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 Comprehensive Gröbner basis package</TITLE>
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34 | <BR>
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35 | <B> Next:</B> <A NAME="tex2html856"
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36 | HREF="node5.html">The Coefficient Ring package</A>
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40 | HREF="node3.html">The String Interface to</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="tex2html857"
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48 | HREF="node4.html#SECTION00040010000000000000">
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49 | <I>*colored<MATH CLASS="INLINE">
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50 | -
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51 | </MATH>poly<MATH CLASS="INLINE">
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52 | -
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53 | </MATH>debug*</I></A>
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54 | <LI><A NAME="tex2html858"
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55 | HREF="node4.html#SECTION00040020000000000000">
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56 | <I>debug<MATH CLASS="INLINE">
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57 | -
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58 | </MATH>cgb</I></A>
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59 | <LI><A NAME="tex2html859"
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60 | HREF="node4.html#SECTION00040030000000000000">
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61 | <I>make<MATH CLASS="INLINE">
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62 | -
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63 | </MATH>colored<MATH CLASS="INLINE">
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64 | -
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65 | </MATH>poly</I></A>
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66 | <LI><A NAME="tex2html860"
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67 | HREF="node4.html#SECTION00040040000000000000">
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68 | <I>make<MATH CLASS="INLINE">
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69 | -
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70 | </MATH>colored<MATH CLASS="INLINE">
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71 | -
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72 | </MATH>poly<MATH CLASS="INLINE">
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73 | -
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74 | </MATH>list</I></A>
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75 | <LI><A NAME="tex2html861"
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76 | HREF="node4.html#SECTION00040050000000000000">
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77 | <I>color<MATH CLASS="INLINE">
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78 | -
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79 | </MATH>poly<MATH CLASS="INLINE">
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80 | -
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81 | </MATH>list</I></A>
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82 | <LI><A NAME="tex2html862"
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83 | HREF="node4.html#SECTION00040060000000000000">
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84 | <I>color<MATH CLASS="INLINE">
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85 | -
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86 | </MATH>poly</I></A>
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87 | <LI><A NAME="tex2html863"
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88 | HREF="node4.html#SECTION00040070000000000000">
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89 | <I>colored<MATH CLASS="INLINE">
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90 | -
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91 | </MATH>poly<MATH CLASS="INLINE">
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92 | -
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93 | </MATH>to<MATH CLASS="INLINE">
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94 | -
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95 | </MATH>poly</I></A>
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96 | <LI><A NAME="tex2html864"
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97 | HREF="node4.html#SECTION00040080000000000000">
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98 | <I>colored<MATH CLASS="INLINE">
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99 | -
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100 | </MATH>poly<MATH CLASS="INLINE">
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101 | -
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102 | </MATH>print</I></A>
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103 | <LI><A NAME="tex2html865"
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104 | HREF="node4.html#SECTION00040090000000000000">
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105 | <I>colored<MATH CLASS="INLINE">
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106 | -
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107 | </MATH>poly<MATH CLASS="INLINE">
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108 | -
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109 | </MATH>print<MATH CLASS="INLINE">
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110 | -
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111 | </MATH>list</I></A>
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112 | <LI><A NAME="tex2html866"
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113 | HREF="node4.html#SECTION000400100000000000000">
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114 | <I>determine</I></A>
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115 | <LI><A NAME="tex2html867"
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116 | HREF="node4.html#SECTION000400110000000000000">
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117 | <I>determine<MATH CLASS="INLINE">
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118 | -
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119 | </MATH>1</I></A>
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120 | <LI><A NAME="tex2html868"
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121 | HREF="node4.html#SECTION000400120000000000000">
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122 | <I>determine<MATH CLASS="INLINE">
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123 | -
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124 | </MATH>white<MATH CLASS="INLINE">
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125 | -
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126 | </MATH>term</I></A>
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127 | <LI><A NAME="tex2html869"
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128 | HREF="node4.html#SECTION000400130000000000000">
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129 | <I>cond<MATH CLASS="INLINE">
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130 | -
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131 | </MATH>system<MATH CLASS="INLINE">
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132 | -
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133 | </MATH>print</I></A>
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134 | <LI><A NAME="tex2html870"
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135 | HREF="node4.html#SECTION000400140000000000000">
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136 | <I>cond<MATH CLASS="INLINE">
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137 | -
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138 | </MATH>print</I></A>
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139 | <LI><A NAME="tex2html871"
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140 | HREF="node4.html#SECTION000400150000000000000">
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141 | <I>add<MATH CLASS="INLINE">
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142 | -
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143 | </MATH>pairs</I></A>
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144 | <LI><A NAME="tex2html872"
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145 | HREF="node4.html#SECTION000400160000000000000">
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146 | <I>cond<MATH CLASS="INLINE">
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147 | -
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148 | </MATH>part</I></A>
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149 | <LI><A NAME="tex2html873"
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150 | HREF="node4.html#SECTION000400170000000000000">
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151 | <I>cond<MATH CLASS="INLINE">
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152 | -
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153 | </MATH>hm</I></A>
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154 | <LI><A NAME="tex2html874"
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155 | HREF="node4.html#SECTION000400180000000000000">
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156 | <I>delete<MATH CLASS="INLINE">
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157 | -
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158 | </MATH>green<MATH CLASS="INLINE">
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159 | -
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160 | </MATH>polys</I></A>
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161 | <LI><A NAME="tex2html875"
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162 | HREF="node4.html#SECTION000400190000000000000">
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163 | <I>grobner<MATH CLASS="INLINE">
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164 | -
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165 | </MATH>system</I></A>
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166 | <LI><A NAME="tex2html876"
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167 | HREF="node4.html#SECTION000400200000000000000">
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168 | <I>reorder<MATH CLASS="INLINE">
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169 | -
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170 | </MATH>pairs</I></A>
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171 | <LI><A NAME="tex2html877"
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172 | HREF="node4.html#SECTION000400210000000000000">
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173 | <I>colored<MATH CLASS="INLINE">
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174 | -
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175 | </MATH>criterion<MATH CLASS="INLINE">
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176 | -
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177 | </MATH>1</I></A>
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178 | <LI><A NAME="tex2html878"
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179 | HREF="node4.html#SECTION000400220000000000000">
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180 | <I>colored<MATH CLASS="INLINE">
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181 | -
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182 | </MATH>criterion<MATH CLASS="INLINE">
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183 | -
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184 | </MATH>2</I></A>
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185 | <LI><A NAME="tex2html879"
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186 | HREF="node4.html#SECTION000400230000000000000">
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187 | <I>cond<MATH CLASS="INLINE">
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188 | -
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189 | </MATH>normal<MATH CLASS="INLINE">
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190 | -
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191 | </MATH>form</I></A>
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192 | <LI><A NAME="tex2html880"
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193 | HREF="node4.html#SECTION000400240000000000000">
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194 | <I>cond<MATH CLASS="INLINE">
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195 | -
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196 | </MATH>spoly</I></A>
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197 | <LI><A NAME="tex2html881"
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198 | HREF="node4.html#SECTION000400250000000000000">
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199 | <I>cond<MATH CLASS="INLINE">
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200 | -
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201 | </MATH>lm</I></A>
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202 | <LI><A NAME="tex2html882"
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203 | HREF="node4.html#SECTION000400260000000000000">
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204 | <I>cond<MATH CLASS="INLINE">
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205 | -
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206 | </MATH>lc</I></A>
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207 | <LI><A NAME="tex2html883"
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208 | HREF="node4.html#SECTION000400270000000000000">
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209 | <I>colored<MATH CLASS="INLINE">
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210 | -
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211 | </MATH>term<MATH CLASS="INLINE">
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212 | -
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213 | </MATH>times<MATH CLASS="INLINE">
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214 | -
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215 | </MATH>poly</I></A>
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216 | <LI><A NAME="tex2html884"
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217 | HREF="node4.html#SECTION000400280000000000000">
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218 | <I>colored<MATH CLASS="INLINE">
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219 | -
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220 | </MATH>scalar<MATH CLASS="INLINE">
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221 | -
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222 | </MATH>times<MATH CLASS="INLINE">
