[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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| 7 | <HEAD>
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| 8 | <TITLE>The Gröbner Basis package</TITLE>
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| 9 | <META NAME="description" CONTENT="The Gröbner Basis package">
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| 10 | <META NAME="keywords" CONTENT="manual">
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| 19 | </HEAD>
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| 20 | <BODY bgcolor="#ffffff">
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| 21 | <!--Navigation Panel-->
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| 22 | <A NAME="tex2html742"
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| 23 | HREF="node3.html">
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| 29 | HREF="node1.html">
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| 32 | HREF="node1.html">
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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="tex2html743"
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| 36 | HREF="node3.html">The String Interface to</A>
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| 37 | <B> Up:</B> <A NAME="tex2html740"
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| 38 | HREF="manual.html">CGBLisp User Guide and</A>
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| 39 | <B> Previous:</B> <A NAME="tex2html734"
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| 40 | HREF="node1.html">Contents</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="tex2html744"
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| 48 | HREF="node2.html#SECTION00020010000000000000">
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| 49 | <I>*grobner<MATH CLASS="INLINE">
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| 50 | -
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| 51 | </MATH>debug*</I></A>
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| 52 | <LI><A NAME="tex2html745"
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| 53 | HREF="node2.html#SECTION00020020000000000000">
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| 54 | <I>*buchberger<MATH CLASS="INLINE">
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| 55 | -
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| 56 | </MATH>merge<MATH CLASS="INLINE">
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| 57 | -
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| 58 | </MATH>pairs*</I></A>
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| 59 | <LI><A NAME="tex2html746"
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| 60 | HREF="node2.html#SECTION00020030000000000000">
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| 61 | <I>*gebauer<MATH CLASS="INLINE">
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| 62 | -
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| 63 | </MATH>moeller<MATH CLASS="INLINE">
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| 64 | -
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| 65 | </MATH>merge<MATH CLASS="INLINE">
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| 66 | -
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| 67 | </MATH>pairs*</I></A>
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| 68 | <LI><A NAME="tex2html747"
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| 69 | HREF="node2.html#SECTION00020040000000000000">
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| 70 | <I>*grobner<MATH CLASS="INLINE">
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| 71 | -
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| 72 | </MATH>function*</I></A>
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| 73 | <LI><A NAME="tex2html748"
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| 74 | HREF="node2.html#SECTION00020050000000000000">
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| 75 | <I>select<MATH CLASS="INLINE">
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| 76 | -
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| 77 | </MATH>grobner<MATH CLASS="INLINE">
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| 78 | -
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| 79 | </MATH>algorithm</I></A>
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| 80 | <LI><A NAME="tex2html749"
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| 81 | HREF="node2.html#SECTION00020060000000000000">
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| 82 | <I>grobner</I></A>
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| 83 | <LI><A NAME="tex2html750"
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| 84 | HREF="node2.html#SECTION00020070000000000000">
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| 85 | <I>debug<MATH CLASS="INLINE">
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| 86 | -
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| 87 | </MATH>cgb</I></A>
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| 88 | <LI><A NAME="tex2html751"
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| 89 | HREF="node2.html#SECTION00020080000000000000">
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| 90 | <I>spoly</I></A>
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| 91 | <LI><A NAME="tex2html752"
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| 92 | HREF="node2.html#SECTION00020090000000000000">
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| 93 | <I>grobner<MATH CLASS="INLINE">
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| 94 | -
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| 95 | </MATH>primitive<MATH CLASS="INLINE">
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| 96 | -
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| 97 | </MATH>part</I></A>
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| 98 | <LI><A NAME="tex2html753"
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| 99 | HREF="node2.html#SECTION000200100000000000000">
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| 100 | <I>grobner<MATH CLASS="INLINE">
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| 101 | -
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| 102 | </MATH>content</I></A>
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| 103 | <LI><A NAME="tex2html754"
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| 104 | HREF="node2.html#SECTION000200110000000000000">
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| 105 | <I>normal<MATH CLASS="INLINE">
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| 106 | -
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| 107 | </MATH>form</I></A>
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| 108 | <LI><A NAME="tex2html755"
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| 109 | HREF="node2.html#SECTION000200120000000000000">
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| 110 | <I>buchberger</I></A>
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| 111 | <LI><A NAME="tex2html756"
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| 112 | HREF="node2.html#SECTION000200130000000000000">
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| 113 | <I>grobner<MATH CLASS="INLINE">
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| 114 | -
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| 115 | </MATH>op</I></A>
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| 116 | <LI><A NAME="tex2html757"
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| 117 | HREF="node2.html#SECTION000200140000000000000">
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| 118 | <I>buchberger<MATH CLASS="INLINE">
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| 119 | -
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| 120 | </MATH>sort<MATH CLASS="INLINE">
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| 121 | -
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| 122 | </MATH>pairs</I></A>
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| 123 | <LI><A NAME="tex2html758"
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| 124 | HREF="node2.html#SECTION000200150000000000000">
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| 125 | <I>mock<MATH CLASS="INLINE">
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| 126 | -
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| 127 | </MATH>spoly</I></A>
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| 128 | <LI><A NAME="tex2html759"
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| 129 | HREF="node2.html#SECTION000200160000000000000">
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| 130 | <I>buchberger<MATH CLASS="INLINE">
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| 131 | -
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| 132 | </MATH>merge<MATH CLASS="INLINE">
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| 133 | -
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| 134 | </MATH>pairs<MATH CLASS="INLINE">
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| 135 | -
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| 136 | </MATH>use<MATH CLASS="INLINE">
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| 137 | -
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| 138 | </MATH>mock<MATH CLASS="INLINE">
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| 139 | -
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| 140 | </MATH>spoly</I></A>
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| 141 | <LI><A NAME="tex2html760"
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| 142 | HREF="node2.html#SECTION000200170000000000000">
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| 143 | <I>buchberger<MATH CLASS="INLINE">
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| 144 | -
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| 145 | </MATH>merge<MATH CLASS="INLINE">
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| 146 | -
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| 147 | </MATH>pairs<MATH CLASS="INLINE">
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| 148 | -
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| 149 | </MATH>smallest<MATH CLASS="INLINE">
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| 150 | -
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| 151 | </MATH>lcm</I></A>
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| 152 | <LI><A NAME="tex2html761"
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| 153 | HREF="node2.html#SECTION000200180000000000000">
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| 154 | <I>buchberger<MATH CLASS="INLINE">
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| 155 | -
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| 156 | </MATH>merge<MATH CLASS="INLINE">
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| 157 | -
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| 158 | </MATH>pairs<MATH CLASS="INLINE">
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| 159 | -
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| 160 | </MATH>use<MATH CLASS="INLINE">
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| 161 | -
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| 162 | </MATH>smallest<MATH CLASS="INLINE">
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| 163 | -
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| 164 | </MATH>degree</I></A>
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| 165 | <LI><A NAME="tex2html762"
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| 166 | HREF="node2.html#SECTION000200190000000000000">
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| 167 | <I>buchberger<MATH CLASS="INLINE">
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| 168 | -
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| 169 | </MATH>merge<MATH CLASS="INLINE">
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| 170 | -
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| 171 | </MATH>pairs<MATH CLASS="INLINE">
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| 172 | -
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| 173 | </MATH>use<MATH CLASS="INLINE">
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| 174 | -
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| 175 | </MATH>smallest<MATH CLASS="INLINE">
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| 176 | -
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| 177 | </MATH>length</I></A>
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| 178 | <LI><A NAME="tex2html763"
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| 179 | HREF="node2.html#SECTION000200200000000000000">
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| 180 | <I>buchberger<MATH CLASS="INLINE">
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| 181 | -
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| 182 | </MATH>merge<MATH CLASS="INLINE">
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| 183 | -
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| 184 | </MATH>pairs<MATH CLASS="INLINE">
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| 185 | -
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| 186 | </MATH>use<MATH CLASS="INLINE">
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| 187 | -
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| 188 | </MATH>smallest<MATH CLASS="INLINE">
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| 189 | -
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| 190 | </MATH>coefficient<MATH CLASS="INLINE">
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| 191 | -
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| 192 | </MATH>length</I></A>
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| 193 | <LI><A NAME="tex2html764"
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| 194 | HREF="node2.html#SECTION000200210000000000000">
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| 195 | <I>buchberger<MATH CLASS="INLINE">
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| 196 | -
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| 197 | </MATH>set<MATH CLASS="INLINE">
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| 198 | -
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| 199 | </MATH>pair<MATH CLASS="INLINE">
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| 200 | -
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| 201 | </MATH>heuristic</I></A>
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| 202 | <LI><A NAME="tex2html765"
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| 203 | HREF="node2.html#SECTION000200220000000000000">
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| 204 | <I>criterion<MATH CLASS="INLINE">
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| 205 | -
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| 206 | </MATH>1</I></A>
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| 207 | <LI><A NAME="tex2html766"
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| 208 | HREF="node2.html#SECTION000200230000000000000">
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| 209 | <I>criterion<MATH CLASS="INLINE">
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| 210 | -
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| 211 | </MATH>2</I></A>
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| 212 | <LI><A NAME="tex2html767"
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| 213 | HREF="node2.html#SECTION000200240000000000000">
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| 214 | <I>normalize<MATH CLASS="INLINE">
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| 215 | -
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| 216 | </MATH>poly</I></A>
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| 217 | <LI><A NAME="tex2html768"
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| 218 | HREF="node2.html#SECTION000200250000000000000">
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| 219 | <I>normalize<MATH CLASS="INLINE">
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| 220 | -
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| 221 | </MATH>basis</I></A>
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| 222 | <LI><A NAME="tex2html769"
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| 223 | HREF="node2.html#SECTION000200260000000000000">
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| 224 | <I>reduction</I></A>
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| 225 | <LI><A NAME="tex2html770"
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| 226 | HREF="node2.html#SECTION000200270000000000000">
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| 227 | <I>reduced<MATH CLASS="INLINE">
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| 228 | -
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| 229 | </MATH>grobner</I></A>
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| 230 | <LI><A NAME="tex2html771"
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| 231 | HREF="node2.html#SECTION000200280000000000000">
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| 232 | <I>monom<MATH CLASS="INLINE">
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| 233 | -
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| 234 | </MATH>depends<MATH CLASS="INLINE">
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| 235 | -
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| 236 | </MATH>p</I></A>
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| 237 | <LI><A NAME="tex2html772"
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| 238 | HREF="node2.html#SECTION000200290000000000000">
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| 239 | <I>term<MATH CLASS="INLINE">
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| 240 | -
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| 241 | </MATH>depends<MATH CLASS="INLINE">
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| 242 | -
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| 243 | </MATH>p</I></A>
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| 244 | <LI><A NAME="tex2html773"
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| 245 | HREF="node2.html#SECTION000200300000000000000">
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| 246 | <I>poly<MATH CLASS="INLINE">
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| 247 | -
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| 248 | </MATH>depends<MATH CLASS="INLINE">
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| 249 | -
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| 250 | </MATH>p</I></A>
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| 251 | <LI><A NAME="tex2html774"
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| 252 | HREF="node2.html#SECTION000200310000000000000">
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| 253 | <I>ring<MATH CLASS="INLINE">
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| 254 | -
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| 255 | </MATH>intersection</I></A>
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| 256 | <LI><A NAME="tex2html775"
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| 257 | HREF="node2.html#SECTION000200320000000000000">
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| 258 | <I>elimination<MATH CLASS="INLINE">
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| 259 | -
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| 260 | </MATH>ideal</I></A>
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| 261 | <LI><A NAME="tex2html776"
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| 262 | HREF="node2.html#SECTION000200330000000000000">
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| 263 | <I>ideal<MATH CLASS="INLINE">
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| 264 | -
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| 265 | </MATH>intersection</I></A>
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| 266 | <LI><A NAME="tex2html777"
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| 267 | HREF="node2.html#SECTION000200340000000000000">
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| 268 | <I>poly<MATH CLASS="INLINE">
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| 269 | -
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| 270 | </MATH>contract</I></A>
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| 271 | <LI><A NAME="tex2html778"
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| 272 | HREF="node2.html#SECTION000200350000000000000">
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| 273 | <I>poly<MATH CLASS="INLINE">
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| 274 | -
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| 275 | </MATH>lcm</I></A>
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| 276 | <LI><A NAME="tex2html779"
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| 277 | HREF="node2.html#SECTION000200360000000000000">
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| 278 | <I>grobner<MATH CLASS="INLINE">
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| 279 | -
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| 280 | </MATH>gcd</I></A>
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| 281 | <LI><A NAME="tex2html780"
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| 282 | HREF="node2.html#SECTION000200370000000000000">
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| 283 | <I>grobner<MATH CLASS="INLINE">
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| 284 | -
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| 285 | </MATH>equal</I></A>
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| 286 | <LI><A NAME="tex2html781"
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| 287 | HREF="node2.html#SECTION000200380000000000000">
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| 288 | <I>grobner<MATH CLASS="INLINE">
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| 289 | -
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| 290 | </MATH>subsetp</I></A>
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| 291 | <LI><A NAME="tex2html782"
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| 292 | HREF="node2.html#SECTION000200390000000000000">
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| 293 | <I>grobner<MATH CLASS="INLINE">
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| 294 | -
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| 295 | </MATH>member</I></A>
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| 296 | <LI><A NAME="tex2html783"
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| 297 | HREF="node2.html#SECTION000200400000000000000">
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| 298 | <I>ideal<MATH CLASS="INLINE">
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| 299 | -
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| 300 | </MATH>equal</I></A>
