringvar\/} by {\it monomial\/} |
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380 | in {\it expression}\cr |
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381 | \endsec |
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382 | |
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383 | \sec Miscellany(1.5cm) |
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384 | \longentry setring({\it ringname\/})& |
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385 | make {\it ringname\/} the current basering\cr |
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386 | \subsec{Data on polynomials} |
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387 | \longentry ord({\it poly\/\alt vector\/})& |
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388 | return (weighted) degree of initial term\cr |
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389 | \longentry deg({\it poly\/\alt vector\/})& |
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390 | return maximal (weighted) degree\cr |
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391 | \longentry |
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392 | size({\it ideal\/\alt module\/})\par |
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393 | size({\it poly\/\alt vector\/})\par |
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394 | size({\it string\/\alt intvec\/\alt list\/})& |
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395 | return (1) number of non-zero generators; |
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396 | (2) number of monomials; (3) length\cr |
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397 | \longentry lead({\it expression\/})& |
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398 | return initial term(s)\cr |
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399 | \subsec{Operations on polynomials} |
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400 | \longentry gcd({\it $\hbox{poly}_1$}, {\it $\hbox{poly}_2$\/})& |
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401 | return greatest common divisor\cr |
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402 | \longentry factorize({\it poly\/}\opt{, {\it int\/}})& |
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403 | return irreducible factors. Return constant |
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404 | factor and multiplicities in dependency on {\it |
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405 | int}.\cr |
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406 | \endsec |
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407 | |
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408 | \sec Differentiation and jets(1.5cm) |
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409 | \longentry |
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410 | diff({\it expression}, {\it ringvar\/})\par |
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411 | diff({\it $\hbox{ideal}_1$}, {\it $\hbox{ideal}_2$\/})& |
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412 | (1) return partial derivation by {\it |
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413 | ringvar\/}; (2) differentiate each elt.\ of {\it |
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414 | $\hbox{ideal}_2$\/} by the differential |
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415 | operators corres\-pon\-ding to the elements of {\it |
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416 | $\hbox{ideal}_1$}\cr |
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417 | \longentry jacob({\it poly\/\alt ideal\/})& |
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418 | return jacobi ideal or matrix, resp.\cr |
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419 | \longentry jet({\it expression}, {\it int\/}\opt{, {\it intvec\/}})& |
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420 | return {\it int\/}-jet of {\it expression}. |
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421 | Return weighted {\it int\/}-jet if {\it |
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422 | intvec\/} is specifified.\cr |
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423 | \endsec |
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424 | |
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425 | \eject |
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426 | |
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427 | \sec Standard bases(1.5cm) |
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428 | \longentry groebner({\it ideal\/\alt module\/}\opt{, {\it int\/}})& |
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429 | compute a standard basis (SB) of {\it ideal\/} |
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430 | resp.\ {\it module\/} using a heuristically |
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431 | chosen method. Delimit com\-pu\-tation time to |
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432 | {\it int\/} seconds.\cr |
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433 | \longentry std({\it ideal\/\alt module\/}\opt{, {\it intvec\/}})& |
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434 | compute a SB. Use first Hilbert series {\it |
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435 | intvec\/} (result from {\tt hilb($\ldots$, 1)}) |
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436 | for Hilbert-driven computation.\cr |
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437 | \longentry stdfglm({\it ideal\/}\opt{, {\it string\/}})& |
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438 | use FGLM algorithm to compute a SB from a SB |
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439 | w.r.t.\ the ``simpler'' ordering {\it string\/} |
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440 | (de\-faults to {\tt dp})\cr |
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441 | \longentry stdhilb({\it ideal\/}\opt{, {\it intvec\/}})& |
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442 | use Hilbert-driven algorithm to compute a SB. |
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443 | If Hil\-bert series {\it intvec\/} is not |
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444 | specified compute it first.\cr |
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445 | \longentry fglm({\it ringname}, {\it idealname\/})& |
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446 | use FGLM algorithm to transform SB {\it |
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447 | idealname\/} from ring {\it ringname\/} to a SB |
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448 | w.r.t.\ the ordering of the current basering\cr |
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449 | \longentry reduce({\it expression}, {\it ideal\/\alt module\/}\opt{, {\it int\/}})& |
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450 | reduce {\it expression\/} w.r.t.\ second |
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451 | argument which should be a SB. Use lazy |
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452 | reduction if {\it int\/} equals one.\cr |
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453 | \endsec |
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454 | |
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455 | \sec Computation of invariants(1.5cm) |
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456 | \sectext |
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457 | Most of the results are meaningful only if the input ideal or module is |
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458 | represented by a standard basis.\cr |
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459 | \longentry degree({\it ideal\/\alt module\/})& |
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460 | display (Krull) dimension, codimension and |
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461 | multiplicity\cr |
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462 | \longentry dim({\it ideal\/\alt module\/})& |
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463 | return (Krull) dimension\cr |
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464 | \longentry hilb({\it ideal\/\alt module\/}\opt{, {\it int\/}})& |
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465 | display first and second Hilbert series with one |
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466 | argument. Return {\it int}-th Hilber series |
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467 | otherwise (${\it \hbox{int}} = 1,2$).\cr |
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468 | \longentry mult({\it ideal\/\alt module\/})& |
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469 | return multiplicity\cr |
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470 | \longentry vdim({\it ideal\/\alt module\/})& |
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471 | return vector space dimension of current |
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472 | basering modulo {\it ideal\/} or {\it module}, |
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473 | resp.\cr |
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474 | \endsec |
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475 | |
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476 | \sec Resolutions(1.5cm) |
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477 | \sectext |
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478 | An integer argument {\it length\/} in the following descriptions specifies the |
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479 | length of the resolution to compute. If {\it length\/} equals zero, the whole |
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480 | resolution is computed.\cr |
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481 | \longentry res({\it ideal\/\alt module}, {\it length\/}\opt{, {\it int\/}})& |
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482 | compute a free resolution (FR) of {\it ideal\/} |
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483 | resp.\ {\it module\/} using a heuristically |
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484 | chosen method. Compute a minimal resolution if |
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485 | a third argument is given.\cr |
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486 | \longentry mres({\it ideal\/\alt module}, {\it length\/})& |
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487 | compute a minimal FR using the standard basis |
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488 | method\cr |
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489 | \longentry lres({\it ideal\/\alt module}, {\it length\/})& |
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490 | compute a FR using LaSacala's method\cr |
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491 | \longentry sres({\it ideal\/\alt module}, {\it length\/})& |
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492 | compute a FR using Schreyer's method\cr |
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493 | \longentry syz({\it ideal\/\alt module\/})& |
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494 | compute the first syzygy\cr |
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495 | \longentry minres({\it resolution\/\alt list\/})& |
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496 | minimize a free resolution\cr |
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497 | \longentry betty({\it resolution\/\alt list\/})& |
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498 | compute the graded Betti numbers of a module |
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499 | represented by a resolution\cr |
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500 | \endsec |
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501 | |
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502 | \bye |
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