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223 | -
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224 | </MATH>poly</I></A>
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225 | <LI><A NAME="tex2html885"
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226 | HREF="node4.html#SECTION000400290000000000000">
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227 | <I>colored<MATH CLASS="INLINE">
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228 | -
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229 | </MATH>term*</I></A>
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230 | <LI><A NAME="tex2html886"
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231 | HREF="node4.html#SECTION000400300000000000000">
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232 | <I>color*</I></A>
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233 | <LI><A NAME="tex2html887"
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234 | HREF="node4.html#SECTION000400310000000000000">
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235 | <I>color+</I></A>
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236 | <LI><A NAME="tex2html888"
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237 | HREF="node4.html#SECTION000400320000000000000">
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238 | <I>color<MATH CLASS="INLINE">
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239 | -
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240 | </MATH></I></A>
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241 | <LI><A NAME="tex2html889"
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242 | HREF="node4.html#SECTION000400330000000000000">
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243 | <I>colored<MATH CLASS="INLINE">
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244 | -
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245 | </MATH>poly+</I></A>
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246 | <LI><A NAME="tex2html890"
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247 | HREF="node4.html#SECTION000400340000000000000">
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248 | <I>colored<MATH CLASS="INLINE">
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249 | -
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250 | </MATH>poly<MATH CLASS="INLINE">
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251 | -
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252 | </MATH></I></A>
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253 | <LI><A NAME="tex2html891"
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254 | HREF="node4.html#SECTION000400350000000000000">
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255 | <I>colored<MATH CLASS="INLINE">
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256 | -
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257 | </MATH>term<MATH CLASS="INLINE">
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258 | -
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259 | </MATH>uminus</I></A>
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260 | <LI><A NAME="tex2html892"
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261 | HREF="node4.html#SECTION000400360000000000000">
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262 | <I>colored<MATH CLASS="INLINE">
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263 | -
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264 | </MATH>minus<MATH CLASS="INLINE">
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265 | -
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266 | </MATH>poly</I></A>
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267 | <LI><A NAME="tex2html893"
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268 | HREF="node4.html#SECTION000400370000000000000">
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269 | <I>string<MATH CLASS="INLINE">
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270 | -
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271 | </MATH>grobner<MATH CLASS="INLINE">
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272 | -
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273 | </MATH>system</I></A>
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274 | <LI><A NAME="tex2html894"
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275 | HREF="node4.html#SECTION000400380000000000000">
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276 | <I>string<MATH CLASS="INLINE">
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277 | -
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278 | </MATH>cond</I></A>
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279 | <LI><A NAME="tex2html895"
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280 | HREF="node4.html#SECTION000400390000000000000">
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281 | <I>string<MATH CLASS="INLINE">
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282 | -
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283 | </MATH>cover</I></A>
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284 | <LI><A NAME="tex2html896"
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285 | HREF="node4.html#SECTION000400400000000000000">
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286 | <I>saturate<MATH CLASS="INLINE">
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287 | -
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288 | </MATH>cover</I></A>
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289 | <LI><A NAME="tex2html897"
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290 | HREF="node4.html#SECTION000400410000000000000">
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291 | <I>saturate<MATH CLASS="INLINE">
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292 | -
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293 | </MATH>cond</I></A>
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294 | <LI><A NAME="tex2html898"
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295 | HREF="node4.html#SECTION000400420000000000000">
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296 | <I>string<MATH CLASS="INLINE">
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297 | -
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298 | </MATH>determine</I></A>
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299 | <LI><A NAME="tex2html899"
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300 | HREF="node4.html#SECTION000400430000000000000">
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301 | <I>tidy<MATH CLASS="INLINE">
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302 | -
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303 | </MATH>grobner<MATH CLASS="INLINE">
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304 | -
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305 | </MATH>system</I></A>
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306 | <LI><A NAME="tex2html900"
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307 | HREF="node4.html#SECTION000400440000000000000">
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308 | <I>tidy<MATH CLASS="INLINE">
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309 | -
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310 | </MATH>pair</I></A>
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311 | <LI><A NAME="tex2html901"
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312 | HREF="node4.html#SECTION000400450000000000000">
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313 | <I>tidy<MATH CLASS="INLINE">
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314 | -
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315 | </MATH>cond</I></A>
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316 | <LI><A NAME="tex2html902"
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317 | HREF="node4.html#SECTION000400460000000000000">
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318 | <I>colored<MATH CLASS="INLINE">
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319 | -
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320 | </MATH>reduction</I></A>
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321 | <LI><A NAME="tex2html903"
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322 | HREF="node4.html#SECTION000400470000000000000">
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323 | <I>green<MATH CLASS="INLINE">
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324 | -
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325 | </MATH>reduce<MATH CLASS="INLINE">
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326 | -
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327 | </MATH>colored<MATH CLASS="INLINE">
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328 | -
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329 | </MATH>poly</I></A>
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330 | <LI><A NAME="tex2html904"
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331 | HREF="node4.html#SECTION000400480000000000000">
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332 | <I>green<MATH CLASS="INLINE">
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333 | -
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334 | </MATH>reduce<MATH CLASS="INLINE">
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335 | -
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336 | </MATH>colored<MATH CLASS="INLINE">
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337 | -
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338 | </MATH>list</I></A>
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339 | <LI><A NAME="tex2html905"
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340 | HREF="node4.html#SECTION000400490000000000000">
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341 | <I>cond<MATH CLASS="INLINE">
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342 | -
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343 | </MATH>system<MATH CLASS="INLINE">
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344 | -
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345 | </MATH>green<MATH CLASS="INLINE">
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346 | -
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347 | </MATH>reduce</I></A>
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348 | <LI><A NAME="tex2html906"
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349 | HREF="node4.html#SECTION000400500000000000000">
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350 | <I>parse<MATH CLASS="INLINE">
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351 | -
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352 | </MATH>to<MATH CLASS="INLINE">
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353 | -
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354 | </MATH>colored<MATH CLASS="INLINE">
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355 | -
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356 | </MATH>poly<MATH CLASS="INLINE">
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357 | -
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358 | </MATH>list</I></A>
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359 | <LI><A NAME="tex2html907"
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360 | HREF="node4.html#SECTION000400510000000000000">
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361 | <I>red<MATH CLASS="INLINE">
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362 | -
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363 | </MATH>reduction</I></A>
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364 | </UL>
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365 | <!--End of Table of Child-Links-->
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366 | <HR>
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367 | <H1><A NAME="SECTION00040000000000000000">
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368 | The Comprehensive Gröbner basis package</A>
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369 | </H1>
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370 | <H4><A NAME="SECTION00040010000000000000">
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371 | <I>*colored<MATH CLASS="INLINE">
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372 | -
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373 | </MATH>poly<MATH CLASS="INLINE">
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374 | -
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375 | </MATH>debug*</I></A>
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376 | </H4>
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377 | <P><IMG WIDTH="495" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
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378 | SRC="img1.gif"
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379 | ALT="$\textstyle\parbox{\pboxargslen}{\em nil \/}$"> [<EM>VARIABLE</EM>]
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380 | <BLOCKQUOTE>
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381 | If true debugging output is on.</BLOCKQUOTE><H4><A NAME="SECTION00040020000000000000">
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382 | <I>debug<MATH CLASS="INLINE">
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383 | -
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384 | </MATH>cgb</I></A>
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385 | </H4>
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386 | <P><IMG WIDTH="576" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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387 | SRC="img7.gif"
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388 | ALT="$\textstyle\parbox{\pboxargslen}{\em {\sf \&rest} args \/}$"> [<EM>MACRO</EM>]
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389 | <BLOCKQUOTE>
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390 | </BLOCKQUOTE><H4><A NAME="SECTION00040030000000000000">
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391 | <I>make<MATH CLASS="INLINE">
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392 | -
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393 | </MATH>colored<MATH CLASS="INLINE">
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394 | -
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395 | </MATH>poly</I></A>
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396 | </H4>
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397 | <P><IMG WIDTH="471" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
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398 | SRC="img67.gif"
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399 | ALT="$\textstyle\parbox{\pboxargslen}{\em poly k {\sf \&key} (key
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400 | \char93 'identity)...
