1 | <!--Converted with LaTeX2HTML 97.1 (release) (July 13th, 1997)
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2 | by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds
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3 | * revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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4 | * with significant contributions from:
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5 | Jens Lippman, Marek Rouchal, Martin Wilck and others -->
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6 | <HTML>
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7 | <HEAD>
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8 | <TITLE>The Coefficient Ring package</TITLE>
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34 | <BR>
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35 | <B> Next:</B> <A NAME="tex2html918"
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36 | HREF="node6.html">The Division Package</A>
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37 | <B> Up:</B> <A NAME="tex2html915"
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38 | HREF="manual.html">CGBLisp User Guide and</A>
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39 | <B> Previous:</B> <A NAME="tex2html909"
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40 | HREF="node4.html">The Comprehensive Gröbner basis</A>
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41 | <BR>
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42 | <BR>
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43 | <!--End of Navigation Panel-->
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44 | <!--Table of Child-Links-->
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45 | <A NAME="CHILD_LINKS"><strong>Subsections</strong></A>
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46 | <UL>
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47 | <LI><A NAME="tex2html919"
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48 | HREF="node5.html#SECTION00050010000000000000">
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49 | <I>ring</I></A>
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50 | <LI><A NAME="tex2html920"
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51 | HREF="node5.html#SECTION00050020000000000000">
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52 | <I>*ring<MATH CLASS="INLINE">
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53 | -
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54 | </MATH>of<MATH CLASS="INLINE">
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55 | -
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56 | </MATH>integers*</I></A>
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57 | <LI><A NAME="tex2html921"
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58 | HREF="node5.html#SECTION00050030000000000000">
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59 | <I>*field<MATH CLASS="INLINE">
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60 | -
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61 | </MATH>of<MATH CLASS="INLINE">
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62 | -
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63 | </MATH>rationals*</I></A>
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64 | <LI><A NAME="tex2html922"
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65 | HREF="node5.html#SECTION00050040000000000000">
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66 | <I>field<MATH CLASS="INLINE">
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67 | -
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68 | </MATH>modulo<MATH CLASS="INLINE">
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69 | -
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70 | </MATH>prime</I></A>
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71 | <LI><A NAME="tex2html923"
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72 | HREF="node5.html#SECTION00050050000000000000">
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73 | <I>*coefficient<MATH CLASS="INLINE">
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74 | -
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75 | </MATH>ring*</I></A>
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76 | </UL>
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77 | <!--End of Table of Child-Links-->
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78 | <HR>
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79 | <H1><A NAME="SECTION00050000000000000000">
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80 | The Coefficient Ring package</A>
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81 | </H1>
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82 | <H4><A NAME="SECTION00050010000000000000">
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83 | <I>ring</I></A>
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84 | </H4>
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85 | <P><IMG WIDTH="565" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
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86 | SRC="img109.gif"
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87 | ALT="$\textstyle\parbox{\pboxargslen}{\em + $-$\space * / gcd lcm zerop unit length signum numerator denominator \/}$"> [<EM>STRUCTURE</EM>]
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88 | <BLOCKQUOTE>
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89 | The structure whose slots are bound to functions
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90 | performing usual ring operations. In addition to usual arithmetical
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91 | operations, bindings for other common operations
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92 | which increase efficiency of Grobner basis calculations are also
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93 | included. They are as follows:
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94 | GCD <MATH CLASS="INLINE">
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95 | -
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96 | </MATH> greatest common divisor;
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97 | LCM <MATH CLASS="INLINE">
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98 | -
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99 | </MATH> least common multiple;
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100 | ZEROP <MATH CLASS="INLINE">
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101 | -
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102 | </MATH> test whether an element is zero;
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103 | SIGNUM <MATH CLASS="INLINE">
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104 | -
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105 | </MATH> the sign of a ring element (+1, <MATH CLASS="INLINE">
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106 | -
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107 | </MATH>1 or zero);
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108 | UNIT <MATH CLASS="INLINE">
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109 | -
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110 | </MATH> the unit of the ring;
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111 | NUMERATOR <MATH CLASS="INLINE">
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112 | -
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113 | </MATH> the numerator, if a ring of fractions
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114 | DENOMINATOR <MATH CLASS="INLINE">
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115 | -
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116 | </MATH> the denominator, if a ring of fractions
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117 | LENGTH <MATH CLASS="INLINE">
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118 | -
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119 | </MATH> an integer giving the approximate length
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120 | of the representation; for example, for integers
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121 | its default binding is #'integer<MATH CLASS="INLINE">
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122 | -
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123 | </MATH>length;</BLOCKQUOTE><H4><A NAME="SECTION00050020000000000000">
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124 | <I>*ring<MATH CLASS="INLINE">
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125 | -
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126 | </MATH>of<MATH CLASS="INLINE">
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127 | -
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128 | </MATH>integers*</I></A>
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129 | </H4>
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130 | <P><IMG WIDTH="482" HEIGHT="90" ALIGN="MIDDLE" BORDER="0"
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131 | SRC="img110.gif"
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132 | ALT="$\textstyle\parbox{\pboxargslen}{\em (make$-$ring :+ \char93 '+ :$-$\space \char...
