dixon/dr.mw at master · mincode/dixon · GitHub

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<Text-field style="_cstyle12" layout="_pstyle10">Introduction to the DR (Dixon Resultant) Package</Text-field>
<Text-field style="_cstyle13" layout="_pstyle11">(C) 2015 Manfred Minimair</Text-field>
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<Text-field style="_cstyle14" layout="_pstyle12">Description</Text-field>
<Text-field style="_pstyle12" layout="_pstyle12"><Font style="_cstyle15">This package provides the function DixonResultant which computes the Dixon projection of a list of multivariate polynomials. The function implements Kapur/Saxena/Yang's approach (see D. Kapur, T. Saxena, L. Yang, </Font><Font style="Text" italic="true">Algebraic and Geometric Reasoning Using Dixon Resultants</Font><Font style="_cstyle15">. International Symposium on Symbolic and Algebraic Computation (ISSAC), July 1994, Oxford University, England).</Font></Text-field>
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<Text-field style="_cstyle14" layout="_pstyle12">Calling Sequence</Text-field>
<Text-field style="_cstyle17" layout="_pstyle13">DR:-DixonResultant(polylist, varlist)</Text-field>
<Text-field style="_cstyle14" size="12" layout="_pstyle12"><Font bold="false" size="12">Additional features of the package DR are shown below.</Font><Font size="12">