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401 | ...er
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402 | \char93 'lex$\gt$) (parameter$-$order
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403 | \char93 'lex$\gt$) {\sf \&aux} l \/}$"> [<EM>FUNCTION</EM>]
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404 | <BLOCKQUOTE>
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405 | Colored poly is represented as a list
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406 | (TERM1 TERM2 ... TERMS)
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407 | where each term is a triple
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408 | (MONOM . (POLY . COLOR))
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409 | where monoms and polys have common number of variables while color is
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410 | one of the three: :RED, :GREEN or :WHITE. This function translates
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411 | an ordinary polynomial into a colored one by dividing variables into
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412 | K and N<MATH CLASS="INLINE">
|
---|
413 | -
|
---|
414 | </MATH>K, where N is the total number of variables in the
|
---|
415 | polynomial poly; the function KEY can be called to select variables
|
---|
416 | in arbitrary order.</BLOCKQUOTE><H4><A NAME="SECTION00040040000000000000">
|
---|
417 | <I>make<MATH CLASS="INLINE">
|
---|
418 | -
|
---|
419 | </MATH>colored<MATH CLASS="INLINE">
|
---|
420 | -
|
---|
421 | </MATH>poly<MATH CLASS="INLINE">
|
---|
422 | -
|
---|
423 | </MATH>list</I></A>
|
---|
424 | </H4>
|
---|
425 | <P><IMG WIDTH="438" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
426 | SRC="img68.gif"
|
---|
427 | ALT="$\textstyle\parbox{\pboxargslen}{\em plist {\sf \&rest} rest \/}$"> [<EM>FUNCTION</EM>]
|
---|
428 | <BLOCKQUOTE>
|
---|
429 | Translate a list of polynomials PLIST into a list of colored
|
---|
430 | polynomials by calling MAKE<MATH CLASS="INLINE">
|
---|
431 | -
|
---|
432 | </MATH>COLORED<MATH CLASS="INLINE">
|
---|
433 | -
|
---|
434 | </MATH>POLY. Returns the resulting
|
---|
435 | list. </BLOCKQUOTE><H4><A NAME="SECTION00040050000000000000">
|
---|
436 | <I>color<MATH CLASS="INLINE">
|
---|
437 | -
|
---|
438 | </MATH>poly<MATH CLASS="INLINE">
|
---|
439 | -
|
---|
440 | </MATH>list</I></A>
|
---|
441 | </H4>
|
---|
442 | <P><IMG WIDTH="503" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
443 | SRC="img69.gif"
|
---|
444 | ALT="$\textstyle\parbox{\pboxargslen}{\em flist {\sf \&optional} (cond (list nil nil)) \/}$"> [<EM>FUNCTION</EM>]
|
---|
445 | <BLOCKQUOTE>
|
---|
446 | Add colors to an ordinary list of polynomials FLIST, according to a
|
---|
447 | condition COND. A condition is a pair of polynomial lists. Each
|
---|
448 | polynomial in COND is a polynomial in parameters only. The list
|
---|
449 | (FIRST COND) is called the ``green list'' and it consists of
|
---|
450 | polynomials which vanish for the parameters associated with the
|
---|
451 | condition. The list (SECOND COND) is called the ``red list</BLOCKQUOTE><H4><A NAME="SECTION00040060000000000000">
|
---|
452 | <I>color<MATH CLASS="INLINE">
|
---|
453 | -
|
---|
454 | </MATH>poly</I></A>
|
---|
455 | </H4>
|
---|
456 | <P><IMG WIDTH="536" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
457 | SRC="img70.gif"
|
---|
458 | ALT="$\textstyle\parbox{\pboxargslen}{\em f {\sf \&optional} (cond (list nil nil)) \/}$"> [<EM>FUNCTION</EM>]
|
---|
459 | <BLOCKQUOTE>
|
---|
460 | Add color to a single polynomial F, according to condition COND.
|
---|
461 | See the documentation of COLOR<MATH CLASS="INLINE">
|
---|
462 | -
|
---|
463 | </MATH>POLY<MATH CLASS="INLINE">
|
---|
464 | -
|
---|
465 | </MATH>LIST.</BLOCKQUOTE><H4><A NAME="SECTION00040070000000000000">
|
---|
466 | <I>colored<MATH CLASS="INLINE">
|
---|
467 | -
|
---|
468 | </MATH>poly<MATH CLASS="INLINE">
|
---|
469 | -
|
---|
470 | </MATH>to<MATH CLASS="INLINE">
|
---|
471 | -
|
---|
472 | </MATH>poly</I></A>
|
---|
473 | </H4>
|
---|
474 | <P><IMG WIDTH="451" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
475 | SRC="img71.gif"
|
---|
476 | ALT="$\textstyle\parbox{\pboxargslen}{\em cpoly \/}$"> [<EM>FUNCTION</EM>]
|
---|
477 | <BLOCKQUOTE>
|
---|
478 | For a given colored polynomial CPOLY, removes the colors and
|
---|
479 | it returns the polynomial as an ordinary polynomial with
|
---|
480 | coefficients which are polynomials in parameters.</BLOCKQUOTE><H4><A NAME="SECTION00040080000000000000">
|
---|
481 | <I>colored<MATH CLASS="INLINE">
|
---|
482 | -
|
---|
483 | </MATH>poly<MATH CLASS="INLINE">
|
---|
484 | -
|
---|
485 | </MATH>print</I></A>
|
---|
486 | </H4>
|
---|
487 | <P><IMG WIDTH="475" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
|
---|
488 | SRC="img72.gif"
|
---|
489 | ALT="$\textstyle\parbox{\pboxargslen}{\em poly vars {\sf \&key} (stream
|
---|
490 | t) (beg
|
---|
491 | t) (print$-$green$-$part
|
---|
492 | nil) (mark$-$coefficients
|
---|
493 | nil) \/}$"> [<EM>FUNCTION</EM>]
|
---|
494 | <BLOCKQUOTE>
|
---|
495 | Print a colored polynomial POLY. Use variables VARS to represent
|
---|
496 | the variables. Some of the variables are going to be used as
|
---|
497 | parameters, according to the length of the monomials in the main
|
---|
498 | monomial and coefficient part of each term in POLY. The key variable
|
---|
499 | STREAM may be used to redirect the output. If parameter
|
---|
500 | PRINT<MATH CLASS="INLINE">
|
---|
501 | -
|
---|
502 | </MATH>GREEN<MATH CLASS="INLINE">
|
---|
503 | -
|
---|
504 | </MATH>PART is set then the coefficients which have color
|
---|
505 | :GREEN will be printed, otherwise they are discarded silently. If
|
---|
506 | MARK<MATH CLASS="INLINE">
|
---|
507 | -
|
---|
508 | </MATH>COEFFICIENTS is not NIL then every coefficient will be marked
|
---|
509 | according to its color, for instance G(U<MATH CLASS="INLINE">
|
---|
510 | -
|
---|
511 | </MATH>1) would mean that U<MATH CLASS="INLINE">
|
---|
512 | -
|
---|
513 | </MATH>1
|
---|
514 | is in the green list. Returns P.</BLOCKQUOTE><H4><A NAME="SECTION00040090000000000000">
|
---|
515 | <I>colored<MATH CLASS="INLINE">
|
---|
516 | -
|
---|
517 | </MATH>poly<MATH CLASS="INLINE">
|
---|
518 | -
|
---|
519 | </MATH>print<MATH CLASS="INLINE">
|
---|
520 | -
|
---|
521 | </MATH>list</I></A>
|
---|
522 | </H4>
|
---|
523 | <P><IMG WIDTH="442" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
|
---|
524 | SRC="img73.gif"
|
---|
525 | ALT="$\textstyle\parbox{\pboxargslen}{\em poly$-$list vars {\sf \&key} (stream
|
---|
526 | t) (beg
|
---|
527 | t) (print$-$green$-$part
|
---|
528 | nil) (mark$-$coefficients
|
---|
529 | nil) \/}$"> [<EM>FUNCTION</EM>]
|
---|
530 | <BLOCKQUOTE>
|
---|
531 | Pring a list of colored polynomials via a call to
|
---|
532 | COLORED<MATH CLASS="INLINE">
|
---|
533 | -
|
---|
534 | </MATH>POLY<MATH CLASS="INLINE">
|
---|
535 | -
|
---|
536 | </MATH>PRINT.</BLOCKQUOTE><H4><A NAME="SECTION000400100000000000000">
|
---|
537 | <I>determine</I></A>
|
---|
538 | </H4>
|
---|
539 | <P><IMG WIDTH="543" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
540 | SRC="img74.gif"
|
---|
541 | ALT="$\textstyle\parbox{\pboxargslen}{\em f {\sf \&optional} (cond (list nil nil)) (order
|
---|
542 | \char93 'lex$\gt$) (ring
|
---|
543 | *coefficient$-$ring*) \/}$"> [<EM>FUNCTION</EM>]
|
---|
544 | <BLOCKQUOTE>
|
---|
545 | This function takes a list of colored polynomials F and a condition
|
---|
546 | COND, and it returns a list of pairs (COND' F') such that COND' cover
|
---|
547 | COND and F' is a ``determined'' version of the colored polynomial
|
---|
548 | list F, i.e. every polynomial has its leading coefficient determined.
|
---|
549 | This means that some of the initial coefficients in each polynomial
|
---|
550 | in F' are in the green list of COND, and the first non<MATH CLASS="INLINE">
|
---|
551 | -
|
---|
552 | </MATH>green
|
---|
553 | coefficient is in the red list of COND. We note that F' differs from
|
---|
554 | F only by different colors: some of the terms marked :WHITE are now
|
---|
555 | marked either :GREEN or :RED. Coloring is done either by explicitly
|
---|
556 | checking membership in red or green list of COND, or implicitly by
|
---|
557 | performing Grobner basis calculations in the polynomial ring over the
|
---|
558 | parameters. The admissible monomial order ORDER is used only in the
|
---|
559 | parameter space. Also, the ring structure RING is used only for
|
---|
560 | calculations on polynomials of the parameters only.</BLOCKQUOTE><H4><A NAME="SECTION000400110000000000000">
|
---|
561 | <I>determine<MATH CLASS="INLINE">
|
---|
562 | -
|
---|
563 | </MATH>1</I></A>
|
---|
564 | </H4>
|
---|
565 | <P><IMG WIDTH="522" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
566 | SRC="img75.gif"
|
---|
567 | ALT="$\textstyle\parbox{\pboxargslen}{\em cond p end gp order ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
568 | <BLOCKQUOTE>
|
---|
569 | Determine a single colored polynomial P according to condition COND.
|
---|
570 | Prepend green part GP to P. Cons the result with END, which should be
|
---|
571 | a list of colored polynomials, and return the resulting list of
|
---|
572 | polynomials. This is an auxillary function of DETERMINE.</BLOCKQUOTE><H4><A NAME="SECTION000400120000000000000">
|
---|
573 | <I>determine<MATH CLASS="INLINE">
|
---|
574 | -
|
---|
575 | </MATH>white<MATH CLASS="INLINE">
|
---|
576 | -
|
---|
577 | </MATH>term</I></A>
|
---|
578 | </H4>
|
---|
579 | <P><IMG WIDTH="448" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
580 | SRC="img76.gif"
|
---|
581 | ALT="$\textstyle\parbox{\pboxargslen}{\em cond term restp end gp order ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
582 | <BLOCKQUOTE>
|
---|
583 | This is an auxillary function of DETERMINE. In this function the
|
---|
584 | parameter COND is a condition. The parameters TERM, RESTP and GP are
|
---|
585 | three parts of a polynomial being processed, where TERM is colored
|
---|
586 | :WHITE. We test the membership in the red and green list of COND we
|
---|
587 | try to determine whether the term is :RED or :GREEN. This is done by
|
---|
588 | performing ideal membership tests in the polynomial ring. Let C be
|
---|
589 | the coefficient of TERM. Thus, C is a polynomial in parameters. We
|
---|
590 | find whether C is in the green list by performing a plain ideal
|
---|
591 | membership test. However, to test properly whether C is in the red
|
---|
592 | list, one needs a different strategy. In fact, we test whether
|
---|
593 | adding C to the red list would produce a non<MATH CLASS="INLINE">
|
---|
594 | -
|
---|
595 | </MATH>empty set of
|
---|
596 | parameters in some algebraic extension. The test is whether 1 belongs
|
---|
597 | to the saturation ideal of (FIRST COND) in (CONS C (SECOND COND)).
|
---|
598 | Thus, we use POLY<MATH CLASS="INLINE">
|
---|
599 | -
|
---|
600 | </MATH>SATURATION. If we are successful in determining
|
---|
601 | the color of TERM, we simply change the color of the term and return
|
---|
602 | the list ((COND P)) where P is obtained by appending GP, (LIST TERM)
|
---|
603 | and RESTP. If we cannot determine whether TERM is :RED or :GREEN, we
|
---|
604 | return the list ((COND' P') (COND'' P</BLOCKQUOTE><H4><A NAME="SECTION000400130000000000000">
|
---|
605 | <I>cond<MATH CLASS="INLINE">
|
---|
606 | -
|
---|
607 | </MATH>system<MATH CLASS="INLINE">
|
---|
608 | -
|
---|
609 | </MATH>print</I></A>
|
---|
610 | </H4>
|
---|
611 | <P><IMG WIDTH="474" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
|
---|
612 | SRC="img77.gif"
|
---|
613 | ALT="$\textstyle\parbox{\pboxargslen}{\em system vars params {\sf \&key} (suppress$-$...
|
---|
614 | ...rint$-$green$-$part
|
---|
615 | nil) (mark$-$coefficients
|
---|
616 | nil) {\sf \&aux} (label
|
---|
617 | 0) \/}$"> [<EM>FUNCTION</EM>]
|
---|
618 | <BLOCKQUOTE>
|
---|
619 | A conditional system SYSTEM is a list of pairs (COND PLIST), where
|
---|
620 | COND is a condition (a pair (GREEN<MATH CLASS="INLINE">
|
---|
621 | -
|
---|
622 | </MATH>LIST RED<MATH CLASS="INLINE">
|
---|
623 | -
|
---|
624 | </MATH>LIST)) and PLIST is a
|
---|
625 | list of colored polynomials. This function pretty<MATH CLASS="INLINE">
|
---|
626 | -
|
---|
627 | </MATH>prints this list
|
---|
628 | of pairs. A conditional system is the data structure returned by
|
---|
629 | GROBNER<MATH CLASS="INLINE">
|
---|
630 | -
|
---|
631 | </MATH>SYSTEM. This function returns SYSTEM, if SUPPRESS<MATH CLASS="INLINE">
|
---|
632 | -
|
---|
633 | </MATH>VALUE
|
---|
634 | is non<MATH CLASS="INLINE">
|
---|
635 | -
|
---|
636 | </MATH>NIL and no value otherwise. If MARK<MATH CLASS="INLINE">
|
---|
637 | -
|
---|
638 | </MATH>COEFFICIENTS is
|
---|
639 | non<MATH CLASS="INLINE">
|
---|
640 | -
|
---|
641 | </MATH>NIL coefficients will be marked as in G(u<MATH CLASS="INLINE">
|
---|
642 | -
|
---|
643 | </MATH>1)*x+R(2)*y, which
|
---|
644 | means that u<MATH CLASS="INLINE">
|
---|
645 | -
|
---|
646 | </MATH>1 is :GREEN and 2 is :RED. </BLOCKQUOTE><H4><A NAME="SECTION000400140000000000000">
|
---|
647 | <I>cond<MATH CLASS="INLINE">
|
---|
648 | -
|
---|
649 | </MATH>print</I></A>
|
---|
650 | </H4>
|
---|
651 | <P><IMG WIDTH="533" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
652 | SRC="img78.gif"
|
---|
653 | ALT="$\textstyle\parbox{\pboxargslen}{\em cond params \/}$"> [<EM>FUNCTION</EM>]
|
---|
654 | <BLOCKQUOTE>
|
---|
655 | Pretty<MATH CLASS="INLINE">
|
---|
656 | -
|
---|
657 | </MATH>print a condition COND, using symbol list PARAMS as
|
---|
658 | parameter names. </BLOCKQUOTE><H4><A NAME="SECTION000400150000000000000">
|
---|
659 | <I>add<MATH CLASS="INLINE">
|
---|
660 | -
|
---|
661 | </MATH>pairs</I></A>
|
---|
662 | </H4>
|
---|
663 | <P><IMG WIDTH="541" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
664 | SRC="img79.gif"
|
---|
665 | ALT="$\textstyle\parbox{\pboxargslen}{\em gs pred \/}$"> [<EM>FUNCTION</EM>]
|
---|
666 | <BLOCKQUOTE>
|
---|
667 | The parameter GS shoud be a Grobner system, i.e. a set of pairs
|
---|
668 | (CONDITION POLY<MATH CLASS="INLINE">
|
---|
669 | -
|
---|
670 | </MATH>LIST) This functions adds the third component: the
|
---|
671 | list of initial critical pairs (I J), as in the ordinary Grobner
|
---|
672 | basis algorithm. In addition, it adds the length of of the
|
---|
673 | POLY<MATH CLASS="INLINE">
|
---|
674 | -
|
---|
675 | </MATH>LIST, less 1, as the fourth component. The resulting list of
|
---|
676 | quadruples is returned.</BLOCKQUOTE><H4><A NAME="SECTION000400160000000000000">
|
---|
677 | <I>cond<MATH CLASS="INLINE">
|
---|
678 | -
|
---|
679 | </MATH>part</I></A>
|
---|
680 | </H4>
|
---|
681 | <P><IMG WIDTH="538" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
|
---|
682 | SRC="img80.gif"
|
---|
683 | ALT="$\textstyle\parbox{\pboxargslen}{\em p \/}$"> [<EM>FUNCTION</EM>]
|
---|
684 | <BLOCKQUOTE>
|
---|
685 | Find the part of a colored polynomial P starting with the first
|
---|
686 | non<MATH CLASS="INLINE">
|
---|
687 | -
|
---|
688 | </MATH>green term.</BLOCKQUOTE><H4><A NAME="SECTION000400170000000000000">
|
---|
689 | <I>cond<MATH CLASS="INLINE">
|
---|
690 | -
|
---|
691 | </MATH>hm</I></A>
|
---|
692 | </H4>
|
---|
693 | <P><IMG WIDTH="538" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
|
---|
694 | SRC="img80.gif"
|
---|
695 | ALT="$\textstyle\parbox{\pboxargslen}{\em p \/}$"> [<EM>FUNCTION</EM>]
|
---|
696 | <BLOCKQUOTE>
|
---|
697 | Return the conditional head monomial of a colored polynomial P.</BLOCKQUOTE><H4><A NAME="SECTION000400180000000000000">
|
---|
698 | <I>delete<MATH CLASS="INLINE">
|
---|
699 | -
|
---|
700 | </MATH>green<MATH CLASS="INLINE">
|
---|
701 | -
|
---|
702 | </MATH>polys</I></A>
|
---|
703 | </H4>
|
---|
704 | <P><IMG WIDTH="472" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
|
---|
705 | SRC="img81.gif"
|
---|
706 | ALT="$\textstyle\parbox{\pboxargslen}{\em gamma \/}$"> [<EM>FUNCTION</EM>]
|
---|
707 | <BLOCKQUOTE>
|
---|
708 | Delete totally green polynomials from in a grobner system GAMMA.</BLOCKQUOTE><H4><A NAME="SECTION000400190000000000000">
|
---|
709 | <I>grobner<MATH CLASS="INLINE">
|
---|
710 | -
|
---|
711 | </MATH>system</I></A>
|
---|
712 | </H4>
|
---|
713 | <P><IMG WIDTH="499" HEIGHT="130" ALIGN="MIDDLE" BORDER="0"
|
---|
714 | SRC="img82.gif"
|
---|
715 | ALT="$\textstyle\parbox{\pboxargslen}{\em f {\sf \&key} (cover (list '(nil nil))) (ma...
|
---|
716 | ...char93 '(lambda (cond) (determine f cond parameter$-$order ring))
|
---|
717 | cover))) \/}$"> [<EM>FUNCTION</EM>]
|
---|
718 | <BLOCKQUOTE>
|
---|
719 | This function returns a grobner system, given a list of colored
|
---|
720 | polynomials F, Other parameters are:
|
---|
721 | A cover COVER, i.e. a list of conditions, i.e. pairs of the form
|
---|
722 | (GREEN<MATH CLASS="INLINE">
|
---|
723 | -
|
---|
724 | </MATH>LIST RED<MATH CLASS="INLINE">
|
---|
725 | -
|
---|
726 | </MATH>LIST), where GREEN<MATH CLASS="INLINE">
|
---|
727 | -
|
---|
728 | </MATH>LIST and RED<MATH CLASS="INLINE">
|
---|
729 | -
|
---|
730 | </MATH>LIST are to
|
---|
731 | lists of ordinary polynomials in parameters. A monomial order
|
---|
732 | MAIN<MATH CLASS="INLINE">
|
---|
733 | -
|
---|
734 | </MATH>ORDER used on main variables (not parameters). A monomial
|
---|
735 | order PARAMETER<MATH CLASS="INLINE">
|
---|
736 | -
|
---|
737 | </MATH>ORDER used in calculations with parameters only.
|
---|
738 | REDUCE, a flag deciding whether COLORED<MATH CLASS="INLINE">
|
---|
739 | -
|
---|
740 | </MATH>REDUCTION will be performed
|
---|
741 | on the resulting grobner system. GREEN<MATH CLASS="INLINE">
|
---|
742 | -
|
---|
743 | </MATH>REDUCE, a flag deciding
|
---|
744 | whether the green list of each condition will be reduced in a form of
|
---|
745 | a reduced Grobner basis. TOP<MATH CLASS="INLINE">
|
---|
746 | -
|
---|
747 | </MATH>REDUCTION<MATH CLASS="INLINE">
|
---|
748 | -
|
---|
749 | </MATH>ONLY, a flag deciding
|
---|
750 | whether in the internal calculations in the space of parameters top
|
---|
751 | reduction only will be used. RING, a structure as in the package
|
---|
752 | COEFFICIENT<MATH CLASS="INLINE">
|
---|
753 | -
|
---|
754 | </MATH>RING, used in operations on the coefficients of the
|
---|
755 | polynomials in parameters. </BLOCKQUOTE><H4><A NAME="SECTION000400200000000000000">
|
---|
756 | <I>reorder<MATH CLASS="INLINE">
|
---|
757 | -
|
---|
758 | </MATH>pairs</I></A>
|
---|
759 | </H4>
|
---|
760 | <P><IMG WIDTH="518" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
761 | SRC="img83.gif"
|
---|
762 | ALT="$\textstyle\parbox{\pboxargslen}{\em b bnew g pred {\sf \&optional} (sort$-$first nil) \/}$"> [<EM>FUNCTION</EM>]
|
---|
763 | <BLOCKQUOTE>
|
---|
764 | Reorder pairs according to some heuristic. The heuristic at this time
|
---|
765 | is ad hoc, in the future it should be replaced with sugar strategy
|
---|
766 | and a mechanism for implementing new heuristic strategies, as in the
|
---|
767 | GROBNER package. </BLOCKQUOTE><H4><A NAME="SECTION000400210000000000000">
|
---|
768 | <I>colored<MATH CLASS="INLINE">
|
---|
769 | -
|
---|
770 | </MATH>criterion<MATH CLASS="INLINE">
|
---|
771 | -
|
---|
772 | </MATH>1</I></A>
|
---|
773 | </H4>
|
---|
774 | <P><IMG WIDTH="471" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
775 | SRC="img84.gif"
|
---|
776 | ALT="$\textstyle\parbox{\pboxargslen}{\em i j f \/}$"> [<EM>FUNCTION</EM>]
|
---|
777 | <BLOCKQUOTE>
|
---|
778 | Buchberger criterion 1 for colored polynomials.</BLOCKQUOTE><H4><A NAME="SECTION000400220000000000000">
|
---|
779 | <I>colored<MATH CLASS="INLINE">
|
---|
780 | -
|
---|
781 | </MATH>criterion<MATH CLASS="INLINE">
|
---|
782 | -
|
---|
783 | </MATH>2</I></A>
|
---|
784 | </H4>
|
---|
785 | <P><IMG WIDTH="471" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
786 | SRC="img85.gif"
|
---|
787 | ALT="$\textstyle\parbox{\pboxargslen}{\em i j f b s \/}$"> [<EM>FUNCTION</EM>]
|
---|
788 | <BLOCKQUOTE>
|
---|
789 | Buchberger criterion 2 for colored polynomials.</BLOCKQUOTE><H4><A NAME="SECTION000400230000000000000">
|
---|
790 | <I>cond<MATH CLASS="INLINE">
|
---|
791 | -
|
---|
792 | </MATH>normal<MATH CLASS="INLINE">
|
---|
793 | -
|
---|
794 | </MATH>form</I></A>
|
---|
795 | </H4>
|
---|
796 | <P><IMG WIDTH="474" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
797 | SRC="img86.gif"
|
---|
798 | ALT="$\textstyle\parbox{\pboxargslen}{\em f fl main$-$order parameter$-$order top$-$reduction$-$only ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
799 | <BLOCKQUOTE>
|
---|
800 | Returns the conditional normal form of a colored polynomial F with
|
---|
801 | respect to the list of colored polynomials FL. The list FL is assumed
|
---|
802 | to consist of determined polynomials, i.e. such that the first term
|
---|
803 | which is not marked :GREEN is :RED.</BLOCKQUOTE><H4><A NAME="SECTION000400240000000000000">
|
---|
804 | <I>cond<MATH CLASS="INLINE">
|
---|
805 | -
|
---|
806 | </MATH>spoly</I></A>
|
---|
807 | </H4>
|
---|
808 | <P><IMG WIDTH="530" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
809 | SRC="img87.gif"
|
---|
810 | ALT="$\textstyle\parbox{\pboxargslen}{\em f g main$-$order parameter$-$order ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
811 | <BLOCKQUOTE>
|
---|
812 | Returns the conditional S<MATH CLASS="INLINE">
|
---|
813 | -
|
---|
814 | </MATH>polynomial of two colored polynomials F
|
---|
815 | and G. Both polynomials are assumed to be determined.</BLOCKQUOTE><H4><A NAME="SECTION000400250000000000000">
|
---|
816 | <I>cond<MATH CLASS="INLINE">
|
---|
817 | -
|
---|
818 | </MATH>lm</I></A>
|
---|
819 | </H4>
|
---|
820 | <P><IMG WIDTH="549" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
821 | SRC="img88.gif"
|
---|
822 | ALT="$\textstyle\parbox{\pboxargslen}{\em f \/}$"> [<EM>FUNCTION</EM>]
|
---|
823 | <BLOCKQUOTE>
|
---|
824 | Returns the conditional leading monomial of a colored polynomial F,
|
---|
825 | which is assumed to be determined.</BLOCKQUOTE><H4><A NAME="SECTION000400260000000000000">
|
---|
826 | <I>cond<MATH CLASS="INLINE">
|
---|
827 | -
|
---|
828 | </MATH>lc</I></A>
|
---|
829 | </H4>
|
---|
830 | <P><IMG WIDTH="549" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
831 | SRC="img88.gif"
|
---|
832 | ALT="$\textstyle\parbox{\pboxargslen}{\em f \/}$"> [<EM>FUNCTION</EM>]
|
---|
833 | <BLOCKQUOTE>
|
---|
834 | Returns the conditional leading coefficient of a colored polynomial
|
---|
835 | F, which is assumed to be determined.</BLOCKQUOTE><H4><A NAME="SECTION000400270000000000000">
|
---|
836 | <I>colored<MATH CLASS="INLINE">
|
---|
837 | -
|
---|
838 | </MATH>term<MATH CLASS="INLINE">
|
---|
839 | -
|
---|
840 | </MATH>times<MATH CLASS="INLINE">
|
---|
841 | -
|
---|
842 | </MATH>poly</I></A>
|
---|
843 | </H4>
|
---|
844 | <P><IMG WIDTH="426" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
845 | SRC="img89.gif"
|
---|
846 | ALT="$\textstyle\parbox{\pboxargslen}{\em term f order ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
847 | <BLOCKQUOTE>
|
---|
848 | Returns the product of a colored term TERM and a colored polynomial
|
---|
849 | F. </BLOCKQUOTE><H4><A NAME="SECTION000400280000000000000">
|
---|
850 | <I>colored<MATH CLASS="INLINE">
|
---|
851 | -
|
---|
852 | </MATH>scalar<MATH CLASS="INLINE">
|
---|
853 | -
|
---|
854 | </MATH>times<MATH CLASS="INLINE">
|
---|
855 | -
|
---|
856 | </MATH>poly</I></A>
|
---|
857 | </H4>
|
---|
858 | <P><IMG WIDTH="419" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
859 | SRC="img90.gif"
|
---|
860 | ALT="$\textstyle\parbox{\pboxargslen}{\em c f ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
861 | <BLOCKQUOTE>
|
---|
862 | Returns the product of an element of the coefficient ring C a colored
|
---|
863 | polynomial F. </BLOCKQUOTE><H4><A NAME="SECTION000400290000000000000">
|
---|
864 | <I>colored<MATH CLASS="INLINE">
|
---|
865 | -
|
---|
866 | </MATH>term*</I></A>
|
---|
867 | </H4>
|
---|
868 | <P><IMG WIDTH="509" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
869 | SRC="img91.gif"
|
---|
870 | ALT="$\textstyle\parbox{\pboxargslen}{\em term1 term2 order ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
871 | <BLOCKQUOTE>
|
---|
872 | Returns the product of two colored terms TERM1 and TERM2.</BLOCKQUOTE><H4><A NAME="SECTION000400300000000000000">
|
---|
873 | <I>color*</I></A>
|
---|
874 | </H4>
|
---|
875 | <P><IMG WIDTH="570" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
|
---|
876 | SRC="img92.gif"
|
---|
877 | ALT="$\textstyle\parbox{\pboxargslen}{\em c1 c2 \/}$"> [<EM>FUNCTION</EM>]
|
---|
878 | <BLOCKQUOTE>
|
---|
879 | Returns a product of two colores. Rules:
|
---|
880 | :red * :red yields :red
|
---|
881 | any * :green yields :green
|
---|
882 | otherwise the result is :white.</BLOCKQUOTE><H4><A NAME="SECTION000400310000000000000">
|
---|
883 | <I>color+</I></A>
|
---|
884 | </H4>
|
---|
885 | <P><IMG WIDTH="570" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
|
---|
886 | SRC="img92.gif"
|
---|
887 | ALT="$\textstyle\parbox{\pboxargslen}{\em c1 c2 \/}$"> [<EM>FUNCTION</EM>]
|
---|
888 | <BLOCKQUOTE>
|
---|
889 | Returns a sum of colors. Rules:
|
---|
890 | :green + :green yields :green,
|
---|
891 | :red + :green yields :red
|
---|
892 | any other result is :white.</BLOCKQUOTE><H4><A NAME="SECTION000400320000000000000">
|
---|
893 | <I>color<MATH CLASS="INLINE">
|
---|
894 | -
|
---|
895 | </MATH></I></A>
|
---|
896 | </H4>
|
---|
897 | <P><IMG WIDTH="570" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
|
---|
898 | SRC="img92.gif"
|
---|
899 | ALT="$\textstyle\parbox{\pboxargslen}{\em c1 c2 \/}$"> [<EM>FUNCTION</EM>]
|
---|
900 | <BLOCKQUOTE>
|
---|
901 | Identical to COLOR+.</BLOCKQUOTE><H4><A NAME="SECTION000400330000000000000">
|
---|
902 | <I>colored<MATH CLASS="INLINE">
|
---|
903 | -
|
---|
904 | </MATH>poly+</I></A>
|
---|
905 | </H4>
|
---|
906 | <P><IMG WIDTH="508" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
907 | SRC="img93.gif"
|
---|
908 | ALT="$\textstyle\parbox{\pboxargslen}{\em p q main$-$order parameter$-$order ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
909 | <BLOCKQUOTE>
|
---|
910 | Returns the sum of colored polynomials P and Q.</BLOCKQUOTE><H4><A NAME="SECTION000400340000000000000">
|
---|
911 | <I>colored<MATH CLASS="INLINE">
|
---|
912 | -
|
---|
913 | </MATH>poly<MATH CLASS="INLINE">
|
---|
914 | -
|
---|
915 | </MATH></I></A>
|
---|
916 | </H4>
|
---|
917 | <P><IMG WIDTH="508" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
918 | SRC="img93.gif"
|
---|
919 | ALT="$\textstyle\parbox{\pboxargslen}{\em p q main$-$order parameter$-$order ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
920 | <BLOCKQUOTE>
|
---|
921 | Returns the difference of colored polynomials P and Q.</BLOCKQUOTE><H4><A NAME="SECTION000400350000000000000">
|
---|
922 | <I>colored<MATH CLASS="INLINE">
|
---|
923 | -
|
---|
924 | </MATH>term<MATH CLASS="INLINE">
|
---|
925 | -
|
---|
926 | </MATH>uminus</I></A>
|
---|
927 | </H4>
|
---|
928 | <P><IMG WIDTH="455" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
|
---|
929 | SRC="img94.gif"
|
---|
930 | ALT="$\textstyle\parbox{\pboxargslen}{\em term ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
931 | <BLOCKQUOTE>
|
---|
932 | Returns the negation of a colored term TERM.</BLOCKQUOTE><H4><A NAME="SECTION000400360000000000000">
|
---|
933 | <I>colored<MATH CLASS="INLINE">
|
---|
934 | -
|
---|
935 | </MATH>minus<MATH CLASS="INLINE">
|
---|
936 | -
|
---|
937 | </MATH>poly</I></A>
|
---|
938 | </H4>
|
---|
939 | <P><IMG WIDTH="446" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
|
---|
940 | SRC="img9.gif"
|
---|
941 | ALT="$\textstyle\parbox{\pboxargslen}{\em p ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
942 | <BLOCKQUOTE>
|
---|
943 | Returns the negation of a colored polynomial P.</BLOCKQUOTE><H4><A NAME="SECTION000400370000000000000">
|
---|
944 | <I>string<MATH CLASS="INLINE">
|
---|
945 | -
|
---|
946 | </MATH>grobner<MATH CLASS="INLINE">
|
---|
947 | -
|
---|
948 | </MATH>system</I></A>
|
---|
949 | </H4>
|
---|
950 | <P><IMG WIDTH="447" HEIGHT="150" ALIGN="MIDDLE" BORDER="0"
|
---|
951 | SRC="img95.gif"
|
---|
952 | ALT="$\textstyle\parbox{\pboxargslen}{\em f vars params {\sf \&key} (cover
|
---|
953 | (list
|
---|
954 | (l...