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| 301 | <LI><A NAME="tex2html784"
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| 302 | HREF="node2.html#SECTION000200410000000000000">
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| 303 | <I>ideal<MATH CLASS="INLINE">
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| 304 | -
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| 305 | </MATH>subsetp</I></A>
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| 306 | <LI><A NAME="tex2html785"
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| 307 | HREF="node2.html#SECTION000200420000000000000">
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| 308 | <I>ideal<MATH CLASS="INLINE">
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| 309 | -
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| 310 | </MATH>member</I></A>
|
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| 311 | <LI><A NAME="tex2html786"
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| 312 | HREF="node2.html#SECTION000200430000000000000">
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| 313 | <I>ideal<MATH CLASS="INLINE">
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| 314 | -
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| 315 | </MATH>saturation<MATH CLASS="INLINE">
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| 316 | -
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| 317 | </MATH>1</I></A>
|
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| 318 | <LI><A NAME="tex2html787"
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| 319 | HREF="node2.html#SECTION000200440000000000000">
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| 320 | <I>add<MATH CLASS="INLINE">
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| 321 | -
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| 322 | </MATH>variables</I></A>
|
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| 323 | <LI><A NAME="tex2html788"
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| 324 | HREF="node2.html#SECTION000200450000000000000">
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| 325 | <I>extend<MATH CLASS="INLINE">
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| 326 | -
|
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| 327 | </MATH>polynomials</I></A>
|
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| 328 | <LI><A NAME="tex2html789"
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| 329 | HREF="node2.html#SECTION000200460000000000000">
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| 330 | <I>saturation<MATH CLASS="INLINE">
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| 331 | -
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| 332 | </MATH>extension</I></A>
|
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| 333 | <LI><A NAME="tex2html790"
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| 334 | HREF="node2.html#SECTION000200470000000000000">
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| 335 | <I>polysaturation<MATH CLASS="INLINE">
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| 336 | -
|
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| 337 | </MATH>extension</I></A>
|
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| 338 | <LI><A NAME="tex2html791"
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| 339 | HREF="node2.html#SECTION000200480000000000000">
|
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| 340 | <I>saturation<MATH CLASS="INLINE">
|
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| 341 | -
|
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| 342 | </MATH>extension<MATH CLASS="INLINE">
|
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| 343 | -
|
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| 344 | </MATH>1</I></A>
|
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| 345 | <LI><A NAME="tex2html792"
|
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| 346 | HREF="node2.html#SECTION000200490000000000000">
|
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| 347 | <I>ideal<MATH CLASS="INLINE">
|
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| 348 | -
|
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| 349 | </MATH>polysaturation<MATH CLASS="INLINE">
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| 350 | -
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| 351 | </MATH>1</I></A>
|
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| 352 | <LI><A NAME="tex2html793"
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| 353 | HREF="node2.html#SECTION000200500000000000000">
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| 354 | <I>ideal<MATH CLASS="INLINE">
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| 355 | -
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| 356 | </MATH>saturation</I></A>
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| 357 | <LI><A NAME="tex2html794"
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| 358 | HREF="node2.html#SECTION000200510000000000000">
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| 359 | <I>ideal<MATH CLASS="INLINE">
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| 360 | -
|
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| 361 | </MATH>polysaturation</I></A>
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| 362 | <LI><A NAME="tex2html795"
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| 363 | HREF="node2.html#SECTION000200520000000000000">
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| 364 | <I>buchberger<MATH CLASS="INLINE">
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| 365 | -
|
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| 366 | </MATH>criterion</I></A>
|
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| 367 | <LI><A NAME="tex2html796"
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| 368 | HREF="node2.html#SECTION000200530000000000000">
|
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| 369 | <I>grobner<MATH CLASS="INLINE">
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| 370 | -
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| 371 | </MATH>test</I></A>
|
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| 372 | <LI><A NAME="tex2html797"
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| 373 | HREF="node2.html#SECTION000200540000000000000">
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| 374 | <I>minimization</I></A>
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| 375 | <LI><A NAME="tex2html798"
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| 376 | HREF="node2.html#SECTION000200550000000000000">
|
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| 377 | <I>add<MATH CLASS="INLINE">
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| 378 | -
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| 379 | </MATH>minimized</I></A>
|
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| 380 | <LI><A NAME="tex2html799"
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| 381 | HREF="node2.html#SECTION000200560000000000000">
|
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| 382 | <I>colon<MATH CLASS="INLINE">
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| 383 | -
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| 384 | </MATH>ideal</I></A>
|
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| 385 | <LI><A NAME="tex2html800"
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| 386 | HREF="node2.html#SECTION000200570000000000000">
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| 387 | <I>colon<MATH CLASS="INLINE">
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| 388 | -
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| 389 | </MATH>ideal<MATH CLASS="INLINE">
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| 390 | -
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| 391 | </MATH>1</I></A>
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| 392 | <LI><A NAME="tex2html801"
|
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| 393 | HREF="node2.html#SECTION000200580000000000000">
|
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| 394 | <I>pseudo<MATH CLASS="INLINE">
|
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| 395 | -
|
---|
| 396 | </MATH>divide</I></A>
|
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| 397 | <LI><A NAME="tex2html802"
|
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| 398 | HREF="node2.html#SECTION000200590000000000000">
|
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| 399 | <I>gebauer<MATH CLASS="INLINE">
|
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| 400 | -
|
---|
| 401 | </MATH>moeller</I></A>
|
---|
| 402 | <LI><A NAME="tex2html803"
|
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| 403 | HREF="node2.html#SECTION000200600000000000000">
|
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| 404 | <I>update</I></A>
|
---|
| 405 | <LI><A NAME="tex2html804"
|
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| 406 | HREF="node2.html#SECTION000200610000000000000">
|
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| 407 | <I>gebauer<MATH CLASS="INLINE">
|
---|
| 408 | -
|
---|
| 409 | </MATH>moeller<MATH CLASS="INLINE">
|
---|
| 410 | -
|
---|
| 411 | </MATH>merge<MATH CLASS="INLINE">
|
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| 412 | -
|
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| 413 | </MATH>pairs<MATH CLASS="INLINE">
|
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| 414 | -
|
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| 415 | </MATH>use<MATH CLASS="INLINE">
|
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| 416 | -
|
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| 417 | </MATH>mock<MATH CLASS="INLINE">
|
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| 418 | -
|
---|
| 419 | </MATH>spoly</I></A>
|
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| 420 | <LI><A NAME="tex2html805"
|
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| 421 | HREF="node2.html#SECTION000200620000000000000">
|
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| 422 | <I>gebauer<MATH CLASS="INLINE">
|
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| 423 | -
|
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| 424 | </MATH>moeller<MATH CLASS="INLINE">
|
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| 425 | -
|
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| 426 | </MATH>merge<MATH CLASS="INLINE">
|
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| 427 | -
|
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| 428 | </MATH>pairs<MATH CLASS="INLINE">
|
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| 429 | -
|
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| 430 | </MATH>smallest<MATH CLASS="INLINE">
|
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| 431 | -
|
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| 432 | </MATH>lcm</I></A>
|
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| 433 | <LI><A NAME="tex2html806"
|
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| 434 | HREF="node2.html#SECTION000200630000000000000">
|
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| 435 | <I>gebauer<MATH CLASS="INLINE">
|
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| 436 | -
|
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| 437 | </MATH>moeller<MATH CLASS="INLINE">
|
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| 438 | -
|
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| 439 | </MATH>merge<MATH CLASS="INLINE">
|
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| 440 | -
|
---|
| 441 | </MATH>pairs<MATH CLASS="INLINE">
|
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| 442 | -
|
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| 443 | </MATH>use<MATH CLASS="INLINE">
|
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| 444 | -
|
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| 445 | </MATH>smallest<MATH CLASS="INLINE">
|
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| 446 | -
|
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| 447 | </MATH>degree</I></A>
|
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| 448 | <LI><A NAME="tex2html807"
|
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| 449 | HREF="node2.html#SECTION000200640000000000000">
|
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| 450 | <I>gebauer<MATH CLASS="INLINE">
|
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| 451 | -
|
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| 452 | </MATH>moeller<MATH CLASS="INLINE">
|
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| 453 | -
|
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| 454 | </MATH>merge<MATH CLASS="INLINE">
|
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| 455 | -
|
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| 456 | </MATH>pairs<MATH CLASS="INLINE">
|
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| 457 | -
|
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| 458 | </MATH>use<MATH CLASS="INLINE">
|
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| 459 | -
|
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| 460 | </MATH>smallest<MATH CLASS="INLINE">
|
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| 461 | -
|
---|
| 462 | </MATH>length</I></A>
|
---|
| 463 | <LI><A NAME="tex2html808"
|
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| 464 | HREF="node2.html#SECTION000200650000000000000">
|
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| 465 | <I>gebauer<MATH CLASS="INLINE">
|
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| 466 | -
|
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| 467 | </MATH>moeller<MATH CLASS="INLINE">
|
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| 468 | -
|
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| 469 | </MATH>merge<MATH CLASS="INLINE">
|
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| 470 | -
|
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| 471 | </MATH>pairs<MATH CLASS="INLINE">
|
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| 472 | -
|
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| 473 | </MATH>use<MATH CLASS="INLINE">
|
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| 474 | -
|
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| 475 | </MATH>smallest<MATH CLASS="INLINE">
|
---|
| 476 | -
|
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| 477 | </MATH>coefficient<MATH CLASS="INLINE">
|
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| 478 | -
|
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| 479 | </MATH>length</I></A>
|
---|
| 480 | <LI><A NAME="tex2html809"
|
---|
| 481 | HREF="node2.html#SECTION000200660000000000000">
|
---|
| 482 | <I>gebauer<MATH CLASS="INLINE">
|
---|
| 483 | -
|
---|
| 484 | </MATH>moeller<MATH CLASS="INLINE">
|
---|
| 485 | -
|
---|
| 486 | </MATH>set<MATH CLASS="INLINE">
|
---|
| 487 | -
|
---|
| 488 | </MATH>pair<MATH CLASS="INLINE">
|
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| 489 | -
|
---|
| 490 | </MATH>heuristic</I></A>
|
---|
| 491 | <LI><A NAME="tex2html810"
|
---|
| 492 | HREF="node2.html#SECTION000200670000000000000">
|
---|
| 493 | <I>spoly<MATH CLASS="INLINE">
|
---|
| 494 | -
|
---|
| 495 | </MATH>sugar</I></A>
|
---|
| 496 | <LI><A NAME="tex2html811"
|
---|
| 497 | HREF="node2.html#SECTION000200680000000000000">
|
---|
| 498 | <I>spoly<MATH CLASS="INLINE">
|
---|
| 499 | -
|
---|
| 500 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 501 | -
|
---|
| 502 | </MATH>sugar</I></A>
|
---|
| 503 | <LI><A NAME="tex2html812"
|
---|
| 504 | HREF="node2.html#SECTION000200690000000000000">
|
---|
| 505 | <I>normal<MATH CLASS="INLINE">
|
---|
| 506 | -
|
---|
| 507 | </MATH>form<MATH CLASS="INLINE">
|
---|
| 508 | -
|
---|
| 509 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 510 | -
|
---|
| 511 | </MATH>sugar</I></A>
|
---|
| 512 | <LI><A NAME="tex2html813"
|
---|
| 513 | HREF="node2.html#SECTION000200700000000000000">
|
---|
| 514 | <I>buchberger<MATH CLASS="INLINE">
|
---|
| 515 | -
|
---|
| 516 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 517 | -
|
---|
| 518 | </MATH>sugar</I></A>
|
---|
| 519 | <LI><A NAME="tex2html814"
|
---|
| 520 | HREF="node2.html#SECTION000200710000000000000">
|
---|
| 521 | <I>buchberger<MATH CLASS="INLINE">
|
---|
| 522 | -
|
---|
| 523 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 524 | -
|
---|
| 525 | </MATH>sugar<MATH CLASS="INLINE">
|
---|
| 526 | -
|
---|
| 527 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 528 | -
|
---|
| 529 | </MATH>pairs</I></A>
|
---|
| 530 | <LI><A NAME="tex2html815"
|
---|
| 531 | HREF="node2.html#SECTION000200720000000000000">
|
---|
| 532 | <I>buchberger<MATH CLASS="INLINE">
|
---|
| 533 | -
|
---|
| 534 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 535 | -
|
---|
| 536 | </MATH>sugar<MATH CLASS="INLINE">
|
---|
| 537 | -
|
---|
| 538 | </MATH>sort<MATH CLASS="INLINE">
|
---|
| 539 | -
|
---|
| 540 | </MATH>pairs</I></A>
|
---|
| 541 | <LI><A NAME="tex2html816"
|
---|
| 542 | HREF="node2.html#SECTION000200730000000000000">
|
---|
| 543 | <I>criterion<MATH CLASS="INLINE">
|
---|
| 544 | -
|
---|
| 545 | </MATH>1<MATH CLASS="INLINE">
|
---|
| 546 | -
|
---|
| 547 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 548 | -
|
---|
| 549 | </MATH>sugar</I></A>
|
---|
| 550 | <LI><A NAME="tex2html817"
|
---|
| 551 | HREF="node2.html#SECTION000200740000000000000">
|
---|
| 552 | <I>criterion<MATH CLASS="INLINE">
|
---|
| 553 | -
|
---|
| 554 | </MATH>2<MATH CLASS="INLINE">
|
---|
| 555 | -
|
---|
| 556 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 557 | -
|
---|
| 558 | </MATH>sugar</I></A>
|
---|
| 559 | <LI><A NAME="tex2html818"
|
---|
| 560 | HREF="node2.html#SECTION000200750000000000000">
|
---|
| 561 | <I>gebauer<MATH CLASS="INLINE">
|
---|
| 562 | -
|
---|
| 563 | </MATH>moeller<MATH CLASS="INLINE">
|
---|
| 564 | -
|
---|
| 565 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 566 | -
|
---|
| 567 | </MATH>sugar</I></A>
|
---|
| 568 | <LI><A NAME="tex2html819"
|
---|
| 569 | HREF="node2.html#SECTION000200760000000000000">
|
---|
| 570 | <I>update<MATH CLASS="INLINE">
|
---|
| 571 | -
|
---|
| 572 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 573 | -
|
---|
| 574 | </MATH>sugar</I></A>
|
---|
| 575 | <LI><A NAME="tex2html820"
|
---|
| 576 | HREF="node2.html#SECTION000200770000000000000">
|
---|
| 577 | <I>gebauer<MATH CLASS="INLINE">
|
---|
| 578 | -
|
---|
| 579 | </MATH>moeller<MATH CLASS="INLINE">
|
---|
| 580 | -
|
---|
| 581 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 582 | -
|
---|
| 583 | </MATH>sugar<MATH CLASS="INLINE">
|
---|
| 584 | -
|
---|
| 585 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 586 | -
|
---|
| 587 | </MATH>pairs</I></A>
|
---|
| 588 | <LI><A NAME="tex2html821"
|
---|
| 589 | HREF="node2.html#SECTION000200780000000000000">
|
---|
| 590 | <I>grobner<MATH CLASS="INLINE">
|
---|
| 591 | -
|
---|
| 592 | </MATH>primitive<MATH CLASS="INLINE">
|
---|
| 593 | -
|
---|
| 594 | </MATH>part<MATH CLASS="INLINE">
|
---|
| 595 | -
|
---|
| 596 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 597 | -
|
---|
| 598 | </MATH>sugar</I></A>
|
---|
| 599 | </UL>
|
---|
| 600 | <!--End of Table of Child-Links-->
|
---|
| 601 | <HR>
|
---|
| 602 | <H1><A NAME="SECTION00020000000000000000">
|
---|
| 603 | The Gröbner Basis package</A>
|
---|
| 604 | </H1>
|
---|
| 605 | <H4><A NAME="SECTION00020010000000000000">
|
---|
| 606 | <I>*grobner<MATH CLASS="INLINE">
|
---|
| 607 | -
|
---|
| 608 | </MATH>debug*</I></A>
|
---|
| 609 | </H4>
|
---|
| 610 | <P><IMG WIDTH="495" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
|
---|
| 611 | SRC="img1.gif"
|
---|
| 612 | ALT="$\textstyle\parbox{\pboxargslen}{\em nil \/}$"> [<EM>VARIABLE</EM>]
|
---|
| 613 | <BLOCKQUOTE>
|
---|
| 614 | If T, produce debugging and tracing output.</BLOCKQUOTE><H4><A NAME="SECTION00020020000000000000">
|
---|
| 615 | <I>*buchberger<MATH CLASS="INLINE">
|
---|
| 616 | -
|
---|
| 617 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 618 | -
|
---|
| 619 | </MATH>pairs*</I></A>
|
---|
| 620 | </H4>
|
---|
| 621 | <P><IMG WIDTH="425" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 622 | SRC="img2.gif"
|
---|
| 623 | ALT="$\textstyle\parbox{\pboxargslen}{\em 'buchberger$-$merge$-$pairs$-$smallest$-$lcm \/}$"> [<EM>VARIABLE</EM>]
|
---|
| 624 | <BLOCKQUOTE>
|
---|
| 625 | A variable holding the predicate used to sort critical pairs
|
---|
| 626 | in the order of decreasing priority for the Buchberger algorithm.</BLOCKQUOTE><H4><A NAME="SECTION00020030000000000000">
|
---|
| 627 | <I>*gebauer<MATH CLASS="INLINE">
|
---|
| 628 | -
|
---|
| 629 | </MATH>moeller<MATH CLASS="INLINE">
|
---|
| 630 | -
|
---|
| 631 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 632 | -
|
---|
| 633 | </MATH>pairs*</I></A>
|
---|
| 634 | </H4>
|
---|
| 635 | <P><IMG WIDTH="384" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 636 | SRC="img3.gif"
|
---|
| 637 | ALT="$\textstyle\parbox{\pboxargslen}{\em 'gebauer$-$moeller$-$merge$-$pairs$-$use$-$mock$-$spoly \/}$"> [<EM>VARIABLE</EM>]
|
---|
| 638 | <BLOCKQUOTE>
|
---|
| 639 | A variable holding the predicate used to sort critical pairs
|
---|
| 640 | in the order of decreasing priority for the Gebauer<MATH CLASS="INLINE">
|
---|
| 641 | -
|
---|
| 642 | </MATH>Moeller
|
---|
| 643 | algorithm </BLOCKQUOTE><H4><A NAME="SECTION00020040000000000000">
|
---|
| 644 | <I>*grobner<MATH CLASS="INLINE">
|
---|
| 645 | -
|
---|
| 646 | </MATH>function*</I></A>
|
---|
| 647 | </H4>
|
---|
| 648 | <P><IMG WIDTH="480" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 649 | SRC="img4.gif"
|
---|
| 650 | ALT="$\textstyle\parbox{\pboxargslen}{\em 'buchberger \/}$"> [<EM>VARIABLE</EM>]
|
---|
| 651 | <BLOCKQUOTE>
|
---|
| 652 | The name of the function used to calculate Grobner basis</BLOCKQUOTE><H4><A NAME="SECTION00020050000000000000">
|
---|
| 653 | <I>select<MATH CLASS="INLINE">
|
---|
| 654 | -
|
---|
| 655 | </MATH>grobner<MATH CLASS="INLINE">
|
---|
| 656 | -
|
---|
| 657 | </MATH>algorithm</I></A>
|
---|
| 658 | </H4>
|
---|
| 659 | <P><IMG WIDTH="429" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 660 | SRC="img5.gif"
|
---|
| 661 | ALT="$\textstyle\parbox{\pboxargslen}{\em algorithm \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 662 | <BLOCKQUOTE>
|
---|
| 663 | Select one of the several implementations of the Grobner basis
|
---|
| 664 | algorithm.</BLOCKQUOTE><H4><A NAME="SECTION00020060000000000000">
|
---|
| 665 | <I>grobner</I></A>
|
---|
| 666 | </H4>
|
---|
| 667 | <P><IMG WIDTH="559" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
|
---|
| 668 | SRC="img6.gif"
|
---|
| 669 | ALT="$\textstyle\parbox{\pboxargslen}{\em f {\sf \&optional} (pred \char93 'lex$\gt$) (start
|
---|
| 670 | 0) (top$-$reduction$-$only
|
---|
| 671 | nil) (ring
|
---|
| 672 | *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 673 | <BLOCKQUOTE>
|
---|
| 674 | Return Grobner basis of an ideal generated by polynomial list F.
|
---|
| 675 | Assume that the monomials of each polynomial are ordered according to
|
---|
| 676 | the admissible monomial order PRED. The result will be a list of
|
---|
| 677 | polynomials ordered as well according to PRED. The polynomials from 0
|
---|
| 678 | to START<MATH CLASS="INLINE">
|
---|
| 679 | -
|
---|
| 680 | </MATH>1 in the list F are assumed to be a Grobner basis, and
|
---|
| 681 | thus certain critical pairs do not have to be calculated. If
|
---|
| 682 | TOP<MATH CLASS="INLINE">
|
---|
| 683 | -
|
---|
| 684 | </MATH>REDUCTION<MATH CLASS="INLINE">
|
---|
| 685 | -
|
---|
| 686 | </MATH>ONLY is not NIL then only top reductions will be
|
---|
| 687 | performed, instead of full division, and thus the returned Grobner
|
---|
| 688 | basis will have its polynomials only top<MATH CLASS="INLINE">
|
---|
| 689 | -
|
---|
| 690 | </MATH>reduced with respect to
|
---|
| 691 | the preceding polynomials. RING should be set to the RING structure
|
---|
| 692 | (see COEFFICIENT<MATH CLASS="INLINE">
|
---|
| 693 | -
|
---|
| 694 | </MATH>RING package) corresponding to the coefficients of
|
---|
| 695 | the polynomials in the list F. This function is currently just an
|
---|
| 696 | interface to several algorithms which fine Grobner bases. Use
|
---|
| 697 | SELECT<MATH CLASS="INLINE">
|
---|
| 698 | -
|
---|
| 699 | </MATH>GROBNER<MATH CLASS="INLINE">
|
---|
| 700 | -
|
---|
| 701 | </MATH>ALGORITHM in order to change the algorithm.</BLOCKQUOTE><H4><A NAME="SECTION00020070000000000000">
|
---|
| 702 | <I>debug<MATH CLASS="INLINE">
|
---|
| 703 | -
|
---|
| 704 | </MATH>cgb</I></A>
|
---|
| 705 | </H4>
|
---|
| 706 | <P><IMG WIDTH="576" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 707 | SRC="img7.gif"
|
---|
| 708 | ALT="$\textstyle\parbox{\pboxargslen}{\em {\sf \&rest} args \/}$"> [<EM>MACRO</EM>]
|
---|
| 709 | <BLOCKQUOTE>
|
---|
| 710 | This macro is used in printing debugging and tracing messages
|
---|
| 711 | and its arguments are analogous to the FORMAT function, except
|
---|
| 712 | that the stream argument is omitted. By default, the output is
|
---|
| 713 | sent to *trace<MATH CLASS="INLINE">
|
---|
| 714 | -
|
---|
| 715 | </MATH>output*, which can be set to produce output
|
---|
| 716 | in a file.</BLOCKQUOTE><H4><A NAME="SECTION00020080000000000000">
|
---|
| 717 | <I>spoly</I></A>
|
---|
| 718 | </H4>
|
---|
| 719 | <P><IMG WIDTH="576" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 720 | SRC="img8.gif"
|
---|
| 721 | ALT="$\textstyle\parbox{\pboxargslen}{\em f g pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 722 | <BLOCKQUOTE>
|
---|
| 723 | Return the S<MATH CLASS="INLINE">
|
---|
| 724 | -
|
---|
| 725 | </MATH>polynomial of F and G, which should
|
---|
| 726 | be two polynomials with terms ordered by PRED and
|
---|
| 727 | coefficients in a ring whose operations are the slots
|
---|
| 728 | of the RING structure.</BLOCKQUOTE><H4><A NAME="SECTION00020090000000000000">
|
---|
| 729 | <I>grobner<MATH CLASS="INLINE">
|
---|
| 730 | -
|
---|
| 731 | </MATH>primitive<MATH CLASS="INLINE">
|
---|
| 732 | -
|
---|
| 733 | </MATH>part</I></A>
|
---|
| 734 | </H4>
|
---|
| 735 | <P><IMG WIDTH="446" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
|
---|
| 736 | SRC="img9.gif"
|
---|
| 737 | ALT="$\textstyle\parbox{\pboxargslen}{\em p ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 738 | <BLOCKQUOTE>
|
---|
| 739 | Divide polynomial P with integer coefficients by gcd of its
|
---|
| 740 | coefficients and return the result. Use the RING structure from the
|
---|
| 741 | COEFFICIENT<MATH CLASS="INLINE">
|
---|
| 742 | -
|
---|
| 743 | </MATH>RING package to perform the operations on the
|
---|
| 744 | coefficients.</BLOCKQUOTE><H4><A NAME="SECTION000200100000000000000">
|
---|
| 745 | <I>grobner<MATH CLASS="INLINE">
|
---|
| 746 | -
|
---|
| 747 | </MATH>content</I></A>
|
---|
| 748 | </H4>
|
---|
| 749 | <P><IMG WIDTH="446" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
|
---|
| 750 | SRC="img9.gif"
|
---|
| 751 | ALT="$\textstyle\parbox{\pboxargslen}{\em p ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 752 | <BLOCKQUOTE>
|
---|
| 753 | Greatest common divisor of the coefficients of the polynomial P. Use
|
---|
| 754 | the RING structure to compute the greatest common divisor.</BLOCKQUOTE><H4><A NAME="SECTION000200110000000000000">
|
---|
| 755 | <I>normal<MATH CLASS="INLINE">
|
---|
| 756 | -
|
---|
| 757 | </MATH>form</I></A>
|
---|
| 758 | </H4>
|
---|
| 759 | <P><IMG WIDTH="520" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 760 | SRC="img10.gif"
|
---|
| 761 | ALT="$\textstyle\parbox{\pboxargslen}{\em f fl pred top$-$reduction$-$only ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 762 | <BLOCKQUOTE>
|
---|
| 763 | Calculate the normal form of the polynomial F with respect to
|
---|
| 764 | the polynomial list FL. Destructive to F; thus, copy<MATH CLASS="INLINE">
|
---|
| 765 | -
|
---|
| 766 | </MATH>tree should be
|
---|
| 767 | used where F needs to be preserved. It returns multiple values.
|
---|
| 768 | The first value is the polynomial R which is reduced (or
|
---|
| 769 | top<MATH CLASS="INLINE">
|
---|
| 770 | -
|
---|
| 771 | </MATH>reduced if TOP<MATH CLASS="INLINE">
|
---|
| 772 | -
|
---|
| 773 | </MATH>REDUCTION<MATH CLASS="INLINE">
|
---|
| 774 | -
|
---|
| 775 | </MATH>ONLY is not NIL). The second value
|
---|
| 776 | is an integer which is the number of reductions actually performed.
|
---|
| 777 | The third value is the coefficient by which F needs to be multiplied
|
---|
| 778 | in order to be able to perform the division without actually having
|
---|
| 779 | to divide in the coefficient ring (a form of pseudo<MATH CLASS="INLINE">
|
---|
| 780 | -
|
---|
| 781 | </MATH>division is
|
---|
| 782 | used). All operations on the coefficients are performed using the
|
---|
| 783 | RING structure from the COEFFICIENT<MATH CLASS="INLINE">
|
---|
| 784 | -
|
---|
| 785 | </MATH>RING package.</BLOCKQUOTE><H4><A NAME="SECTION000200120000000000000">
|
---|
| 786 | <I>buchberger</I></A>
|
---|
| 787 | </H4>
|
---|
| 788 | <P><IMG WIDTH="535" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
| 789 | SRC="img11.gif"
|
---|
| 790 | ALT="$\textstyle\parbox{\pboxargslen}{\em f pred start top$-$reduction$-$only ring {\sf \&aux} (s (1$-$\space (length f))) b m \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 791 | <BLOCKQUOTE>
|
---|
| 792 | An implementation of the Buchberger algorithm. Return Grobner
|
---|
| 793 | basis of the ideal generated by the polynomial list F.
|
---|
| 794 | The terms of each polynomial are ordered by the admissible
|
---|
| 795 | monomial order PRED. Polynomials 0 to START<MATH CLASS="INLINE">
|
---|
| 796 | -
|
---|
| 797 | </MATH>1 are assumed to
|
---|
| 798 | be a Grobner basis already, so that certain critical pairs
|
---|
| 799 | will not be examined. If TOP<MATH CLASS="INLINE">
|
---|
| 800 | -
|
---|
| 801 | </MATH>REDUCTION<MATH CLASS="INLINE">
|
---|
| 802 | -
|
---|
| 803 | </MATH>ONLY set, top reduction
|
---|
| 804 | will be preformed. The RING structure is used to perform operations
|
---|
| 805 | on coefficents (see COEFFICIENT<MATH CLASS="INLINE">
|
---|
| 806 | -
|
---|
| 807 | </MATH>RING package).</BLOCKQUOTE><H4><A NAME="SECTION000200130000000000000">
|
---|
| 808 | <I>grobner<MATH CLASS="INLINE">
|
---|
| 809 | -
|
---|
| 810 | </MATH>op</I></A>
|
---|
| 811 | </H4>
|
---|
| 812 | <P><IMG WIDTH="529" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 813 | SRC="img12.gif"
|
---|
| 814 | ALT="$\textstyle\parbox{\pboxargslen}{\em c1 c2 m a b pred ring {\sf \&aux} n a1 a2 \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 815 | <BLOCKQUOTE>
|
---|
| 816 | Perform a calculation equivalent to
|
---|
| 817 | (POLY<MATH CLASS="INLINE">
|
---|
| 818 | -
|
---|
| 819 | </MATH> (SCALAR<MATH CLASS="INLINE">
|
---|
| 820 | -
|
---|
| 821 | </MATH>TIMES<MATH CLASS="INLINE">
|
---|
| 822 | -
|
---|
| 823 | </MATH>POLY C2 A)
|
---|
| 824 | (SCALAR<MATH CLASS="INLINE">
|
---|
| 825 | -
|
---|
| 826 | </MATH>TIMES<MATH CLASS="INLINE">
|
---|
| 827 | -
|
---|
| 828 | </MATH>POLY C1 (MONOM<MATH CLASS="INLINE">
|
---|
| 829 | -
|
---|
| 830 | </MATH>TIMES<MATH CLASS="INLINE">
|
---|
| 831 | -
|
---|
| 832 | </MATH>POLY M B))
|
---|
| 833 | PRED)
|
---|
| 834 | but more efficiently; it destructively modifies A and stores the
|
---|
| 835 | result in it; A is returned.</BLOCKQUOTE><H4><A NAME="SECTION000200140000000000000">
|
---|
| 836 | <I>buchberger<MATH CLASS="INLINE">
|
---|
| 837 | -
|
---|
| 838 | </MATH>sort<MATH CLASS="INLINE">
|
---|
| 839 | -
|
---|
| 840 | </MATH>pairs</I></A>
|
---|
| 841 | </H4>
|
---|
| 842 | <P><IMG WIDTH="451" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 843 | SRC="img13.gif"
|
---|
| 844 | ALT="$\textstyle\parbox{\pboxargslen}{\em b g pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 845 | <BLOCKQUOTE>
|
---|
| 846 | Sort critical pairs B by the function which is
|
---|
| 847 | the value of the vairable *buchberger<MATH CLASS="INLINE">
|
---|
| 848 | -
|
---|
| 849 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 850 | -
|
---|
| 851 | </MATH>pairs*.
|
---|
| 852 | G is the list of polynomials which the elemnts
|
---|
| 853 | of B point into. PRED is the admissible monomial order.
|
---|
| 854 | The RING structure holds operations on the coefficients
|
---|
| 855 | as described in the COEFFICIENT<MATH CLASS="INLINE">
|
---|
| 856 | -
|
---|
| 857 | </MATH>RING package.</BLOCKQUOTE><H4><A NAME="SECTION000200150000000000000">
|
---|
| 858 | <I>mock<MATH CLASS="INLINE">
|
---|
| 859 | -
|
---|
| 860 | </MATH>spoly</I></A>
|
---|
| 861 | </H4>
|
---|
| 862 | <P><IMG WIDTH="526" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 863 | SRC="img14.gif"
|
---|
| 864 | ALT="$\textstyle\parbox{\pboxargslen}{\em f g pred \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 865 | <BLOCKQUOTE>
|
---|
| 866 | Returns an upper<MATH CLASS="INLINE">
|
---|
| 867 | -
|
---|
| 868 | </MATH>bound on the leading
|
---|
| 869 | monomial of the S<MATH CLASS="INLINE">
|
---|
| 870 | -
|
---|
| 871 | </MATH>polynomial of F and G, which are two polynomials
|
---|
| 872 | sorted by the admissible monomial order PRED.</BLOCKQUOTE><H4><A NAME="SECTION000200160000000000000">
|
---|
| 873 | <I>buchberger<MATH CLASS="INLINE">
|
---|
| 874 | -
|
---|
| 875 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 876 | -
|
---|
| 877 | </MATH>pairs<MATH CLASS="INLINE">
|
---|
| 878 | -
|
---|
| 879 | </MATH>use<MATH CLASS="INLINE">
|
---|
| 880 | -
|
---|
| 881 | </MATH>mock<MATH CLASS="INLINE">
|
---|
| 882 | -
|
---|
| 883 | </MATH>spoly</I></A>
|
---|
| 884 | </H4>
|
---|
| 885 | <P><IMG WIDTH="301" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 886 | SRC="img15.gif"
|
---|
| 887 | ALT="$\textstyle\parbox{\pboxargslen}{\em b c g pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 888 | <BLOCKQUOTE>
|
---|
| 889 | Merges lists of critical pairs B and C into one list.
|
---|
| 890 | The pairs are assumed to be ordered according to the
|
---|
| 891 | increasing value of MOCK<MATH CLASS="INLINE">
|
---|
| 892 | -
|
---|
| 893 | </MATH>SPOLY, as determined by the admissible
|
---|
| 894 | monomial order PRED. G is the list of polynomials which the pairs
|
---|
| 895 | of integers in B and C point into, and RING is the ring structure
|
---|
| 896 | defined in the COEFFICIENT<MATH CLASS="INLINE">
|
---|
| 897 | -
|
---|
| 898 | </MATH>RING package.</BLOCKQUOTE><H4><A NAME="SECTION000200170000000000000">
|
---|
| 899 | <I>buchberger<MATH CLASS="INLINE">
|
---|
| 900 | -
|
---|
| 901 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 902 | -
|
---|
| 903 | </MATH>pairs<MATH CLASS="INLINE">
|
---|
| 904 | -
|
---|
| 905 | </MATH>smallest<MATH CLASS="INLINE">
|
---|
| 906 | -
|
---|
| 907 | </MATH>lcm</I></A>
|
---|
| 908 | </H4>
|
---|
| 909 | <P><IMG WIDTH="301" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 910 | SRC="img15.gif"
|
---|
| 911 | ALT="$\textstyle\parbox{\pboxargslen}{\em b c g pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 912 | <BLOCKQUOTE>
|
---|
| 913 | Merges lists of critical pairs B and C into a single ordered
|
---|
| 914 | list. Implements a strategy of ordering pairs according to the
|
---|
| 915 | smallest lcm of the leading monomials of the two polynomials <MATH CLASS="INLINE">
|
---|
| 916 | -
|
---|
| 917 | </MATH> so
|
---|
| 918 | called normal strategy.</BLOCKQUOTE><H4><A NAME="SECTION000200180000000000000">
|
---|
| 919 | <I>buchberger<MATH CLASS="INLINE">
|
---|
| 920 | -
|
---|
| 921 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 922 | -
|
---|
| 923 | </MATH>pairs<MATH CLASS="INLINE">
|
---|
| 924 | -
|
---|
| 925 | </MATH>use<MATH CLASS="INLINE">
|
---|
| 926 | -
|
---|
| 927 | </MATH>smallest<MATH CLASS="INLINE">
|
---|
| 928 | -
|
---|
| 929 | </MATH>degree</I></A>
|
---|
| 930 | </H4>
|
---|
| 931 | <P><IMG WIDTH="301" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 932 | SRC="img15.gif"
|
---|
| 933 | ALT="$\textstyle\parbox{\pboxargslen}{\em b c g pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 934 | <BLOCKQUOTE>
|
---|
| 935 | Mergest lists B and C of critical pairs. This strategy is based on
|
---|
| 936 | ordering the pairs according to the smallest degree of the lcm of
|
---|
| 937 | the leading monomials of the two polynomials.</BLOCKQUOTE><H4><A NAME="SECTION000200190000000000000">
|
---|
| 938 | <I>buchberger<MATH CLASS="INLINE">
|
---|
| 939 | -
|
---|
| 940 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 941 | -
|
---|
| 942 | </MATH>pairs<MATH CLASS="INLINE">
|
---|
| 943 | -
|
---|
| 944 | </MATH>use<MATH CLASS="INLINE">
|
---|
| 945 | -
|
---|
| 946 | </MATH>smallest<MATH CLASS="INLINE">
|
---|
| 947 | -
|
---|
| 948 | </MATH>length</I></A>
|
---|
| 949 | </H4>
|
---|
| 950 | <P><IMG WIDTH="301" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 951 | SRC="img15.gif"
|
---|
| 952 | ALT="$\textstyle\parbox{\pboxargslen}{\em b c g pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 953 | <BLOCKQUOTE>
|
---|
| 954 | Mergest lists B and C of critical pairs. This strategy is based on
|
---|
| 955 | ordering the pairs according to the smallest total length of the
|
---|
| 956 | two polynomials, where length is the number of terms.</BLOCKQUOTE><H4><A NAME="SECTION000200200000000000000">
|
---|
| 957 | <I>buchberger<MATH CLASS="INLINE">
|
---|
| 958 | -
|
---|
| 959 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 960 | -
|
---|
| 961 | </MATH>pairs<MATH CLASS="INLINE">
|
---|
| 962 | -
|
---|
| 963 | </MATH>use<MATH CLASS="INLINE">
|
---|
| 964 | -
|
---|
| 965 | </MATH>smallest<MATH CLASS="INLINE">
|
---|
| 966 | -
|
---|
| 967 | </MATH>coefficient<MATH CLASS="INLINE">
|
---|
| 968 | -
|
---|
| 969 | </MATH>length</I></A>
|
---|
| 970 | </H4>
|
---|
| 971 | <P><IMG WIDTH="301" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 972 | SRC="img15.gif"
|
---|
| 973 | ALT="$\textstyle\parbox{\pboxargslen}{\em b c g pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 974 | <BLOCKQUOTE>
|
---|
| 975 | Mergest lists B and C of critical pairs. This strategy is based on
|
---|
| 976 | ordering the pairs according to the smallest combined length of the
|
---|
| 977 | coefficients of the two polynomials.</BLOCKQUOTE><H4><A NAME="SECTION000200210000000000000">
|
---|
| 978 | <I>buchberger<MATH CLASS="INLINE">
|
---|
| 979 | -
|
---|
| 980 | </MATH>set<MATH CLASS="INLINE">
|
---|
| 981 | -
|
---|
| 982 | </MATH>pair<MATH CLASS="INLINE">
|
---|
| 983 | -
|
---|
| 984 | </MATH>heuristic</I></A>
|
---|
| 985 | </H4>
|
---|
| 986 | <P><IMG WIDTH="551" HEIGHT="173" ALIGN="MIDDLE" BORDER="0"
|
---|
| 987 | SRC="img16.gif"
|
---|
| 988 | ALT="$\textstyle\parbox{\pboxargslen}{\em method {\sf \&aux} (strategy$-$fn
|
---|
| 989 | (ecase
|
---|
| 990 | ...
|
---|
| 991 | ...ar93 'buchberger$-$merge$-$pairs$-$use$-$smallest$-$coefficient$-$length))) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 992 | <BLOCKQUOTE>
|
---|
| 993 | Simply sets the variable *buchberger<MATH CLASS="INLINE">
|
---|
| 994 | -
|
---|
| 995 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 996 | -
|
---|
| 997 | </MATH>pairs* to the
|
---|
| 998 | heuristic function STRATEGY<MATH CLASS="INLINE">
|
---|
| 999 | -
|
---|
| 1000 | </MATH>FN implementing one of several
|
---|
| 1001 | strategies introduces before of selecting the most promising. METHOD
|
---|
| 1002 | should be one of the listed keywords.</BLOCKQUOTE><H4><A NAME="SECTION000200220000000000000">
|
---|
| 1003 | <I>criterion<MATH CLASS="INLINE">
|
---|
| 1004 | -
|
---|
| 1005 | </MATH>1</I></A>
|
---|
| 1006 | </H4>
|
---|
| 1007 | <P><IMG WIDTH="533" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1008 | SRC="img17.gif"
|
---|
| 1009 | ALT="$\textstyle\parbox{\pboxargslen}{\em pair g {\sf \&aux} (i (first pair)) (j (second pair)) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1010 | <BLOCKQUOTE>
|
---|
| 1011 | Returns T if the leading monomials of the two polynomials
|
---|
| 1012 | in G pointed to by the integers in PAIR have disjoint (relatively
|
---|
| 1013 | prime) monomials. This test is known as the first Buchberger
|
---|
| 1014 | criterion. </BLOCKQUOTE><H4><A NAME="SECTION000200230000000000000">
|
---|
| 1015 | <I>criterion<MATH CLASS="INLINE">
|
---|
| 1016 | -
|
---|
| 1017 | </MATH>2</I></A>
|
---|
| 1018 | </H4>
|
---|
| 1019 | <P><IMG WIDTH="533" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1020 | SRC="img18.gif"
|
---|
| 1021 | ALT="$\textstyle\parbox{\pboxargslen}{\em pair g m {\sf \&aux} (i (first pair)) (j (second pair)) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1022 | <BLOCKQUOTE>
|
---|
| 1023 | Returns T if the leading monomial of some element P of G divides
|
---|
| 1024 | the LCM of the leading monomials of the two polynomials in the
|
---|
| 1025 | polynomial list G, pointed to by the two integers in PAIR, and P
|
---|
| 1026 | paired with each of the polynomials pointed to by the the PAIR has
|
---|
| 1027 | already been treated, as indicated by the hash table M, which stores
|
---|
| 1028 | value T for every treated pair so far.</BLOCKQUOTE><H4><A NAME="SECTION000200240000000000000">
|
---|
| 1029 | <I>normalize<MATH CLASS="INLINE">
|
---|
| 1030 | -
|
---|
| 1031 | </MATH>poly</I></A>
|
---|
| 1032 | </H4>
|
---|
| 1033 | <P><IMG WIDTH="446" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1034 | SRC="img9.gif"
|
---|
| 1035 | ALT="$\textstyle\parbox{\pboxargslen}{\em p ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1036 | <BLOCKQUOTE>
|
---|
| 1037 | Divide a polynomial by its leading coefficient. It assumes
|
---|
| 1038 | that the division is possible, which may not always be the
|
---|
| 1039 | case in rings which are not fields. The exact division operator
|
---|
| 1040 | is assumed to be provided by the RING structure of the
|
---|
| 1041 | COEFFICIENT<MATH CLASS="INLINE">
|
---|
| 1042 | -
|
---|
| 1043 | </MATH>RING package.</BLOCKQUOTE><H4><A NAME="SECTION000200250000000000000">
|
---|
| 1044 | <I>normalize<MATH CLASS="INLINE">
|
---|
| 1045 | -
|
---|
| 1046 | </MATH>basis</I></A>
|
---|
| 1047 | </H4>
|
---|
| 1048 | <P><IMG WIDTH="500" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1049 | SRC="img19.gif"
|
---|
| 1050 | ALT="$\textstyle\parbox{\pboxargslen}{\em plist ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1051 | <BLOCKQUOTE>
|
---|
| 1052 | Divide every polynomial in a list PLIST by its leading coefficient.
|
---|
| 1053 | Use RING structure to perform the division of the coefficients.</BLOCKQUOTE><H4><A NAME="SECTION000200260000000000000">
|
---|
| 1054 | <I>reduction</I></A>
|
---|
| 1055 | </H4>
|
---|
| 1056 | <P><IMG WIDTH="547" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1057 | SRC="img20.gif"
|
---|
| 1058 | ALT="$\textstyle\parbox{\pboxargslen}{\em p pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1059 | <BLOCKQUOTE>
|
---|
| 1060 | Reduce a list of polynomials P, so that non of the terms in any of
|
---|
| 1061 | the polynomials is divisible by a leading monomial of another
|
---|
| 1062 | polynomial. Return the reduced list.</BLOCKQUOTE><H4><A NAME="SECTION000200270000000000000">
|
---|
| 1063 | <I>reduced<MATH CLASS="INLINE">
|
---|
| 1064 | -
|
---|
| 1065 | </MATH>grobner</I></A>
|
---|
| 1066 | </H4>
|
---|
| 1067 | <P><IMG WIDTH="559" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1068 | SRC="img6.gif"
|
---|
| 1069 | ALT="$\textstyle\parbox{\pboxargslen}{\em f {\sf \&optional} (pred \char93 'lex$\gt$) (start
|
---|
| 1070 | 0) (top$-$reduction$-$only
|
---|
| 1071 | nil) (ring
|
---|
| 1072 | *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1073 | <BLOCKQUOTE>
|
---|
| 1074 | Return the reduced Grobner basis of the ideal generated by a
|
---|
| 1075 | polynomial list F. This combines calls to two functions: GROBNER and
|
---|
| 1076 | REDUCTION. The parameters have the same meaning as in GROBNER or
|
---|
| 1077 | BUCHBERGER. </BLOCKQUOTE><H4><A NAME="SECTION000200280000000000000">
|
---|
| 1078 | <I>monom<MATH CLASS="INLINE">
|
---|
| 1079 | -
|
---|
| 1080 | </MATH>depends<MATH CLASS="INLINE">
|
---|
| 1081 | -
|
---|
| 1082 | </MATH>p</I></A>
|
---|
| 1083 | </H4>
|
---|
| 1084 | <P><IMG WIDTH="470" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1085 | SRC="img21.gif"
|
---|
| 1086 | ALT="$\textstyle\parbox{\pboxargslen}{\em m k \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1087 | <BLOCKQUOTE>
|
---|
| 1088 | Return T if the monomial M depends on variable number K.</BLOCKQUOTE><H4><A NAME="SECTION000200290000000000000">
|
---|
| 1089 | <I>term<MATH CLASS="INLINE">
|
---|
| 1090 | -
|
---|
| 1091 | </MATH>depends<MATH CLASS="INLINE">
|
---|
| 1092 | -
|
---|
| 1093 | </MATH>p</I></A>
|
---|
| 1094 | </H4>
|
---|
| 1095 | <P><IMG WIDTH="489" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1096 | SRC="img22.gif"
|
---|
| 1097 | ALT="$\textstyle\parbox{\pboxargslen}{\em term k \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1098 | <BLOCKQUOTE>
|
---|
| 1099 | Return T if the term TERM depends on variable number K.</BLOCKQUOTE><H4><A NAME="SECTION000200300000000000000">
|
---|
| 1100 | <I>poly<MATH CLASS="INLINE">
|
---|
| 1101 | -
|
---|
| 1102 | </MATH>depends<MATH CLASS="INLINE">
|
---|
| 1103 | -
|
---|
| 1104 | </MATH>p</I></A>
|
---|
| 1105 | </H4>
|
---|
| 1106 | <P><IMG WIDTH="492" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1107 | SRC="img23.gif"
|
---|
| 1108 | ALT="$\textstyle\parbox{\pboxargslen}{\em p k \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1109 | <BLOCKQUOTE>
|
---|
| 1110 | Return T if the term polynomial P depends on variable number K.</BLOCKQUOTE><H4><A NAME="SECTION000200310000000000000">
|
---|
| 1111 | <I>ring<MATH CLASS="INLINE">
|
---|
| 1112 | -
|
---|
| 1113 | </MATH>intersection</I></A>
|
---|
| 1114 | </H4>
|
---|
| 1115 | <P><IMG WIDTH="493" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1116 | SRC="img24.gif"
|
---|
| 1117 | ALT="$\textstyle\parbox{\pboxargslen}{\em plist k {\sf \&key} (key \char93 'identity) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1118 | <BLOCKQUOTE>
|
---|
| 1119 | This function assumes that polynomial list PLIST is a Grobner basis
|
---|
| 1120 | and it calculates the intersection with the ring R[x[k+1],...,x[n]],
|
---|
| 1121 | i.e. it discards polynomials which depend on variables x[0], x[1],
|
---|
| 1122 | ..., x[k]. </BLOCKQUOTE><H4><A NAME="SECTION000200320000000000000">
|
---|
| 1123 | <I>elimination<MATH CLASS="INLINE">
|
---|
| 1124 | -
|
---|
| 1125 | </MATH>ideal</I></A>
|
---|
| 1126 | </H4>
|
---|
| 1127 | <P><IMG WIDTH="491" HEIGHT="90" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1128 | SRC="img25.gif"
|
---|
| 1129 | ALT="$\textstyle\parbox{\pboxargslen}{\em flist k {\sf \&key} (primary$-$order
|
---|
| 1130 | \char...
|
---|
| 1131 | ...ondary$-$order)) (top$-$reduction$-$only
|
---|
| 1132 | nil) (ring
|
---|
| 1133 | *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1134 | <BLOCKQUOTE>
|
---|
| 1135 | Returns the K<MATH CLASS="INLINE">
|
---|
| 1136 | -
|
---|
| 1137 | </MATH>th elimination ideal of the ideal generated by
|
---|
| 1138 | polynomial list FLIST. Thus, a Grobner basis of the ideal generated
|
---|
| 1139 | by FLIST is calculated and polynomials depending on variables 0<MATH CLASS="INLINE">
|
---|
| 1140 | -
|
---|
| 1141 | </MATH>K
|
---|
| 1142 | are discarded. The monomial order in the calculation is an
|
---|
| 1143 | elimination order formed by combining two orders: PRIMARY<MATH CLASS="INLINE">
|
---|
| 1144 | -
|
---|
| 1145 | </MATH>ORDER
|
---|
| 1146 | used on variables 0<MATH CLASS="INLINE">
|
---|
| 1147 | -
|
---|
| 1148 | </MATH>K and SECONDARY<MATH CLASS="INLINE">
|
---|
| 1149 | -
|
---|
| 1150 | </MATH>ORDER used on variables
|
---|
| 1151 | starting from variable K+1. Thus, if both orders are set to the
|
---|
| 1152 | lexicographic order #'LEX<MATH CLASS="INLINE">
|
---|
| 1153 | >
|
---|
| 1154 | </MATH> then the resulting order is simply
|
---|
| 1155 | equivalent to #'LEX<MATH CLASS="INLINE">
|
---|
| 1156 | >
|
---|
| 1157 | </MATH>. But one could use arbitrary two orders in
|
---|
| 1158 | this function, for instance, two copies of #'GREVLEX<MATH CLASS="INLINE">
|
---|
| 1159 | >
|
---|
| 1160 | </MATH> (see ORDER
|
---|
| 1161 | package). When doing so, the caller must ensure that the same order
|
---|
| 1162 | has been used to sort the terms of the polynomials FLIST. </BLOCKQUOTE><H4><A NAME="SECTION000200330000000000000">
|
---|
| 1163 | <I>ideal<MATH CLASS="INLINE">
|
---|
| 1164 | -
|
---|
| 1165 | </MATH>intersection</I></A>
|
---|
| 1166 | </H4>
|
---|
| 1167 | <P><IMG WIDTH="487" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1168 | SRC="img26.gif"
|
---|
| 1169 | ALT="$\textstyle\parbox{\pboxargslen}{\em f g pred top$-$reduction$-$only ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1170 | <BLOCKQUOTE>
|
---|
| 1171 | Return the Grobner basis of the intersection of the ideals
|
---|
| 1172 | generated by the polynomial lists F and G. PRED is an admissible
|
---|
| 1173 | monomial order with respect to which the terms of F and G have been
|
---|
| 1174 | sorted. This order is going to be used in the Grobner basis
|
---|
| 1175 | calculation. If TOP<MATH CLASS="INLINE">
|
---|
| 1176 | -
|
---|
| 1177 | </MATH>REDUCTION<MATH CLASS="INLINE">
|
---|
| 1178 | -
|
---|
| 1179 | </MATH>ONLY is not NIL, internally the
|
---|
| 1180 | Grobner basis algorithm will perform top reduction only. The RING
|
---|
| 1181 | parameter, as usual, should be set to a structure defined in the
|
---|
| 1182 | package COEFFICIENT<MATH CLASS="INLINE">
|
---|
| 1183 | -
|
---|
| 1184 | </MATH>RING, and it will be used to perform all
|
---|
| 1185 | operations on coefficients.</BLOCKQUOTE><H4><A NAME="SECTION000200340000000000000">
|
---|
| 1186 | <I>poly<MATH CLASS="INLINE">
|
---|
| 1187 | -
|
---|
| 1188 | </MATH>contract</I></A>
|
---|
| 1189 | </H4>
|
---|
| 1190 | <P><IMG WIDTH="512" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1191 | SRC="img27.gif"
|
---|
| 1192 | ALT="$\textstyle\parbox{\pboxargslen}{\em f {\sf \&optional} (k 1) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1193 | <BLOCKQUOTE>
|
---|
| 1194 | Return a polynomial obtained from polynomial F by dropping the
|
---|
| 1195 | first K variables, if F does not depend on them. Note that this
|
---|
| 1196 | operation makes the polynomial incompatible for certain arithmetical
|
---|
| 1197 | operations with the original polynomial F. Calling this function on a
|
---|
| 1198 | polynomial F which does depend on variables 0<MATH CLASS="INLINE">
|
---|
| 1199 | -
|
---|
| 1200 | </MATH>K will result in
|
---|
| 1201 | error. </BLOCKQUOTE><H4><A NAME="SECTION000200350000000000000">
|
---|
| 1202 | <I>poly<MATH CLASS="INLINE">
|
---|
| 1203 | -
|
---|
| 1204 | </MATH>lcm</I></A>
|
---|
| 1205 | </H4>
|
---|
| 1206 | <P><IMG WIDTH="545" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1207 | SRC="img28.gif"
|
---|
| 1208 | ALT="$\textstyle\parbox{\pboxargslen}{\em f g {\sf \&optional} (pred \char93 'lex$\gt$) (ring
|
---|
| 1209 | *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1210 | <BLOCKQUOTE>
|
---|
| 1211 | Return LCM (least common multiple) of two polynomials F and G.
|
---|
| 1212 | The polynomials must be ordered according to monomial order PRED
|
---|
| 1213 | and their coefficients must be compatible with the RING structure
|
---|
| 1214 | defined in the COEFFICIENT<MATH CLASS="INLINE">
|
---|
| 1215 | -
|
---|
| 1216 | </MATH>RING package.</BLOCKQUOTE><H4><A NAME="SECTION000200360000000000000">
|
---|
| 1217 | <I>grobner<MATH CLASS="INLINE">
|
---|
| 1218 | -
|
---|
| 1219 | </MATH>gcd</I></A>
|
---|
| 1220 | </H4>
|
---|
| 1221 | <P><IMG WIDTH="545" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1222 | SRC="img28.gif"
|
---|
| 1223 | ALT="$\textstyle\parbox{\pboxargslen}{\em f g {\sf \&optional} (pred \char93 'lex$\gt$) (ring
|
---|
| 1224 | *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1225 | <BLOCKQUOTE>
|
---|
| 1226 | Return GCD (greatest common divisor) of two polynomials F and G.
|
---|
| 1227 | The polynomials must be ordered according to monomial order PRED
|
---|
| 1228 | and their coefficients must be compatible with the RING structure
|
---|
| 1229 | defined in the COEFFICIENT<MATH CLASS="INLINE">
|
---|
| 1230 | -
|
---|
| 1231 | </MATH>RING package.</BLOCKQUOTE><H4><A NAME="SECTION000200370000000000000">
|
---|
| 1232 | <I>grobner<MATH CLASS="INLINE">
|
---|
| 1233 | -
|
---|
| 1234 | </MATH>equal</I></A>
|
---|
| 1235 | </H4>
|
---|
| 1236 | <P><IMG WIDTH="510" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1237 | SRC="img29.gif"
|
---|
| 1238 | ALT="$\textstyle\parbox{\pboxargslen}{\em g1 g2 {\sf \&optional} (pred
|
---|
| 1239 | \char93 'lex$\gt$) (ring
|
---|
| 1240 | *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1241 | <BLOCKQUOTE>
|
---|
| 1242 | Returns T if two lists of polynomials G1 and G2, assumed to be
|
---|
| 1243 | Grobner bases, generate the same ideal, and NIL otherwise.</BLOCKQUOTE><H4><A NAME="SECTION000200380000000000000">
|
---|
| 1244 | <I>grobner<MATH CLASS="INLINE">
|
---|
| 1245 | -
|
---|
| 1246 | </MATH>subsetp</I></A>
|
---|
| 1247 | </H4>
|
---|
| 1248 | <P><IMG WIDTH="510" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1249 | SRC="img29.gif"
|
---|
| 1250 | ALT="$\textstyle\parbox{\pboxargslen}{\em g1 g2 {\sf \&optional} (pred
|
---|
| 1251 | \char93 'lex$\gt$) (ring
|
---|
| 1252 | *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1253 | <BLOCKQUOTE>
|
---|
| 1254 | Returns T if a list of polynomials G1 generates
|
---|
| 1255 | an ideal contained in the ideal generated by a polynomial list G2,
|
---|
| 1256 | both G1 and G2 assumed to be Grobner bases. Returns NIL otherwise.</BLOCKQUOTE><H4><A NAME="SECTION000200390000000000000">
|
---|
| 1257 | <I>grobner<MATH CLASS="INLINE">
|
---|
| 1258 | -
|
---|
| 1259 | </MATH>member</I></A>
|
---|
| 1260 | </H4>
|
---|
| 1261 | <P><IMG WIDTH="490" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1262 | SRC="img30.gif"
|
---|
| 1263 | ALT="$\textstyle\parbox{\pboxargslen}{\em p g {\sf \&optional} (pred
|
---|
| 1264 | \char93 'lex$\gt$) (ring
|
---|
| 1265 | *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1266 | <BLOCKQUOTE>
|
---|
| 1267 | Returns T if a polynomial P belongs to the ideal generated by the
|
---|
| 1268 | polynomial list G, which is assumed to be a Grobner basis. Returns
|
---|
| 1269 | NIL otherwise. </BLOCKQUOTE><H4><A NAME="SECTION000200400000000000000">
|
---|
| 1270 | <I>ideal<MATH CLASS="INLINE">
|
---|
| 1271 | -
|
---|
| 1272 | </MATH>equal</I></A>
|
---|
| 1273 | </H4>
|
---|
| 1274 | <P><IMG WIDTH="530" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1275 | SRC="img31.gif"
|
---|
| 1276 | ALT="$\textstyle\parbox{\pboxargslen}{\em f1 f2 pred top$-$reduction$-$only ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1277 | <BLOCKQUOTE>
|
---|
| 1278 | Returns T if two ideals generated by polynomial lists F1 and F2 are
|
---|
| 1279 | identical. Returns NIL otherwise.</BLOCKQUOTE><H4><A NAME="SECTION000200410000000000000">
|
---|
| 1280 | <I>ideal<MATH CLASS="INLINE">
|
---|
| 1281 | -
|
---|
| 1282 | </MATH>subsetp</I></A>
|
---|
| 1283 | </H4>
|
---|
| 1284 | <P><IMG WIDTH="515" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1285 | SRC="img32.gif"
|
---|
| 1286 | ALT="$\textstyle\parbox{\pboxargslen}{\em f1 f2 {\sf \&optional} (pred
|
---|
| 1287 | \char93 'lex$\gt$) (ring
|
---|
| 1288 | *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1289 | <BLOCKQUOTE>
|
---|
| 1290 | Returns T if the ideal spanned by the polynomial list F1 is contained
|
---|
| 1291 | in the ideal spanned by the polynomial list F2. Returns NIL
|
---|
| 1292 | otherwise. </BLOCKQUOTE><H4><A NAME="SECTION000200420000000000000">
|
---|
| 1293 | <I>ideal<MATH CLASS="INLINE">
|
---|
| 1294 | -
|
---|
| 1295 | </MATH>member</I></A>
|
---|
| 1296 | </H4>
|
---|
| 1297 | <P><IMG WIDTH="511" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1298 | SRC="img33.gif"
|
---|
| 1299 | ALT="$\textstyle\parbox{\pboxargslen}{\em p plist pred top$-$reduction$-$only ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1300 | <BLOCKQUOTE>
|
---|
| 1301 | Returns T if the polynomial P belongs to the ideal spanned by the
|
---|
| 1302 | polynomial list PLIST, and NIL otherwise.</BLOCKQUOTE><H4><A NAME="SECTION000200430000000000000">
|
---|
| 1303 | <I>ideal<MATH CLASS="INLINE">
|
---|
| 1304 | -
|
---|
| 1305 | </MATH>saturation<MATH CLASS="INLINE">
|
---|
| 1306 | -
|
---|
| 1307 | </MATH>1</I></A>
|
---|
| 1308 | </H4>
|
---|
| 1309 | <P><IMG WIDTH="476" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1310 | SRC="img34.gif"
|
---|
| 1311 | ALT="$\textstyle\parbox{\pboxargslen}{\em f p pred start top$-$reduction$-$only ring {\sf \&aux} (pred
|
---|
| 1312 | (elimination$-$order$-$1
|
---|
| 1313 | pred)) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1314 | <BLOCKQUOTE>
|
---|
| 1315 | Returns the reduced Grobner basis of the saturation of the ideal
|
---|
| 1316 | generated by a polynomial list F in the ideal generated by a single
|
---|
| 1317 | polynomial P. The saturation ideal is defined as the set of
|
---|
| 1318 | polynomials H such for some natural number n (* (EXPT P N) H) is in
|
---|
| 1319 | the ideal F. Geometrically, over an algebraically closed field, this
|
---|
| 1320 | is the set of polynomials in the ideal generated by F which do not
|
---|
| 1321 | identically vanish on the variety of P.</BLOCKQUOTE><H4><A NAME="SECTION000200440000000000000">
|
---|
| 1322 | <I>add<MATH CLASS="INLINE">
|
---|
| 1323 | -
|
---|
| 1324 | </MATH>variables</I></A>
|
---|
| 1325 | </H4>
|
---|
| 1326 | <P><IMG WIDTH="514" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1327 | SRC="img35.gif"
|
---|
| 1328 | ALT="$\textstyle\parbox{\pboxargslen}{\em plist n \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1329 | <BLOCKQUOTE>
|
---|
| 1330 | Add N new variables, adn the first N variables, to every polynomial
|
---|
| 1331 | in polynomial list PLIST. The resulting polynomial will not depend on
|
---|
| 1332 | these N variables. </BLOCKQUOTE><H4><A NAME="SECTION000200450000000000000">
|
---|
| 1333 | <I>extend<MATH CLASS="INLINE">
|
---|
| 1334 | -
|
---|
| 1335 | </MATH>polynomials</I></A>
|
---|
| 1336 | </H4>
|
---|
| 1337 | <P><IMG WIDTH="472" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1338 | SRC="img36.gif"
|
---|
| 1339 | ALT="$\textstyle\parbox{\pboxargslen}{\em plist {\sf \&aux} (k (length plist)) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1340 | <BLOCKQUOTE>
|
---|
| 1341 | Returns polynomial list {U1*P1', U2*P2', ... , UK*PK'}
|
---|
| 1342 | where Ui are new variables and PLIST={P1, P2, ... , PK} is a
|
---|
| 1343 | polynomial list. PI' is obtained from PI by adding new variables U1,
|
---|
| 1344 | U2, ..., UN UK as the first K variables. Thus, the resulting list is
|
---|
| 1345 | consists of polynomials whose terms are ordered by the lexicographic
|
---|
| 1346 | order on variables UI, with ties resolved by the monomial order which
|
---|
| 1347 | was used to order the terms of the polynomials in PLIST. We note that
|
---|
| 1348 | the monomial order does not explicitly participate in this
|
---|
| 1349 | calculation, and neither do the variable names.</BLOCKQUOTE><H4><A NAME="SECTION000200460000000000000">
|
---|
| 1350 | <I>saturation<MATH CLASS="INLINE">
|
---|
| 1351 | -
|
---|
| 1352 | </MATH>extension</I></A>
|
---|
| 1353 | </H4>
|
---|
| 1354 | <P><IMG WIDTH="465" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1355 | SRC="img37.gif"
|
---|
| 1356 | ALT="$\textstyle\parbox{\pboxargslen}{\em f plist ring {\sf \&aux} (k
|
---|
| 1357 | (length plist)) (d
|
---|
| 1358 | (+
|
---|
| 1359 | k
|
---|
| 1360 | (length
|
---|
| 1361 | (caaar
|
---|
| 1362 | plist)))) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1363 | <BLOCKQUOTE>
|
---|
| 1364 | Returns F' union {U1*P1<MATH CLASS="INLINE">
|
---|
| 1365 | -
|
---|
| 1366 | </MATH>1,U2*P2<MATH CLASS="INLINE">
|
---|
| 1367 | -
|
---|
| 1368 | </MATH>1,...,UK*PK<MATH CLASS="INLINE">
|
---|
| 1369 | -
|
---|
| 1370 | </MATH>1} where Ui are
|
---|
| 1371 | new variables and F' is a polynomial list obtained from F by adding
|
---|
| 1372 | variables U1, U2, ..., Uk as the first K variables to each polynomial
|
---|
| 1373 | in F. Thus, the resulting list is consists of polynomials whose terms
|
---|
| 1374 | are ordered by the lexicographic order on variables Ui, with ties
|
---|
| 1375 | resolved by the monomial order which was used to order the terms of
|
---|
| 1376 | the polynomials in F. We note that the monomial order does not
|
---|
| 1377 | explicitly participate in this calculation, and neither do the
|
---|
| 1378 | variable names. </BLOCKQUOTE><H4><A NAME="SECTION000200470000000000000">
|
---|
| 1379 | <I>polysaturation<MATH CLASS="INLINE">
|
---|
| 1380 | -
|
---|
| 1381 | </MATH>extension</I></A>
|
---|
| 1382 | </H4>
|
---|
| 1383 | <P><IMG WIDTH="465" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1384 | SRC="img37.gif"
|
---|
| 1385 | ALT="$\textstyle\parbox{\pboxargslen}{\em f plist ring {\sf \&aux} (k
|
---|
| 1386 | (length plist)) (d
|
---|
| 1387 | (+
|
---|
| 1388 | k
|
---|
| 1389 | (length
|
---|
| 1390 | (caaar
|
---|
| 1391 | plist)))) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1392 | <BLOCKQUOTE>
|
---|
| 1393 | Returns F' union {U1*P1+U2*P2+UK*PK<MATH CLASS="INLINE">
|
---|
| 1394 | -
|
---|
| 1395 | </MATH>1} where Ui are new variables
|
---|
| 1396 | and F' is a polynomial list obtained from F by adding variables U1,
|
---|
| 1397 | U2, ..., Uk as the first K variables to each polynomial in F. Thus,
|
---|
| 1398 | the resulting list is consists of polynomials whose terms are ordered
|
---|
| 1399 | by the lexicographic order on variables Ui, with ties resolved by the
|
---|
| 1400 | monomial order which was used to order the terms of the polynomials
|
---|
| 1401 | in F. We note that the monomial order does not explicitly participate
|
---|
| 1402 | in this calculation, and neither do the variable names. </BLOCKQUOTE><H4><A NAME="SECTION000200480000000000000">
|
---|
| 1403 | <I>saturation<MATH CLASS="INLINE">
|
---|
| 1404 | -
|
---|
| 1405 | </MATH>extension<MATH CLASS="INLINE">
|
---|
| 1406 | -
|
---|
| 1407 | </MATH>1</I></A>
|
---|
| 1408 | </H4>
|
---|
| 1409 | <P><IMG WIDTH="444" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1410 | SRC="img38.gif"
|
---|
| 1411 | ALT="$\textstyle\parbox{\pboxargslen}{\em f p ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1412 | <BLOCKQUOTE>
|
---|
| 1413 | Return the list F' union {U*P<MATH CLASS="INLINE">
|
---|
| 1414 | -
|
---|
| 1415 | </MATH>1} where U is a new variable,
|
---|
| 1416 | where F' is obtained from the polynomial list F by adding U as the
|
---|
| 1417 | first variable. The variable U will be added as the first variable,
|
---|
| 1418 | so that it can be easily eliminated.</BLOCKQUOTE><H4><A NAME="SECTION000200490000000000000">
|
---|
| 1419 | <I>ideal<MATH CLASS="INLINE">
|
---|
| 1420 | -
|
---|
| 1421 | </MATH>polysaturation<MATH CLASS="INLINE">
|
---|
| 1422 | -
|
---|
| 1423 | </MATH>1</I></A>
|
---|
| 1424 | </H4>
|
---|
| 1425 | <P><IMG WIDTH="447" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1426 | SRC="img39.gif"
|
---|
| 1427 | ALT="$\textstyle\parbox{\pboxargslen}{\em f plist pred start top$-$reduction$-$only ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1428 | <BLOCKQUOTE>
|
---|
| 1429 | Returns the reduced Grobner basis of the ideal obtained by a
|
---|
| 1430 | sequence of successive saturations in the polynomials
|
---|
| 1431 | of the polynomial list PLIST of the ideal generated by the
|
---|
| 1432 | polynomial list F.</BLOCKQUOTE><H4><A NAME="SECTION000200500000000000000">
|
---|
| 1433 | <I>ideal<MATH CLASS="INLINE">
|
---|
| 1434 | -
|
---|
| 1435 | </MATH>saturation</I></A>
|
---|
| 1436 | </H4>
|
---|
| 1437 | <P><IMG WIDTH="497" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1438 | SRC="img40.gif"
|
---|
| 1439 | ALT="$\textstyle\parbox{\pboxargslen}{\em f g pred start top$-$reduction$-$only ring ...
|
---|
| 1440 | ...$order k :primary$-$order \char93 'lex$\gt$\space :secondary$-$order pred)) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1441 | <BLOCKQUOTE>
|
---|
| 1442 | Returns the reduced Grobner basis of the saturation of the ideal
|
---|
| 1443 | generated by a polynomial list F in the ideal generated a polynomial
|
---|
| 1444 | list G The saturation ideal is defined as the set of polynomials H
|
---|
| 1445 | such for some natural number n and some P in the ideal generated by G
|
---|
| 1446 | the polynomial P**N * H is in the ideal spanned by F. Geometrically,
|
---|
| 1447 | over an algebraically closed field, this is the set of polynomials in
|
---|
| 1448 | the ideal generated by F which do not identically vanish on the
|
---|
| 1449 | variety of G.</BLOCKQUOTE><H4><A NAME="SECTION000200510000000000000">
|
---|
| 1450 | <I>ideal<MATH CLASS="INLINE">
|
---|
| 1451 | -
|
---|
| 1452 | </MATH>polysaturation</I></A>
|
---|
| 1453 | </H4>
|
---|
| 1454 | <P><IMG WIDTH="468" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1455 | SRC="img41.gif"
|
---|
| 1456 | ALT="$\textstyle\parbox{\pboxargslen}{\em f ideal$-$list pred start top$-$reduction$-$only ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1457 | <BLOCKQUOTE>
|
---|
| 1458 | Returns the reduced Grobner basis of the ideal obtained by a
|
---|
| 1459 | successive applications of IDEAL<MATH CLASS="INLINE">
|
---|
| 1460 | -
|
---|
| 1461 | </MATH>SATURATIONS to F and
|
---|
| 1462 | lists of polynomials in the list IDEAL<MATH CLASS="INLINE">
|
---|
| 1463 | -
|
---|
| 1464 | </MATH>LIST.</BLOCKQUOTE><H4><A NAME="SECTION000200520000000000000">
|
---|
| 1465 | <I>buchberger<MATH CLASS="INLINE">
|
---|
| 1466 | -
|
---|
| 1467 | </MATH>criterion</I></A>
|
---|
| 1468 | </H4>
|
---|
| 1469 | <P><IMG WIDTH="465" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1470 | SRC="img42.gif"
|
---|
| 1471 | ALT="$\textstyle\parbox{\pboxargslen}{\em g pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1472 | <BLOCKQUOTE>
|
---|
| 1473 | Returns T if G is a Grobner basis, by using the Buchberger
|
---|
| 1474 | criterion: for every two polynomials h1 and h2 in G the
|
---|
| 1475 | S<MATH CLASS="INLINE">
|
---|
| 1476 | -
|
---|
| 1477 | </MATH>polynomial S(h1,h2) reduces to 0 modulo G.</BLOCKQUOTE><H4><A NAME="SECTION000200530000000000000">
|
---|
| 1478 | <I>grobner<MATH CLASS="INLINE">
|
---|
| 1479 | -
|
---|
| 1480 | </MATH>test</I></A>
|
---|
| 1481 | </H4>
|
---|
| 1482 | <P><IMG WIDTH="520" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1483 | SRC="img43.gif"
|
---|
| 1484 | ALT="$\textstyle\parbox{\pboxargslen}{\em g f pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1485 | <BLOCKQUOTE>
|
---|
| 1486 | Test whether G is a Grobner basis and F is contained in G. Return T
|
---|
| 1487 | upon success and NIL otherwise.</BLOCKQUOTE><H4><A NAME="SECTION000200540000000000000">
|
---|
| 1488 | <I>minimization</I></A>
|
---|
| 1489 | </H4>
|
---|
| 1490 | <P><IMG WIDTH="523" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1491 | SRC="img44.gif"
|
---|
| 1492 | ALT="$\textstyle\parbox{\pboxargslen}{\em p pred \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1493 | <BLOCKQUOTE>
|
---|
| 1494 | Returns a sublist of the polynomial list P spanning the same
|
---|
| 1495 | monomial ideal as P but minimal, i.e. no leading monomial
|
---|
| 1496 | of a polynomial in the sublist divides the leading monomial
|
---|
| 1497 | of another polynomial.</BLOCKQUOTE><H4><A NAME="SECTION000200550000000000000">
|
---|
| 1498 | <I>add<MATH CLASS="INLINE">
|
---|
| 1499 | -
|
---|
| 1500 | </MATH>minimized</I></A>
|
---|
| 1501 | </H4>
|
---|
| 1502 | <P><IMG WIDTH="503" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1503 | SRC="img45.gif"
|
---|
| 1504 | ALT="$\textstyle\parbox{\pboxargslen}{\em f gred pred \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1505 | <BLOCKQUOTE>
|
---|
| 1506 | Adds a polynomial f to GRED, reduced Grobner basis, preserving the
|
---|
| 1507 | property described in the documentation for MINIMIZATION.</BLOCKQUOTE><H4><A NAME="SECTION000200560000000000000">
|
---|
| 1508 | <I>colon<MATH CLASS="INLINE">
|
---|
| 1509 | -
|
---|
| 1510 | </MATH>ideal</I></A>
|
---|
| 1511 | </H4>
|
---|
| 1512 | <P><IMG WIDTH="487" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1513 | SRC="img26.gif"
|
---|
| 1514 | ALT="$\textstyle\parbox{\pboxargslen}{\em f g pred top$-$reduction$-$only ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1515 | <BLOCKQUOTE>
|
---|
| 1516 | Returns the reduced Grobner basis of the colon ideal Id(F):Id(G),
|
---|
| 1517 | where F and G are two lists of polynomials. The colon ideal I:J is
|
---|
| 1518 | defined as the set of polynomials H such that there is a polynomial W
|
---|
| 1519 | in J for which W*H belongs to I.</BLOCKQUOTE><H4><A NAME="SECTION000200570000000000000">
|
---|
| 1520 | <I>colon<MATH CLASS="INLINE">
|
---|
| 1521 | -
|
---|
| 1522 | </MATH>ideal<MATH CLASS="INLINE">
|
---|
| 1523 | -
|
---|
| 1524 | </MATH>1</I></A>
|
---|
| 1525 | </H4>
|
---|
| 1526 | <P><IMG WIDTH="487" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1527 | SRC="img26.gif"
|
---|
| 1528 | ALT="$\textstyle\parbox{\pboxargslen}{\em f g pred top$-$reduction$-$only ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1529 | <BLOCKQUOTE>
|
---|
| 1530 | Returns the reduced Grobner basis of the colon ideal Id(F):Id({G}),
|
---|
| 1531 | where F is a list of polynomials and G is a polynomial.</BLOCKQUOTE><H4><A NAME="SECTION000200580000000000000">
|
---|
| 1532 | <I>pseudo<MATH CLASS="INLINE">
|
---|
| 1533 | -
|
---|
| 1534 | </MATH>divide</I></A>
|
---|
| 1535 | </H4>
|
---|
| 1536 | <P><IMG WIDTH="511" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1537 | SRC="img46.gif"
|
---|
| 1538 | ALT="$\textstyle\parbox{\pboxargslen}{\em f fl pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1539 | <BLOCKQUOTE>
|
---|
| 1540 | Pseudo<MATH CLASS="INLINE">
|
---|
| 1541 | -
|
---|
| 1542 | </MATH>divide a polynomial F by the list of polynomials FL. Return
|
---|
| 1543 | multiple values. The first value is a list of quotients A.