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133 | ...$-$length :numerator \char93 'numerator
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134 | :denominator \char93 'denominator) \/}$"> [<EM>VARIABLE</EM>]
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135 | <BLOCKQUOTE>
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136 | Operations in the ring of integers.</BLOCKQUOTE><H4><A NAME="SECTION00050030000000000000">
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137 | <I>*field<MATH CLASS="INLINE">
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138 | -
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139 | </MATH>of<MATH CLASS="INLINE">
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140 | -
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141 | </MATH>rationals*</I></A>
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142 | </H4>
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143 | <P><IMG WIDTH="475" HEIGHT="110" ALIGN="MIDDLE" BORDER="0"
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144 | SRC="img111.gif"
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145 | ALT="$\textstyle\parbox{\pboxargslen}{\em (make$-$ring :+ \char93 '+ :$-$\space \char...
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146 | ...or x))))
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147 | :numerator \char93 'numerator :denominator
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148 | \char93 'denominator) \/}$"> [<EM>VARIABLE</EM>]
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149 | <BLOCKQUOTE>
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150 | Operations on the field of rational numbers.</BLOCKQUOTE><H4><A NAME="SECTION00050040000000000000">
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151 | <I>field<MATH CLASS="INLINE">
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152 | -
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153 | </MATH>modulo<MATH CLASS="INLINE">
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154 | -
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155 | </MATH>prime</I></A>
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156 | </H4>
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157 | <P><IMG WIDTH="467" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
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158 | SRC="img112.gif"
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159 | ALT="$\textstyle\parbox{\pboxargslen}{\em modulus \/}$"> [<EM>FUNCTION</EM>]
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160 | <BLOCKQUOTE>
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161 | Return a RING structure with operations bound
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162 | to the arithmetical operations modulo MODULUS, which
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163 | should be a prime.</BLOCKQUOTE><H4><A NAME="SECTION00050050000000000000">
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164 | <I>*coefficient<MATH CLASS="INLINE">
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165 | -
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166 | </MATH>ring*</I></A>
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167 | </H4>
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168 | <P><IMG WIDTH="492" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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169 | SRC="img113.gif"
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170 | ALT="$\textstyle\parbox{\pboxargslen}{\em *ring$-$of$-$integers* \/}$"> [<EM>VARIABLE</EM>]
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171 | <BLOCKQUOTE>
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172 | The default RING structure, used in most operations
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173 | on the coefficients of polynomials. It should be carefully
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174 | set if rings other than the default ring is used.</BLOCKQUOTE><HR>
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188 | <BR>
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189 | <B> Next:</B> <A NAME="tex2html918"
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190 | HREF="node6.html">The Division Package</A>
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191 | <B> Up:</B> <A NAME="tex2html915"
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192 | HREF="manual.html">CGBLisp User Guide and</A>
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193 | <B> Previous:</B> <A NAME="tex2html909"
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194 | HREF="node4.html">The Comprehensive Gröbner basis</A>
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195 | <!--End of Navigation Panel-->
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196 | <ADDRESS>
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197 | <I>Marek Rychlik</I>
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198 | <BR><I>3/21/1998</I>
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199 | </ADDRESS>
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