|
---|
955 | ...meter$-$order)) (cover
|
---|
956 | (string$-$cover
|
---|
957 | cover
|
---|
958 | params
|
---|
959 | parameter$-$order)) \/}$"> [<EM>FUNCTION</EM>]
|
---|
960 | <BLOCKQUOTE>
|
---|
961 | An interface to GROBNER<MATH CLASS="INLINE">
|
---|
962 | -
|
---|
963 | </MATH>SYSTEM in which polynomials can be
|
---|
964 | specified in infix notations as strings. Lists of polynomials are
|
---|
965 | comma<MATH CLASS="INLINE">
|
---|
966 | -
|
---|
967 | </MATH>separated list marked by a matchfix operators [] </BLOCKQUOTE><H4><A NAME="SECTION000400380000000000000">
|
---|
968 | <I>string<MATH CLASS="INLINE">
|
---|
969 | -
|
---|
970 | </MATH>cond</I></A>
|
---|
971 | </H4>
|
---|
972 | <P><IMG WIDTH="527" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
973 | SRC="img96.gif"
|
---|
974 | ALT="$\textstyle\parbox{\pboxargslen}{\em cond params {\sf \&optional} (order \char93 'lex$\gt$) \/}$"> [<EM>FUNCTION</EM>]
|
---|
975 | <BLOCKQUOTE>
|
---|
976 | Return the internal representation of a condition COND, specified
|
---|
977 | as pairs of strings (GREEN<MATH CLASS="INLINE">
|
---|
978 | -
|
---|
979 | </MATH>LIST RED<MATH CLASS="INLINE">
|
---|
980 | -
|
---|
981 | </MATH>LIST). GREEN<MATH CLASS="INLINE">
|
---|
982 | -
|
---|
983 | </MATH>LIST and
|
---|
984 | RED<MATH CLASS="INLINE">
|
---|
985 | -
|
---|
986 | </MATH>LIST in the input are assumed to be strings which parse to two
|
---|
987 | lists of polynomials with respect to variables whose names are in the
|
---|
988 | list of symbols PARAMS. ORDER is the predicate used to sort the terms
|
---|
989 | of the polynomials.</BLOCKQUOTE><H4><A NAME="SECTION000400390000000000000">
|
---|
990 | <I>string<MATH CLASS="INLINE">
|
---|
991 | -
|
---|
992 | </MATH>cover</I></A>
|
---|
993 | </H4>
|
---|
994 | <P><IMG WIDTH="523" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
995 | SRC="img97.gif"
|
---|
996 | ALT="$\textstyle\parbox{\pboxargslen}{\em cover params {\sf \&optional} (order \char93 'lex$\gt$) \/}$"> [<EM>FUNCTION</EM>]
|
---|
997 | <BLOCKQUOTE>
|
---|
998 | Returns the internal representation of COVER, given in the form of
|
---|
999 | a list of conditions. See STRING<MATH CLASS="INLINE">
|
---|
1000 | -
|
---|
1001 | </MATH>COND for description of a
|
---|
1002 | condition. </BLOCKQUOTE><H4><A NAME="SECTION000400400000000000000">
|
---|
1003 | <I>saturate<MATH CLASS="INLINE">
|
---|
1004 | -
|
---|
1005 | </MATH>cover</I></A>
|
---|
1006 | </H4>
|
---|
1007 | <P><IMG WIDTH="506" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
1008 | SRC="img98.gif"
|
---|
1009 | ALT="$\textstyle\parbox{\pboxargslen}{\em cover order ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
1010 | <BLOCKQUOTE>
|
---|
1011 | Brings every condition of a list of conditions COVER to the form (G
|
---|
1012 | R) where G is saturated with respect to R and G is a Grobner basis
|
---|
1013 | We could reduce R so that the elements of R are relatively prime,
|
---|
1014 | but this is not currently done.</BLOCKQUOTE><H4><A NAME="SECTION000400410000000000000">
|
---|
1015 | <I>saturate<MATH CLASS="INLINE">
|
---|
1016 | -
|
---|
1017 | </MATH>cond</I></A>
|
---|
1018 | </H4>
|
---|
1019 | <P><IMG WIDTH="510" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
1020 | SRC="img99.gif"
|
---|
1021 | ALT="$\textstyle\parbox{\pboxargslen}{\em cond order ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
1022 | <BLOCKQUOTE>
|
---|
1023 | Saturate a single condition COND. An auxillary function of
|
---|
1024 | SATURATE<MATH CLASS="INLINE">
|
---|
1025 | -
|
---|
1026 | </MATH>COVER. </BLOCKQUOTE><H4><A NAME="SECTION000400420000000000000">
|
---|
1027 | <I>string<MATH CLASS="INLINE">
|
---|
1028 | -
|
---|
1029 | </MATH>determine</I></A>
|
---|
1030 | </H4>
|
---|
1031 | <P><IMG WIDTH="492" HEIGHT="130" ALIGN="MIDDLE" BORDER="0"
|
---|
1032 | SRC="img100.gif"
|
---|
1033 | ALT="$\textstyle\parbox{\pboxargslen}{\em f vars params {\sf \&key} (cond
|
---|
1034 | '([] [])) ...
|
---|
1035 | ...arameter$-$order)) (cond
|
---|
1036 | (string$-$cond
|
---|
1037 | cond
|
---|
1038 | params
|
---|
1039 | parameter$-$order)) \/}$"> [<EM>FUNCTION</EM>]
|
---|
1040 | <BLOCKQUOTE>
|
---|
1041 | A string interface to DETERMINE. See the documentation of
|
---|
1042 | STRING<MATH CLASS="INLINE">
|
---|
1043 | -
|
---|
1044 | </MATH>GROBNER<MATH CLASS="INLINE">
|
---|
1045 | -
|
---|
1046 | </MATH>SYSTEM. </BLOCKQUOTE><H4><A NAME="SECTION000400430000000000000">
|
---|
1047 | <I>tidy<MATH CLASS="INLINE">
|
---|
1048 | -
|
---|
1049 | </MATH>grobner<MATH CLASS="INLINE">
|
---|
1050 | -
|
---|
1051 | </MATH>system</I></A>
|
---|
1052 | </H4>
|
---|
1053 | <P><IMG WIDTH="460" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
1054 | SRC="img101.gif"
|
---|
1055 | ALT="$\textstyle\parbox{\pboxargslen}{\em gs main$-$order parameter$-$order reduce green$-$reduce ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
1056 | <BLOCKQUOTE>
|
---|
1057 | Apply TIDY<MATH CLASS="INLINE">
|
---|
1058 | -
|
---|
1059 | </MATH>PAIR to every pair of a Grobner system.</BLOCKQUOTE><H4><A NAME="SECTION000400440000000000000">
|
---|
1060 | <I>tidy<MATH CLASS="INLINE">
|
---|
1061 | -
|
---|
1062 | </MATH>pair</I></A>
|
---|
1063 | </H4>
|
---|
1064 | <P><IMG WIDTH="546" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
1065 | SRC="img102.gif"
|
---|
1066 | ALT="$\textstyle\parbox{\pboxargslen}{\em pair main$-$order parameter$-$order reduce green$-$reduce ring {\sf \&aux} gs \/}$"> [<EM>FUNCTION</EM>]
|
---|
1067 | <BLOCKQUOTE>
|
---|
1068 | Make the output of Grobner system more readable by performing
|
---|
1069 | certain simplifications on an element of a Grobner system.
|
---|
1070 | If REDUCE is non<MATH CLASS="INLINE">
|
---|
1071 | -
|
---|
1072 | </MATH>NIL then COLORED<MATH CLASS="INLINE">
|
---|
1073 | -
|
---|
1074 | </MATH>reduction will be performed.
|
---|
1075 | In addition TIDY<MATH CLASS="INLINE">
|
---|
1076 | -
|
---|
1077 | </MATH>COND is called on the condition part of the pair
|
---|
1078 | PAIR. </BLOCKQUOTE><H4><A NAME="SECTION000400450000000000000">
|
---|
1079 | <I>tidy<MATH CLASS="INLINE">
|
---|
1080 | -
|
---|
1081 | </MATH>cond</I></A>
|
---|
1082 | </H4>
|
---|
1083 | <P><IMG WIDTH="510" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
1084 | SRC="img99.gif"
|
---|
1085 | ALT="$\textstyle\parbox{\pboxargslen}{\em cond order ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
1086 | <BLOCKQUOTE>
|
---|
1087 | Currently saturates condition COND and does RED<MATH CLASS="INLINE">
|
---|
1088 | -
|
---|
1089 | </MATH>REDUCTION on the
|
---|
1090 | red list. </BLOCKQUOTE><H4><A NAME="SECTION000400460000000000000">
|
---|
1091 | <I>colored<MATH CLASS="INLINE">
|
---|
1092 | -
|
---|
1093 | </MATH>reduction</I></A>
|
---|
1094 | </H4>
|
---|
1095 | <P><IMG WIDTH="485" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
|
---|
1096 | SRC="img103.gif"
|
---|
1097 | ALT="$\textstyle\parbox{\pboxargslen}{\em cond p main$-$order parameter$-$order ring {\sf \&aux} (open
|
---|
1098 | (list
|
---|
1099 | (list
|
---|
1100 | cond
|
---|
1101 | nil
|
---|
1102 | p))) closed \/}$"> [<EM>FUNCTION</EM>]
|
---|
1103 | <BLOCKQUOTE>
|
---|
1104 | Reduce a list of colored polynomials P. The difficulty as compared
|
---|
1105 | to the usual Buchberger algorithm is that the polys may have the same
|
---|
1106 | leading monomial which may result in cancellations and polynomials
|
---|
1107 | which may not be determined. Thus, when we find those, we will have
|
---|
1108 | to split the condition by calling determine. Returns a list of pairs
|
---|
1109 | (COND' P') where P' is a reduced grobner basis with respect to any
|
---|
1110 | parameter choice compatible with condition COND'. Moreover, COND'
|
---|
1111 | form a cover of COND.</BLOCKQUOTE><H4><A NAME="SECTION000400470000000000000">
|
---|
1112 | <I>green<MATH CLASS="INLINE">
|
---|
1113 | -
|
---|
1114 | </MATH>reduce<MATH CLASS="INLINE">
|
---|
1115 | -
|
---|
1116 | </MATH>colored<MATH CLASS="INLINE">
|
---|
1117 | -
|
---|
1118 | </MATH>poly</I></A>
|
---|
1119 | </H4>
|
---|
1120 | <P><IMG WIDTH="413" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
1121 | SRC="img104.gif"
|
---|
1122 | ALT="$\textstyle\parbox{\pboxargslen}{\em cond f parameter$-$order ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
1123 | <BLOCKQUOTE>
|
---|
1124 | It takes a colored polynomial F and it returns a modified
|
---|
1125 | polynomial obtained by reducing coefficient of F modulo green list of
|
---|
1126 | the condition COND.</BLOCKQUOTE><H4><A NAME="SECTION000400480000000000000">
|
---|
1127 | <I>green<MATH CLASS="INLINE">
|
---|
1128 | -
|
---|
1129 | </MATH>reduce<MATH CLASS="INLINE">
|
---|
1130 | -
|
---|
1131 | </MATH>colored<MATH CLASS="INLINE">
|
---|
1132 | -
|
---|
1133 | </MATH>list</I></A>
|
---|
1134 | </H4>
|
---|
1135 | <P><IMG WIDTH="421" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
1136 | SRC="img105.gif"
|
---|
1137 | ALT="$\textstyle\parbox{\pboxargslen}{\em cond fl parameter$-$order ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
1138 | <BLOCKQUOTE>
|
---|
1139 | Apply GREEN<MATH CLASS="INLINE">
|
---|
1140 | -
|
---|
1141 | </MATH>REDUCE<MATH CLASS="INLINE">
|
---|
1142 | -
|
---|
1143 | </MATH>COLORED<MATH CLASS="INLINE">
|
---|
1144 | -
|
---|
1145 | </MATH>POLY to a list of polynomials FL.</BLOCKQUOTE><H4><A NAME="SECTION000400490000000000000">
|
---|
1146 | <I>cond<MATH CLASS="INLINE">
|
---|
1147 | -
|
---|
1148 | </MATH>system<MATH CLASS="INLINE">
|
---|
1149 | -
|
---|
1150 | </MATH>green<MATH CLASS="INLINE">
|
---|
1151 | -
|
---|
1152 | </MATH>reduce</I></A>
|
---|
1153 | </H4>
|
---|
1154 | <P><IMG WIDTH="411" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
---|
1155 | SRC="img106.gif"
|
---|
1156 | ALT="$\textstyle\parbox{\pboxargslen}{\em gs parameter$-$order ring \/}$"> [<EM>FUNCTION</EM>]
|
---|
1157 | <BLOCKQUOTE>
|
---|
1158 | Apply GREEN<MATH CLASS="INLINE">
|
---|
1159 | -
|
---|
1160 | </MATH>REDUCE<MATH CLASS="INLINE">
|
---|
1161 | -
|
---|
1162 | </MATH>COLORED<MATH CLASS="INLINE">
|
---|
1163 | -
|
---|
1164 | </MATH>LIST to every pair of
|
---|
1165 | a grobner system GS.</BLOCKQUOTE><H4><A NAME="SECTION000400500000000000000">
|
---|
1166 | <I>parse<MATH CLASS="INLINE">
|
---|
1167 | -
|
---|
1168 | </MATH>to<MATH CLASS="INLINE">
|
---|
1169 | -
|
---|
1170 | </MATH>colored<MATH CLASS="INLINE">
|
---|
1171 | -
|
---|
1172 | </MATH>poly<MATH CLASS="INLINE">
|
---|
1173 | -
|
---|
1174 | </MATH>list</I></A>
|
---|
1175 | </H4>
|
---|
1176 | <P><IMG WIDTH="412" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
|
---|
1177 | SRC="img107.gif"
|
---|
1178 | ALT="$\textstyle\parbox{\pboxargslen}{\em f vars params main$-$order parameter$-$order {\sf \&aux} (k
|
---|
1179 | (length
|
---|
1180 | vars)) (vars$-$params
|
---|
1181 | (append
|
---|
1182 | vars
|
---|
1183 | params)) \/}$"> [<EM>FUNCTION</EM>]
|
---|
1184 | <BLOCKQUOTE>
|
---|
1185 | Parse a list of polynomials F, given as a string, with respect to
|
---|
1186 | a list of variables VARS, given as a list of symbols, to the internal
|
---|
1187 | representation of a colored polynomial. The polynomials will be
|
---|
1188 | properly sorted by MAIN<MATH CLASS="INLINE">
|
---|
1189 | -
|
---|
1190 | </MATH>ORDER, with the coefficients, which are
|
---|
1191 | polynomials in parameters, sorted by PARAMETER<MATH CLASS="INLINE">
|
---|
1192 | -
|
---|
1193 | </MATH>ORDER. Both orders
|
---|
1194 | must be admissible monomial orders. This form is suitable for parsing
|
---|
1195 | polynomials with integer coefficients.</BLOCKQUOTE><H4><A NAME="SECTION000400510000000000000">
|
---|
1196 | <I>red<MATH CLASS="INLINE">
|
---|
1197 | -
|
---|
1198 | </MATH>reduction</I></A>
|
---|
1199 | </H4>
|
---|
1200 | <P><IMG WIDTH="513" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
---|
1201 | SRC="img108.gif"
|
---|
1202 | ALT="$\textstyle\parbox{\pboxargslen}{\em p pred ring {\sf \&aux} (p
|
---|
1203 | (remove$-$if
|
---|
1204 | \char93 'poly$-$constant$-$p
|
---|
1205 | p)) \/}$"> [<EM>FUNCTION</EM>]
|
---|
1206 | <BLOCKQUOTE>
|
---|
1207 | Takes a family of polynomials and produce a list whose prime factors
|
---|
1208 | are the same but they are relatively prime
|
---|
1209 | Repetitively used the following procedure: it finds two elements f, g
|
---|
1210 | of P which are not relatively prime; it replaces f and g with
|
---|
1211 | f/GCD(f,g), g/ GCD(f,f) and GCD(f,g).</BLOCKQUOTE><HR>
|
---|
1212 | <!--Navigation Panel-->
|
---|
1213 | <A NAME="tex2html855"
|
---|
1214 | HREF="node5.html">
|
---|
1215 | <IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="next_motif.gif"></A>
|
---|
1216 | <A NAME="tex2html852"
|
---|
1217 | HREF="manual.html">
|
---|
1218 | <IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="up_motif.gif"></A>
|
---|
1219 | <A NAME="tex2html846"
|
---|
1220 | HREF="node3.html">
|
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1221 | <IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="previous_motif.gif"></A>
|
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1222 | <A NAME="tex2html854"
|
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1223 | HREF="node1.html">
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1224 | <IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents" SRC="contents_motif.gif"></A>
|
---|
1225 | <BR>
|
---|
1226 | <B> Next:</B> <A NAME="tex2html856"
|
---|
1227 | HREF="node5.html">The Coefficient Ring package</A>
|
---|
1228 | <B> Up:</B> <A NAME="tex2html853"
|
---|
1229 | HREF="manual.html">CGBLisp User Guide and</A>
|
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1230 | <B> Previous:</B> <A NAME="tex2html847"
|
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1231 | HREF="node3.html">The String Interface to</A>
|
---|
1232 | <!--End of Navigation Panel-->
|
---|
1233 | <ADDRESS>
|
---|
1234 | <I>Marek Rychlik</I>
|
---|
1235 | <BR><I>3/21/1998</I>
|
---|
1236 | </ADDRESS>
|
---|
1237 | </BODY>
|
---|
1238 | </HTML>
|
---|