|
---|
| 1544 | The second value is the remainder R. The third value is an integer
|
---|
| 1545 | count of the number of reductions performed. Finally, the fourth
|
---|
| 1546 | argument is a scalar coefficient C, such that C*F can be divided by
|
---|
| 1547 | FL within the ring of coefficients, which is not necessarily a field.
|
---|
| 1548 | The resulting objects satisfy the equation:
|
---|
| 1549 | C*F= sum A[i]*FL[i] + R</BLOCKQUOTE><H4><A NAME="SECTION000200590000000000000">
|
---|
| 1550 | <I>gebauer<MATH CLASS="INLINE">
|
---|
| 1551 | -
|
---|
| 1552 | </MATH>moeller</I></A>
|
---|
| 1553 | </H4>
|
---|
| 1554 | <P><IMG WIDTH="494" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1555 | SRC="img47.gif"
|
---|
| 1556 | ALT="$\textstyle\parbox{\pboxargslen}{\em f pred start top$-$reduction$-$only ring {\sf \&aux} b g f1 \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1557 | <BLOCKQUOTE>
|
---|
| 1558 | Compute Grobner basis by using the algorithm of Gebauer and Moeller.
|
---|
| 1559 | This algorithm is described as BUCHBERGERNEW2 in the book by
|
---|
| 1560 | Becker<MATH CLASS="INLINE">
|
---|
| 1561 | -
|
---|
| 1562 | </MATH>Weispfenning entitled ``Grobner Bases''</BLOCKQUOTE><H4><A NAME="SECTION000200600000000000000">
|
---|
| 1563 | <I>update</I></A>
|
---|
| 1564 | </H4>
|
---|
| 1565 | <P><IMG WIDTH="564" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1566 | SRC="img48.gif"
|
---|
| 1567 | ALT="$\textstyle\parbox{\pboxargslen}{\em g b h pred ring {\sf \&aux} c d e b$-$new g$-$new pair \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1568 | <BLOCKQUOTE>
|
---|
| 1569 | An implementation of the auxillary UPDATE algorithm used by the
|
---|
| 1570 | Gebauer<MATH CLASS="INLINE">
|
---|
| 1571 | -
|
---|
| 1572 | </MATH>Moeller algorithm. G is a list of polynomials, B is a list
|
---|
| 1573 | of critical pairs and H is a new polynomial which possibly will be
|
---|
| 1574 | added to G. The naming conventions used are very close to the one
|
---|
| 1575 | used in the book of Becker<MATH CLASS="INLINE">
|
---|
| 1576 | -
|
---|
| 1577 | </MATH>Weispfenning.</BLOCKQUOTE><H4><A NAME="SECTION000200610000000000000">
|
---|
| 1578 | <I>gebauer<MATH CLASS="INLINE">
|
---|
| 1579 | -
|
---|
| 1580 | </MATH>moeller<MATH CLASS="INLINE">
|
---|
| 1581 | -
|
---|
| 1582 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 1583 | -
|
---|
| 1584 | </MATH>pairs<MATH CLASS="INLINE">
|
---|
| 1585 | -
|
---|
| 1586 | </MATH>use<MATH CLASS="INLINE">
|
---|
| 1587 | -
|
---|
| 1588 | </MATH>mock<MATH CLASS="INLINE">
|
---|
| 1589 | -
|
---|
| 1590 | </MATH>spoly</I></A>
|
---|
| 1591 | </H4>
|
---|
| 1592 | <P><IMG WIDTH="261" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1593 | SRC="img49.gif"
|
---|
| 1594 | ALT="$\textstyle\parbox{\pboxargslen}{\em b c pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1595 | <BLOCKQUOTE>
|
---|
| 1596 | The merging strategy used by the Gebauer<MATH CLASS="INLINE">
|
---|
| 1597 | -
|
---|
| 1598 | </MATH>Moeller algorithm, based
|
---|
| 1599 | on MOCK<MATH CLASS="INLINE">
|
---|
| 1600 | -
|
---|
| 1601 | </MATH>SPOLY; see the documentation of
|
---|
| 1602 | BUCHBERGER<MATH CLASS="INLINE">
|
---|
| 1603 | -
|
---|
| 1604 | </MATH>MERGE<MATH CLASS="INLINE">
|
---|
| 1605 | -
|
---|
| 1606 | </MATH>PAIRS<MATH CLASS="INLINE">
|
---|
| 1607 | -
|
---|
| 1608 | </MATH>USE<MATH CLASS="INLINE">
|
---|
| 1609 | -
|
---|
| 1610 | </MATH>MOCK<MATH CLASS="INLINE">
|
---|
| 1611 | -
|
---|
| 1612 | </MATH>SPOLY.</BLOCKQUOTE><H4><A NAME="SECTION000200620000000000000">
|
---|
| 1613 | <I>gebauer<MATH CLASS="INLINE">
|
---|
| 1614 | -
|
---|
| 1615 | </MATH>moeller<MATH CLASS="INLINE">
|
---|
| 1616 | -
|
---|
| 1617 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 1618 | -
|
---|
| 1619 | </MATH>pairs<MATH CLASS="INLINE">
|
---|
| 1620 | -
|
---|
| 1621 | </MATH>smallest<MATH CLASS="INLINE">
|
---|
| 1622 | -
|
---|
| 1623 | </MATH>lcm</I></A>
|
---|
| 1624 | </H4>
|
---|
| 1625 | <P><IMG WIDTH="261" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1626 | SRC="img49.gif"
|
---|
| 1627 | ALT="$\textstyle\parbox{\pboxargslen}{\em b c pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1628 | <BLOCKQUOTE>
|
---|
| 1629 | The merging strategy based on the smallest<MATH CLASS="INLINE">
|
---|
| 1630 | -
|
---|
| 1631 | </MATH>lcm (normal strategy);
|
---|
| 1632 | see the documentation of BUCHBERGER<MATH CLASS="INLINE">
|
---|
| 1633 | -
|
---|
| 1634 | </MATH>MERGE<MATH CLASS="INLINE">
|
---|
| 1635 | -
|
---|
| 1636 | </MATH>PAIRS<MATH CLASS="INLINE">
|
---|
| 1637 | -
|
---|
| 1638 | </MATH>SMALLEST<MATH CLASS="INLINE">
|
---|
| 1639 | -
|
---|
| 1640 | </MATH>LCM.</BLOCKQUOTE><H4><A NAME="SECTION000200630000000000000">
|
---|
| 1641 | <I>gebauer<MATH CLASS="INLINE">
|
---|
| 1642 | -
|
---|
| 1643 | </MATH>moeller<MATH CLASS="INLINE">
|
---|
| 1644 | -
|
---|
| 1645 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 1646 | -
|
---|
| 1647 | </MATH>pairs<MATH CLASS="INLINE">
|
---|
| 1648 | -
|
---|
| 1649 | </MATH>use<MATH CLASS="INLINE">
|
---|
| 1650 | -
|
---|
| 1651 | </MATH>smallest<MATH CLASS="INLINE">
|
---|
| 1652 | -
|
---|
| 1653 | </MATH>degree</I></A>
|
---|
| 1654 | </H4>
|
---|
| 1655 | <P><IMG WIDTH="261" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1656 | SRC="img49.gif"
|
---|
| 1657 | ALT="$\textstyle\parbox{\pboxargslen}{\em b c pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1658 | <BLOCKQUOTE>
|
---|
| 1659 | The merging strategy based on the smallest<MATH CLASS="INLINE">
|
---|
| 1660 | -
|
---|
| 1661 | </MATH>lcm (normal strategy);
|
---|
| 1662 | see the documentation of
|
---|
| 1663 | BUCHBERGER<MATH CLASS="INLINE">
|
---|
| 1664 | -
|
---|
| 1665 | </MATH>MERGE<MATH CLASS="INLINE">
|
---|
| 1666 | -
|
---|
| 1667 | </MATH>PAIRS<MATH CLASS="INLINE">
|
---|
| 1668 | -
|
---|
| 1669 | </MATH>USE<MATH CLASS="INLINE">
|
---|
| 1670 | -
|
---|
| 1671 | </MATH>SMALLEST<MATH CLASS="INLINE">
|
---|
| 1672 | -
|
---|
| 1673 | </MATH>DEGREE. </BLOCKQUOTE><H4><A NAME="SECTION000200640000000000000">
|
---|
| 1674 | <I>gebauer<MATH CLASS="INLINE">
|
---|
| 1675 | -
|
---|
| 1676 | </MATH>moeller<MATH CLASS="INLINE">
|
---|
| 1677 | -
|
---|
| 1678 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 1679 | -
|
---|
| 1680 | </MATH>pairs<MATH CLASS="INLINE">
|
---|
| 1681 | -
|
---|
| 1682 | </MATH>use<MATH CLASS="INLINE">
|
---|
| 1683 | -
|
---|
| 1684 | </MATH>smallest<MATH CLASS="INLINE">
|
---|
| 1685 | -
|
---|
| 1686 | </MATH>length</I></A>
|
---|
| 1687 | </H4>
|
---|
| 1688 | <P><IMG WIDTH="261" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1689 | SRC="img49.gif"
|
---|
| 1690 | ALT="$\textstyle\parbox{\pboxargslen}{\em b c pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1691 | <BLOCKQUOTE>
|
---|
| 1692 | </BLOCKQUOTE><H4><A NAME="SECTION000200650000000000000">
|
---|
| 1693 | <I>gebauer<MATH CLASS="INLINE">
|
---|
| 1694 | -
|
---|
| 1695 | </MATH>moeller<MATH CLASS="INLINE">
|
---|
| 1696 | -
|
---|
| 1697 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 1698 | -
|
---|
| 1699 | </MATH>pairs<MATH CLASS="INLINE">
|
---|
| 1700 | -
|
---|
| 1701 | </MATH>use<MATH CLASS="INLINE">
|
---|
| 1702 | -
|
---|
| 1703 | </MATH>smallest<MATH CLASS="INLINE">
|
---|
| 1704 | -
|
---|
| 1705 | </MATH>coefficient<MATH CLASS="INLINE">
|
---|
| 1706 | -
|
---|
| 1707 | </MATH>length</I></A>
|
---|
| 1708 | </H4>
|
---|
| 1709 | <P><IMG WIDTH="261" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1710 | SRC="img49.gif"
|
---|
| 1711 | ALT="$\textstyle\parbox{\pboxargslen}{\em b c pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1712 | <BLOCKQUOTE>
|
---|
| 1713 | The merging strategy based on the smallest<MATH CLASS="INLINE">
|
---|
| 1714 | -
|
---|
| 1715 | </MATH>lcm (normal strategy);
|
---|
| 1716 | see the documentation of
|
---|
| 1717 | BUCHBERGER<MATH CLASS="INLINE">
|
---|
| 1718 | -
|
---|
| 1719 | </MATH>MERGE<MATH CLASS="INLINE">
|
---|
| 1720 | -
|
---|
| 1721 | </MATH>PAIRS<MATH CLASS="INLINE">
|
---|
| 1722 | -
|
---|
| 1723 | </MATH>USE<MATH CLASS="INLINE">
|
---|
| 1724 | -
|
---|
| 1725 | </MATH>SMALLEST<MATH CLASS="INLINE">
|
---|
| 1726 | -
|
---|
| 1727 | </MATH>COEFFICIENT<MATH CLASS="INLINE">
|
---|
| 1728 | -
|
---|
| 1729 | </MATH>LENGTH. </BLOCKQUOTE><H4><A NAME="SECTION000200660000000000000">
|
---|
| 1730 | <I>gebauer<MATH CLASS="INLINE">
|
---|
| 1731 | -
|
---|
| 1732 | </MATH>moeller<MATH CLASS="INLINE">
|
---|
| 1733 | -
|
---|
| 1734 | </MATH>set<MATH CLASS="INLINE">
|
---|
| 1735 | -
|
---|
| 1736 | </MATH>pair<MATH CLASS="INLINE">
|
---|
| 1737 | -
|
---|
| 1738 | </MATH>heuristic</I></A>
|
---|
| 1739 | </H4>
|
---|
| 1740 | <P><IMG WIDTH="551" HEIGHT="173" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1741 | SRC="img16.gif"
|
---|
| 1742 | ALT="$\textstyle\parbox{\pboxargslen}{\em method {\sf \&aux} (strategy$-$fn
|
---|
| 1743 | (ecase
|
---|
| 1744 | ...