</Font><Font size="14">Parameters</Font></Text-field>
<Text-field style="_cstyle17" layout="_pstyle13">polylist     - list of (multivariate) polynomials</Text-field>
<Text-field style="_cstyle17" layout="_pstyle13">varlist       - list of variables of the polynomials in polylist</Text-field>
<Text-field style="_pstyle12" layout="_pstyle12"></Text-field>
<Text-field style="Normal" layout="Normal">infolevel[DR] := 2;</Text-field>
<Text-field style="Normal" layout="Normal">causes the package to print some progress information for the computation</Text-field>
<Text-field style="Normal" layout="Normal"></Text-field>
<Text-field style="Normal" layout="Normal"><Font style="_cstyle14">How to cite the package</Font></Text-field>
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<Text-field style="Normal" layout="Normal">@Misc{minimair15:_dr,</Text-field>
<Text-field style="Normal" layout="Normal">  author =	 {M. Minimair},</Text-field>
<Text-field style="Normal" layout="Normal">  title =	 {DR: Dixon Resultant Package for Maple},</Text-field>
<Text-field style="Normal" layout="Normal">  howpublished = {http://minimair.org/dr},</Text-field>
<Text-field style="Normal" layout="Normal">  year =	 2015</Text-field>
<Text-field style="Normal" layout="Normal">}</Text-field>
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<Text-field style="_cstyle14" layout="_pstyle12">Examples</Text-field>
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<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">NihRKEVubmVwZXI2IjclLCoqJEkkdnNzR0YkIiIkISIiKiZGKCIiIkkkdnR0R0YkIiIjRilGKEYpSSJ4R0YkISIkLCoqJkYoRi5GLUYsRjAqJEYtRilGLEYtRjBJInlHRiRGMCwoKiRGKEYuRiwqJEYtRi5GKkkiekdGJEYqNyVGKUYpRi43JEYoRi1GLDclRi9GNEY4</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L19" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field style="Text" layout="Normal">Compute the Dixon resultant.</Text-field>
</Input>
</Group>
<Group labelreference="L16" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="DR:-DixonResultant(f, vars)" display="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">LV9JI0RSRzYiSS9EaXhvblJlc3VsdGFudEdGJTYkSSJmR0YlSSV2YXJzR0Yl</Equation></Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZQLUYjNiYtSSNtbkdGJDYkUSYzNjg2NEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RMSZJbnZpc2libGVUaW1lcztGJ0YxLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkktSSVtc3VwR0YkNiUtSSNtaUdGJDYlUSJ6RicvJSdpdGFsaWNHUSV0cnVlRicvRjJRJ2l0YWxpY0YnLUYuNiRRIjlGJ0YxLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0YxLUY1Ni1RKCZtaW51cztGJ0YxRjhGO0Y9Rj9GQUZDRkUvRkhRLDAuMjIyMjIyMmVtRicvRktGXG8tRiM2KC1GLjYkUScyNDg4MzJGJ0YxRjQtRk02JS1GUDYlUSJ4RidGU0ZWLUYuNiRRIjJGJ0YxRmVuRjQtRk02JUZPLUYuNiRRIjZGJ0YxRmVuRjEtRjU2LVEiK0YnRjFGOEY7Rj1GP0ZBRkNGRUZbb0Zdby1GIzYoRmBvRjQtRk02JS1GUDYlUSJ5RidGU0ZWRmhvRmVuRjRGW3BGMUZobi1GIzYoLUYuNiRRJzY5OTg0MEYnRjFGNC1GTTYlRmVvLUYuNiRRIjRGJ0YxRmVuRjQtRk02JUZPLUYuNiRRIjNGJ0YxRmVuRjFGaG4tRiM2Ki1GLjYkUSgzNjM5MTY4RidGMUY0RmNvRjRGZXBGNEZkcUYxRmhuLUYjNigtRi42JFEoMjIzOTQ4OEYnRjFGNEZjb0Y0LUZNNiVGTy1GLjYkUSI1RidGMUZlbkYxRmhuLUYjNihGXHFGNC1GTTYlRmdwRmFxRmVuRjRGZHFGMUZobi1GIzYoRmByRjRGZXBGNEZjckYxRmhuLUYjNiYtRi42JFEnNjYzNTUyRidGMUY0LUZNNiVGTy1GLjYkUSI3RidGMUZlbkYxRmhuLUYjNiYtRi42JFEnNDE5OTA0RidGMUY0LUZNNiVGZW9GXXBGZW5GMUZgcC1GIzYoLUYuNiRRKDEyNTk3MTJGJ0YxRjRGX3FGNEZlcEYxRmhuLUYjNigtRi42JFEoMjUxOTQyNEYnRjFGNEZfcUY0LUZNNiVGT0Zob0ZlbkYxRmhuLUYjNihGYXRGNEZjb0Y0RmpyRjFGaG4tRiM2KC1GLjYkUSgzNzMyNDgwRidGMUY0RmNvRjQtRk02JUZPRmFxRmVuRjFGYHAtRiM2JkZqc0Y0LUZNNiVGZ3BGXXBGZW5GMUZgcC1GIzYoRmZ0RjRGanJGNEZpdEYxRmBwLUYjNihGX3VGNEZlcEY0RmJ1RjFGYHAtRiM2KEZqc0Y0Rl9xRjRGT0YxRmhuLUYjNiotRi42JFEnODM5ODA4RidGMUY0RmNvRjRGZXBGNEZPRjFGYHAtRiM2KEZgckY0RmNvRjRGZHFGMUZgcC1GIzYoRmpzRjRGanJGNEZPRjFGYHAtRiM2KEZgckY0RmVwRjRGZHFGMUZgcC1GIzYmLUYuNiRRKDI5ODU5ODRGJ0YxRjRGY3JGMUYx">LFAqJEkiekc2IiIiKiIma28kKiZJInhHRiUiIiNGJCIiJyEnSylbIyomSSJ5R0YlRipGJEYrIidLKVsjKiZGKSIiJUYkIiIkISdTKSpwKihGKUYqRi5GKkYkRjIhKG8iUk8qJkYpRipGJCIiJiEoKVtSQSomRi5GMUYkRjJGMyomRi5GKkYkRjdGOCokRiQiIighJ19ObSokRilGKyEnLyo+JSomRilGMUYuRioiKDcoZjcqJkYpRjFGJEYqIShDJT5EKiZGKUYqRi5GMSEoNyhmNyomRilGKkYkRjEhKCFbS1AqJEYuRisiJy8qPiUqJkYuRjFGJEYqIihDJT5EKiZGLkYqRiRGMSIoIVtLUComRilGMUYkIiIiRkkqKEYpRipGLkYqRiRGTyEnMylSKSomRilGKkYkRjIiKClbUkEqJkYuRjFGJEZPRkkqJkYuRipGJEYyRlMqJEYkRjciKCUpZilI</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L22" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field style="Text" layout="Normal">Alternatively:</Text-field>
<Text-field style="Text" layout="Normal">Compute the Dixon polynomial.</Text-field>
</Input>
</Group>
<Group labelreference="L21" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="dp := DR:-DixonPolynomial(f, vars)" display="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">PkkjZHBHNiItX0kjRFJHRiRJMERpeG9uUG9seW5vbWlhbEdGJDYkSSJmR0YkSSV2YXJzR0Yk</Equation></Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="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">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</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L24" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field style="Text" layout="Normal">Compute the Dixon matrix.</Text-field>
</Input>
</Group>
<Group labelreference="L23" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="dm := DR:-DixonMatrix(dp)" display="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">PkkjZG1HNiItX0kjRFJHRiRJLERpeG9uTWF0cml4R0YkNiNJI2RwR0Yk</Equation></Text-field>
</Input>
<Output>
<Text-field style="2D Output" layout="Maple Output"><Equation executable="false" style="2D Output" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEjZG1GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSM6PUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYjNi0tSShtYWN0aW9uR0YkNiUtSShtZmVuY2VkR0YkNictSSdtdGFibGVHRiQ2OC1JJG10ckdGJDYmLUkkbXRkR0YkNigtRiM2Jy1GLDYmUSp+MTF+eH4xMX5GJ0YvLyUpcmVhZG9ubHlHRjFGMi1GLDYmUSdNYXRyaXhGJ0YvRl9vRjIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnRl9vRjkvJSlyb3dhbGlnbkdRIUYnLyUsY29sdW1uYWxpZ25HRmlvLyUrZ3JvdXBhbGlnbkdGaW8vJShyb3dzcGFuR1EiMUYnLyUrY29sdW1uc3BhbkdGYHBGZ29Gam9GXHAtRmVuNiYtRmhuNigtRiM2Jy1GLDYmUSxEYXRhflR5cGU6fkYnRi9GX29GMi1GLDYmUSlhbnl0aGluZ0YnRi9GX29GMkZkb0Zfb0Y5RmdvRmpvRlxwRl5wRmFwRmdvRmpvRlxwLUZlbjYmLUZobjYoLUYjNictRiw2JlEqU3RvcmFnZTp+RidGL0Zfb0YyLUYsNiZRLHJlY3Rhbmd1bGFyRidGL0Zfb0YyRmRvRl9vRjlGZ29Gam9GXHBGXnBGYXBGZ29Gam9GXHAtRmVuNiYtRmhuNigtRiM2Jy1GLDYmUShPcmRlcjp+RidGL0Zfb0YyLUYsNiZRLkZvcnRyYW5fb3JkZXJGJ0YvRl9vRjJGZG9GX29GOUZnb0Zqb0ZccEZecEZhcEZnb0Zqb0ZccC8lJmFsaWduR1ElYXhpc0YnL0Zob1EpYmFzZWxpbmVGJy9GW3BRJ2NlbnRlckYnL0ZdcFEnfGZybGVmdHxockYnLyUvYWxpZ25tZW50c2NvcGVHRjEvJSxjb2x1bW53aWR0aEdRJWF1dG9GJy8lJndpZHRoR0Zkcy8lK3Jvd3NwYWNpbmdHUSYxLjBleEYnLyUuY29sdW1uc3BhY2luZ0dRJjAuOGVtRicvJSlyb3dsaW5lc0dRJW5vbmVGJy8lLGNvbHVtbmxpbmVzR0ZfdC8lJmZyYW1lR0ZfdC8lLWZyYW1lc3BhY2luZ0dRLDAuNGVtfjAuNWV4RicvJSplcXVhbHJvd3NHRj0vJS1lcXVhbGNvbHVtbnNHRj0vJS1kaXNwbGF5c3R5bGVHRj0vJSVzaWRlR1EmcmlnaHRGJy8lMG1pbmxhYmVsc3BhY2luZ0dGXHRGX29GOS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJy8lK2FjdGlvbnR5cGVHUS1icm93c2VydGFibGVGJy8lKXJ0YWJsZWlkR1E1MTg0NDY3NDQwNzQyMzU2MTk1NThGJy1GNjYtUSIsRidGOUY7L0Y/RjFGQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZOUSwwLjMzMzMzMzNlbUYnLUZVNiYtRiM2OS1JI21uR0YkNiRGYHBGOUZedi1GLDYlUSV2dHRfRidGL0YyRl52LUYsNiVRJXZzc19GJ0YvRjJGXnYtRiM2JC1JJW1zdXBHRiQ2JUZddy1GW3c2JFEiMkYnRjkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRjlGXnYtRiM2JkZgdy1GNjYtUTEmSW52aXNpYmxlVGltZXM7RidGOUY7Rj5GQEZCRkRGRkZIRmJ2L0ZORmN2Rl13RjlGXnYtRiM2JC1GZnc2JUZgd0Zod0ZbeEY5Rl52LUYjNiZGYHdGYHhGZXdGOUZedi1GIzYmRmZ4RmB4Rl13RjlGXnYtRiM2JC1GZnc2JUZgdy1GW3c2JFEiM0YnRjlGW3hGOUZedi1GIzYmRmZ4RmB4RmV3RjlGXnYtRiM2JC1GZnc2JUZgdy1GW3c2JFEiNEYnRjlGW3hGOUZkb0Y5RjlGYnVGZXVGXnYtRlU2Ji1GIzYnRmB3Rl52Rl13RmRvRjlGOUZidUZldUZedi1GVTYmLUYjNjlGanZGXnYtRiw2JVEkdnR0RidGL0YyRl52LUYsNiVRJHZzc0YnRi9GMkZedi1GIzYkLUZmdzYlRmR6Rmh3Rlt4RjlGXnYtRiM2JkZnekZgeEZkekY5Rl52LUYjNiQtRmZ3NiVGZ3pGaHdGW3hGOUZedi1GIzYkLUZmdzYlRmR6RmB5Rlt4RjlGXnYtRiM2JkZnekZgeEZcW2xGOUZedi1GIzYmRmJbbEZgeEZkekY5Rl52LUYjNiQtRmZ3NiVGZHpGaXlGW3hGOUZedi1GIzYmRmJbbEZgeEZcW2xGOUZkb0Y5RjlGYnVGZXVGXnYtRlU2Ji1GIzYnRmd6Rl52RmR6RmRvRjlGOUZidUZldUZkb0Y5RmRvRjk=">NictSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JJCVpZEdGKCI1ZSY+Y0J1U3VZJT03LSIiIkkldnR0X0dGKEkldnNzX0dGKCokRi8iIiMqJkYwRi5GL0YuKiRGMEYyKiZGMEYuRi9GMiomRjBGMkYvRi4qJEYwIiIkKiZGMEYyRi9GMiokRjAiIiU3JEYwRi83LUYuSSR2dHRHRihJJHZzc0dGKCokRj5GMiomRj9GLkY+Ri4qJEY/RjIqJEY+RjgqJkY/Ri5GPkYyKiZGP0YyRj5GLiokRj5GOyomRj9GMkY+RjI3JEY/Rj4=</Equation></Text-field>
</Output>
</Group>
<Group labelreference="L25" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field style="Text" layout="Normal">Compute a maximal-rank submatrix of the Dixon matrix.</Text-field>
</Input>
</Group>
<Group labelreference="L26" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="rs := DR:-RankSubMatrix(dm)" display="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">PkkjcnNHNiItX0kjRFJHRiRJLlJhbmtTdWJNYXRyaXhHRiQ2I0kjZG1HRiQ=</Equation></Text-field>
</Input>
<Output>
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</Output>
</Group>
<Group labelreference="L27" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field style="Text" layout="Normal">Compute the Dixon resultant.</Text-field>
</Input>
</Group>
<Group labelreference="L28" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="DR:-DixonExtract(rs)" display="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">LV9JI0RSRzYiSS1EaXhvbkV4dHJhY3RHRiU2I0kjcnNHRiU=</Equation></Text-field>
</Input>
<Output>
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</Output>
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<Group labelreference="L29" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field style="Text" layout="Normal">The empty string &quot;&quot; confirms that everything went well. There is a miniscule chance that DR:-RankSubMatrix does not return a maximal-rank submatrix which would be detected by DR:-DixonExtract. In this case, the string contains an error message.</Text-field>
<Text-field style="Text" layout="Normal"></Text-field>
<Text-field style="Text" bold="true" layout="Normal"><Font bold="true">Further details and functions</Font></Text-field>
</Input>
</Group>
<Group labelreference="L30" drawlabel="true" applyint="true" applyrational="true" applyexponent="false">
<Input>
<Text-field prompt="&gt; " style="Maple Input" layout="Normal"><Equation executable="true" style="2D Input" input-equation="Describe(DR)" display="LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEpRGVzY3JpYmVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2Ji1GLDYlUSNEUkYnRi9GMi8lK2ZvcmVncm91bmRHUShbMCwwLDBdRicvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQ0Y9RkBGQw==">LUkpRGVzY3JpYmVHNiI2I0kjRFJHRiQ=</Equation></Text-field>
</Input>
<Output>
<Text-field style="Line Printed Output" layout="Line Printed Output">
# Dixon Resultant Computation.
# Date: 10/9/15, Version 2.1
# Author: Manfred Minimair
# Additional contributors: Arthur Cherba, Hoon Hong
# E-mail: manfred@minimair.org
# Released under GPL Version 3
# Copyright (C) Manfred Minimair, 2015
module DR:

    # Compute the Dixon resultant.
    # pols::list(polynom) ... list of (d+1) polynomials
    # vars::list(symbol) ... list d variables of polsReturn the Dixon resultant
    # of pols.
    DixonResultant( pols::list(polynom), vars::list(symbol) )

    # Compute the Dixon polynomial.
    # pols ... list of (d+1) polynomials (different order just changes sign)
    # vars ... list d variables (different order produce completely different
    # polynomials!)
    # auxVars ... list of d new variables. If it is empty, the new variables
    # will be automatically generated.
    # Returns dixPol,newVarList, vars,
    # where dixPol is the expanded Dixon polynomial, vars is copied from the
    # input parameter vars and newVarList is the list of d new variables used
    # by the Dixon polynomial.
    DixonPolynomial( pols::list(polynom), vars::list(symbol),
                     auxVars::list(symbol) := [] )

    # Compute the Dixon matrix of a Dixon polynomial.
    # dixonPoly ... Dixon polynomial
    # rowVars ... variables for monomials to index the Dixon matrix rows
    # colVars ... variables for monomials to index the Dixon matrix columns
    # roworder ... ordering of the row monomials indexing the Dixon matrix,
    # compatible with Groebner:-MonomialOrder
    # colorder ... ordering of the column monomials indexing the Dixon matrix,
    # compatible with Groebner:-MonomialOrder
    # Return matrix, rowMons, rowVars, colMons, colVars,
    # where matrix is the Dixon matrix, rowMons and colMons are the monomials
    # indexing the rows and respectively columns of the matrix
    DixonMatrix( dixonPoly, rowVars, colVars, roworder := tdeg,
                 colorder := tdeg )

    # Extract a maximal-rank minor of a parametric matrix.
    # M ... parametric matrix
    # params ... parameters of M
    # Return a maximal-rank minor of the parametric matrix M with high
    # likelihood. If params is empty, the parameters will be automatically
    # determined.
    RankSubMatrix( M::Matrix,
                   { params::list := [] } )

    # Extract the Dixon resultant from a Dixon matrix.M ... Dixon matrix
    # Return message, expression. If message is empty then the expression is
    # the Dixon resultant of pols Otherwise message is an error message and the
    # expression is only a factor of the Dixon resultant.
    DixonExtract( M::Matrix )

    # Various functions to gather information about the computation.
    # Author: Manfred Minimair
    # E-mail: manfred@minimair.org
    # Released under GPL Version 3
    # Copyright (C) Manfred Minimair, 2015
    module Info:

        # Polynomial system information.
        # numVars ... number of variables
        # degrees ... list of degrees or degree vectors of the system
        # numParams ... number of parameters
        # paramDegrees ... list of degrees or degree vectors of the parameters
        # of the system
        # Return a table with the inormation.
        System( sysName::string, numVars::integer, degrees,
                numParams::integer := undefined, paramDegrees := undefined )

        # OPen file to write computational data.
        # fname ... name of the file
        # header ... if true, print a csv header into the file
        # discard ... if true, delete the file if it exists before opening a
        # new file
        # Return the file descriptor of the file.
        OpenFile( fname::string,
                  { discard::truefalse := true, header::truefalse := true } )
                :: filedesc

        # Save initial data.
        # fh ... file descriptor of output file
        # t ... table wit data
        SaveStart( fh::filedesc, t::table )

        # Save remaining data.
        # fh ... file descriptor of output file
        # t ... table wit data
        SaveEnd( fh::filedesc, t::table )

        # Run a function and store performance measures in a file.
        # fh ... file descriptor of csv file to append peformance measures,
        # according to SaveStart and SaveEnd
        # sysName ... name of the polynomial system
        # fun ... function to run, usually Dixon resultant computation
        # f ... polynomial system
        # deg ... degree list or degree in the variables of the system
        # vars ... list of variables of the system
        # paramDeg ... total degree of the parameters of the system
        # params ... list of parameters of the system
        # Return a table with the performace measure, as prduced by
        # DRes:-ComposeDixon.
        RunMeasure( fh::filedesc, fun::{function, procedure}, sysName::string,
                    f::list, deg::{integer, list}, vars::list,
                    paramDeg::integer, params::list )

        # Gather and store information about the Dixon matrix.
        # measures ... table where the information about the Dixon matrix will
        # be stored: number of rows and columns and rank
        # Return a function with input matrix M that returns M.
        DixonMatrix( measures::{name, table, undefined} := undefined )

        # Store the extraction time and the name of the function used for Dixon
        # resultant extraction.
        # extractorList ... list of functions that are composed to extract the
        # Dixon resultant from the Dixon matrix.
        # measures ... table where the information about the Dixon matrix will
        # be stored: number of rows and columns and rank
        # Return a function that returns its own input.
        Extraction( extractorList::list,
                    measures::{name, table, undefined} := undefined )

        # fun ... procedure to be timed and whose byte usage is to be collected
        # Measure the execution time of a function.
        # measures ... output parameter, a table. If measures is a name then a
        # table will be assigned to it. If measures is undefined then no timing
        # will be produced.
        # other paramters ... will be passed as inputs to fun
        # Return a function that returns the output of fun and assigns the
        # timing to the table measures with key convert(fun, string).
        Measure( fun, measures::{name, table, undefined} := undefined )

        # Determine the rank of a parametric matrix.
        # M ... matrix with parametric entries
        # tries ... number of times the matrix parameters should be
        # instantiated with random numbers and instantiated matrix should have
        # the same rank
        # params ... optional list of parameters of matrix, if absent then
        # automatically determined
        # Return the rank (with high probability) of the parametric matrix M
        Rank( M::Matrix, tries::integer := 10, params::list(name) := [] )

    # Various functions used in extracting the Dixon resultant from the Dixon
    # matrix.
    # Author: Manfred Minimair
    # E-mail: manfred@minimair.org
    # Released under GPL Version 3
    # Copyright (C) Manfred Minimair, 2015
    module Extract:

        # Extract a maximal-rank minor of a parametric matrix.
        # M ... parametric matrix
        # parameters ... parameters of M
        # Return a maximal-rank minor of the parametric matrix M with high
        # likelihood. If params is empty, the parameters will be automatically
        # determined.
        RankSubMatrix( M::Matrix,
                       { parameters::list := [] } )

        # Compute the determinant of a matrix with Maple's determinant
        # function.
        # A ... given matrix
        # Return lowerRank::string, the determinant of A. non-empty message iff
        # the determinant is 0 and A is not empty.
        MapleDet( A::Matrix )

        # Compute a maximal-rank minor of a matrix via Gaussian elimination.
        # A ... given matrix
        # Named optional arguments:
        # return_fullrank ... indicate whether a non-zero minor of full rank
        # was found
        # moreRows ... transpose A such that it has at least as many rows as
        # columns
        # *row* ... extract common factors of rows
        # *col* ... extract common factors of columns
        # *Numers ... extract common factors from numerators of matrix entries
        # *Denoms ... extract common factors from denominators of matrix
        # entries
        # pre* ... extract common factors before the Gaussian elimination
        # starts
        # pivot* ... extract common factors from the pivot row during Gaussian
        # elimination
        # rest* ... extract common factors from rows other than the pivot row
        # during Gaussian elimination
        # Return lowerRank::string, a maximal-rank minor of A.
        # If moreRows=false and rest* is activated then the output can be a
        # multiple of a maximal-rank minor, because the output may contain
        # factors of rows that are not needed for the determinant computation.
        MaxMinorElim( A::Matrix,
                      { colDenoms::truefalse := false,
                        colNumers::truefalse := false,
                        moreRows::truefalse := true,
                        pivotRowDenoms::truefalse := false,
                        pivotRowNumers::truefalse := false,
                        preColDenoms::truefalse := false,
                        preColNumers::truefalse := false,
                        preRowDenoms::truefalse := false,
                        preRowNumers::truefalse := false,
                        restRowDenoms::truefalse := false,
                        restRowNumers::truefalse := false,
                        tellFullrank::truefalse := false } )

        # Compute a maximal-rank minor of a matrix via Gaussian elimination.
        # A ... given matrix
        # Named optional arguments: like MaxMinorElim, except tellFullrank
        # which is not allowed.
        # Return lowerRank::string, output from MaxMinorElim
        # If lowerRank is empty then return a maximal minor of full rank.
        # lowerRank is not empty iff A is not square of full rank and is not
        # empty.
        MaxMinorElimFull( A::Matrix )

        # Randomly instantiate a parametric matrix.M ... matrix with parametric
        # entries
        # varLst ... list of variables in M
        # Return M with parameters varLst randomly specialized.
        InstantiateMatrix( M::Matrix, varLst )

    # Versions of maximal minor with factor computation during elimination. DF
    # = Factor Detection
    # See also: Lewis, Rober H., Heuristics to accelerate the Dixon resultant,
    # Mathematics and Computers in Simulation, 77, 4 (2008), 400-407All
    # functions work accordingly:
    # Compute a maximal-rank minor of a matrix via Gaussian elimination.
    # A ... given matrix
    # Return a maximal-rank minor of A.
    # Author: Manfred Minimair
    # E-mail: manfred@minimair.org
    # Released under GPL Version 3
    # Copyright (C) Manfred Minimair, 2015
    module DF:

        ColNumers( A )

        ColDenoms( A )

        Col( A )

        PivotRowNumers( A )

        PivotRowDenoms( A )

        PivotRow( A )

        ColPivotRow( A )

        Row( A )

        ColRow( A )

        RowFull( A )

        ColRowFull( A )

        PivotRowFull( A )

        ColPivotRowFull( A )

    # Versions of Dixon resultant computation.
    # All functions work accordingly:
    #  pols ... list of (d+1) polynomials
    #  vars ... list d variables of pols measures ... output parameter, a
    # table. If measures is a name then a table will be assigned to it. If
    # measures is undefined then no timing will be produced.
    #  Return the Dixon resultant of pols.
    #  If randomized maximal-rank submatrix extraction is used, it is possible,
    # however extremely unlikely, that Dixon resultant computation fails.
    #  In this case, the computation terminates with an error message
    # indicating to run the computation again or not to use randomized
    # maximal-rank submatrix extraction.
    # Author: Manfred Minimair
    # E-mail: manfred@minimair.org
    # Released under GPL Version 3
    # