[1] | 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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| 8 | <TITLE>The Coefficient Ring package</TITLE>
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| 33 | <IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents_motif.gif"></A>
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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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| 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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