|
---|
| 1745 | ...ar93 'buchberger$-$merge$-$pairs$-$use$-$smallest$-$coefficient$-$length))) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1746 | <BLOCKQUOTE>
|
---|
| 1747 | </BLOCKQUOTE><H4><A NAME="SECTION000200670000000000000">
|
---|
| 1748 | <I>spoly<MATH CLASS="INLINE">
|
---|
| 1749 | -
|
---|
| 1750 | </MATH>sugar</I></A>
|
---|
| 1751 | </H4>
|
---|
| 1752 | <P><IMG WIDTH="527" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1753 | SRC="img50.gif"
|
---|
| 1754 | ALT="$\textstyle\parbox{\pboxargslen}{\em f$-$with$-$sugar g$-$with$-$sugar ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1755 | <BLOCKQUOTE>
|
---|
| 1756 | Calculate the ``sugar'' of the S<MATH CLASS="INLINE">
|
---|
| 1757 | -
|
---|
| 1758 | </MATH>polynomial of two polynomials with
|
---|
| 1759 | sugar. A polynomial with sugar is simply a pair (P . SUGAR) where
|
---|
| 1760 | SUGAR is an integer constant defined according to the following
|
---|
| 1761 | algorighm: the sugar of sum or difference of two polynomials with
|
---|
| 1762 | sugar is the MAX of the sugars of those two polynomials. The sugar of
|
---|
| 1763 | a product of a term and a polynomial is the sum of the degree of the
|
---|
| 1764 | term and the sugar of the polynomial. The idea is to accumulate
|
---|
| 1765 | sugar as we perform the arithmetic operations, and that polynomials
|
---|
| 1766 | with small (little) sugar should be given priority in the
|
---|
| 1767 | calculations. Thus, the ``sugar strategy'' of the critical pair
|
---|
| 1768 | selection is to select a pair with the smallest value of sugar of the
|
---|
| 1769 | resulting S<MATH CLASS="INLINE">
|
---|
| 1770 | -
|
---|
| 1771 | </MATH>polynomial.</BLOCKQUOTE><H4><A NAME="SECTION000200680000000000000">
|
---|
| 1772 | <I>spoly<MATH CLASS="INLINE">
|
---|
| 1773 | -
|
---|
| 1774 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 1775 | -
|
---|
| 1776 | </MATH>sugar</I></A>
|
---|
| 1777 | </H4>
|
---|
| 1778 | <P><IMG WIDTH="484" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1779 | SRC="img51.gif"
|
---|
| 1780 | ALT="$\textstyle\parbox{\pboxargslen}{\em f$-$with$-$sugar g$-$with$-$sugar pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1781 | <BLOCKQUOTE>
|
---|
| 1782 | The S<MATH CLASS="INLINE">
|
---|
| 1783 | -
|
---|
| 1784 | </MATH>polynomials of two polynomials with SUGAR strategy.</BLOCKQUOTE><H4><A NAME="SECTION000200690000000000000">
|
---|
| 1785 | <I>normal<MATH CLASS="INLINE">
|
---|
| 1786 | -
|
---|
| 1787 | </MATH>form<MATH CLASS="INLINE">
|
---|
| 1788 | -
|
---|
| 1789 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 1790 | -
|
---|
| 1791 | </MATH>sugar</I></A>
|
---|
| 1792 | </H4>
|
---|
| 1793 | <P><IMG WIDTH="520" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1794 | SRC="img10.gif"
|
---|
| 1795 | ALT="$\textstyle\parbox{\pboxargslen}{\em f fl pred top$-$reduction$-$only ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1796 | <BLOCKQUOTE>
|
---|
| 1797 | Normal form of the polynomial with sugar F with respect to
|
---|
| 1798 | a list of polynomials with sugar FL. Assumes that FL is not empty.
|
---|
| 1799 | Parameters PRED, TOP<MATH CLASS="INLINE">
|
---|
| 1800 | -
|
---|
| 1801 | </MATH>REDUCTION<MATH CLASS="INLINE">
|
---|
| 1802 | -
|
---|
| 1803 | </MATH>ONLY and RING have the same
|
---|
| 1804 | meaning as in NORMAL<MATH CLASS="INLINE">
|
---|
| 1805 | -
|
---|
| 1806 | </MATH>FORM. The parameter SUGAR<MATH CLASS="INLINE">
|
---|
| 1807 | -
|
---|
| 1808 | </MATH>LIMIT should be
|
---|
| 1809 | set to a positive integer. If the sugar limit of the partial
|
---|
| 1810 | remainder exceeds SUGAR<MATH CLASS="INLINE">
|
---|
| 1811 | -
|
---|
| 1812 | </MATH>LIMIT then the calculation is stopped and
|
---|
| 1813 | the partial remainder is returned, although it is not fully reduced
|
---|
| 1814 | with respect to FL. </BLOCKQUOTE><H4><A NAME="SECTION000200700000000000000">
|
---|
| 1815 | <I>buchberger<MATH CLASS="INLINE">
|
---|
| 1816 | -
|
---|
| 1817 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 1818 | -
|
---|
| 1819 | </MATH>sugar</I></A>
|
---|
| 1820 | </H4>
|
---|
| 1821 | <P><IMG WIDTH="443" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1822 | SRC="img52.gif"
|
---|
| 1823 | ALT="$\textstyle\parbox{\pboxargslen}{\em f$-$no$-$sugar pred start top$-$reduction$-$only ring {\sf \&aux} (s (1$-$\space (length f$-$no$-$sugar))) b m f \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1824 | <BLOCKQUOTE>
|
---|
| 1825 | Returns a Grobner basis of an ideal generated by a list of ordinary
|
---|
| 1826 | polynomials F<MATH CLASS="INLINE">
|
---|
| 1827 | -
|
---|
| 1828 | </MATH>NO<MATH CLASS="INLINE">
|
---|
| 1829 | -
|
---|
| 1830 | </MATH>SUGAR. Adds initial sugar to the polynomials and
|
---|
| 1831 | performs the Buchberger algorithm with ``sugar strategy''. It returns
|
---|
| 1832 | an ordinary list of polynomials with no sugar. One of the most
|
---|
| 1833 | effective algorithms for Grobner basis calculation.</BLOCKQUOTE><H4><A NAME="SECTION000200710000000000000">
|
---|
| 1834 | <I>buchberger<MATH CLASS="INLINE">
|
---|
| 1835 | -
|
---|
| 1836 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 1837 | -
|
---|
| 1838 | </MATH>sugar<MATH CLASS="INLINE">
|
---|
| 1839 | -
|
---|
| 1840 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 1841 | -
|
---|
| 1842 | </MATH>pairs</I></A>
|
---|
| 1843 | </H4>
|
---|
| 1844 | <P><IMG WIDTH="343" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1845 | SRC="img53.gif"
|
---|
| 1846 | ALT="$\textstyle\parbox{\pboxargslen}{\em b c g m pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1847 | <BLOCKQUOTE>
|
---|
| 1848 | Merges lists of critical pairs. It orders pairs according to
|
---|
| 1849 | increasing sugar, with ties broken by smaller MOCK<MATH CLASS="INLINE">
|
---|
| 1850 | -
|
---|
| 1851 | </MATH>SPOLY value of
|
---|
| 1852 | the two polynomials. In this function B must already be sorted.</BLOCKQUOTE><H4><A NAME="SECTION000200720000000000000">
|
---|
| 1853 | <I>buchberger<MATH CLASS="INLINE">
|
---|
| 1854 | -
|
---|
| 1855 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 1856 | -
|
---|
| 1857 | </MATH>sugar<MATH CLASS="INLINE">
|
---|
| 1858 | -
|
---|
| 1859 | </MATH>sort<MATH CLASS="INLINE">
|
---|
| 1860 | -
|
---|
| 1861 | </MATH>pairs</I></A>
|
---|
| 1862 | </H4>
|
---|
| 1863 | <P><IMG WIDTH="359" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1864 | SRC="img54.gif"
|
---|
| 1865 | ALT="$\textstyle\parbox{\pboxargslen}{\em c g m pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1866 | <BLOCKQUOTE>
|
---|
| 1867 | Sorts critical pairs C according to sugar strategy.</BLOCKQUOTE><H4><A NAME="SECTION000200730000000000000">
|
---|
| 1868 | <I>criterion<MATH CLASS="INLINE">
|
---|
| 1869 | -
|
---|
| 1870 | </MATH>1<MATH CLASS="INLINE">
|
---|
| 1871 | -
|
---|
| 1872 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 1873 | -
|
---|
| 1874 | </MATH>sugar</I></A>
|
---|
| 1875 | </H4>
|
---|
| 1876 | <P><IMG WIDTH="533" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1877 | SRC="img17.gif"
|
---|
| 1878 | ALT="$\textstyle\parbox{\pboxargslen}{\em pair g {\sf \&aux} (i (first pair)) (j (second pair)) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1879 | <BLOCKQUOTE>
|
---|
| 1880 | An implementation of Buchberger's first criterion for polynomials
|
---|
| 1881 | with sugar. See the documentation of BUCHBERGER<MATH CLASS="INLINE">
|
---|
| 1882 | -
|
---|
| 1883 | </MATH>CRITERION<MATH CLASS="INLINE">
|
---|
| 1884 | -
|
---|
| 1885 | </MATH>1.</BLOCKQUOTE><H4><A NAME="SECTION000200740000000000000">
|
---|
| 1886 | <I>criterion<MATH CLASS="INLINE">
|
---|
| 1887 | -
|
---|
| 1888 | </MATH>2<MATH CLASS="INLINE">
|
---|
| 1889 | -
|
---|
| 1890 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 1891 | -
|
---|
| 1892 | </MATH>sugar</I></A>
|
---|
| 1893 | </H4>
|
---|
| 1894 | <P><IMG WIDTH="533" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1895 | SRC="img18.gif"
|
---|
| 1896 | ALT="$\textstyle\parbox{\pboxargslen}{\em pair g m {\sf \&aux} (i (first pair)) (j (second pair)) \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1897 | <BLOCKQUOTE>
|
---|
| 1898 | An implementation of Buchberger's first criterion for polynomials
|
---|
| 1899 | with sugar. See the documentation of BUCHBERGER<MATH CLASS="INLINE">
|
---|
| 1900 | -
|
---|
| 1901 | </MATH>CRITERION<MATH CLASS="INLINE">
|
---|
| 1902 | -
|
---|
| 1903 | </MATH>2.</BLOCKQUOTE><H4><A NAME="SECTION000200750000000000000">
|
---|
| 1904 | <I>gebauer<MATH CLASS="INLINE">
|
---|
| 1905 | -
|
---|
| 1906 | </MATH>moeller<MATH CLASS="INLINE">
|
---|
| 1907 | -
|
---|
| 1908 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 1909 | -
|
---|
| 1910 | </MATH>sugar</I></A>
|
---|
| 1911 | </H4>
|
---|
| 1912 | <P><IMG WIDTH="494" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1913 | SRC="img47.gif"
|
---|
| 1914 | ALT="$\textstyle\parbox{\pboxargslen}{\em f pred start top$-$reduction$-$only ring {\sf \&aux} b g f1 \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1915 | <BLOCKQUOTE>
|
---|
| 1916 | Compute Grobner basis by using the algorithm of Gebauer and Moeller.
|
---|
| 1917 | This algorithm is described as BUCHBERGERNEW2 in the book by
|
---|
| 1918 | Becker<MATH CLASS="INLINE">
|
---|
| 1919 | -
|
---|
| 1920 | </MATH>Weispfenning entitled ``Grobner Bases''</BLOCKQUOTE><H4><A NAME="SECTION000200760000000000000">
|
---|
| 1921 | <I>update<MATH CLASS="INLINE">
|
---|
| 1922 | -
|
---|
| 1923 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 1924 | -
|
---|
| 1925 | </MATH>sugar</I></A>
|
---|
| 1926 | </H4>
|
---|
| 1927 | <P><IMG WIDTH="564" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1928 | SRC="img48.gif"
|
---|
| 1929 | ALT="$\textstyle\parbox{\pboxargslen}{\em g b h pred ring {\sf \&aux} c d e b$-$new g$-$new pair \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1930 | <BLOCKQUOTE>
|
---|
| 1931 | An implementation of the auxillary UPDATE algorithm used by the
|
---|
| 1932 | Gebauer<MATH CLASS="INLINE">
|
---|
| 1933 | -
|
---|
| 1934 | </MATH>Moeller algorithm. G is a list of polynomials, B is a list
|
---|
| 1935 | of critical pairs and H is a new polynomial which possibly will be
|
---|
| 1936 | added to G. The naming conventions used are very close to the one
|
---|
| 1937 | used in the book of Becker<MATH CLASS="INLINE">
|
---|
| 1938 | -
|
---|
| 1939 | </MATH>Weispfenning. Operates on polynomials
|
---|
| 1940 | with sugar. </BLOCKQUOTE><H4><A NAME="SECTION000200770000000000000">
|
---|
| 1941 | <I>gebauer<MATH CLASS="INLINE">
|
---|
| 1942 | -
|
---|
| 1943 | </MATH>moeller<MATH CLASS="INLINE">
|
---|
| 1944 | -
|
---|
| 1945 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 1946 | -
|
---|
| 1947 | </MATH>sugar<MATH CLASS="INLINE">
|
---|
| 1948 | -
|
---|
| 1949 | </MATH>merge<MATH CLASS="INLINE">
|
---|
| 1950 | -
|
---|
| 1951 | </MATH>pairs</I></A>
|
---|
| 1952 | </H4>
|
---|
| 1953 | <P><IMG WIDTH="261" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1954 | SRC="img49.gif"
|
---|
| 1955 | ALT="$\textstyle\parbox{\pboxargslen}{\em b c pred ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1956 | <BLOCKQUOTE>
|
---|
| 1957 | Merges lists of critical pairs. It orders pairs according to
|
---|
| 1958 | increasing sugar, with ties broken by smaller lcm of head monomials
|
---|
| 1959 | In this function B must already be sorted. Operates on polynomials
|
---|
| 1960 | with sugar. </BLOCKQUOTE><H4><A NAME="SECTION000200780000000000000">
|
---|
| 1961 | <I>grobner<MATH CLASS="INLINE">
|
---|
| 1962 | -
|
---|
| 1963 | </MATH>primitive<MATH CLASS="INLINE">
|
---|
| 1964 | -
|
---|
| 1965 | </MATH>part<MATH CLASS="INLINE">
|
---|
| 1966 | -
|
---|
| 1967 | </MATH>with<MATH CLASS="INLINE">
|
---|
| 1968 | -
|
---|
| 1969 | </MATH>sugar</I></A>
|
---|
| 1970 | </H4>
|
---|
| 1971 | <P><IMG WIDTH="354" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
| 1972 | SRC="img55.gif"
|
---|
| 1973 | ALT="$\textstyle\parbox{\pboxargslen}{\em h ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
| 1974 | <BLOCKQUOTE>
|
---|
| 1975 | </BLOCKQUOTE><HR>
|
---|
| 1976 | <!--Navigation Panel-->
|
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| 1990 | <B> Next:</B> <A NAME="tex2html743"
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| 1996 | <!--End of Navigation Panel-->
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| 1997 | <ADDRESS>
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| 1998 | <I>Marek Rychlik</I>
|
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| 1999 | <BR><I>3/21/1998</I>
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| 2000 | </ADDRESS>
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| 2001 | </BODY>
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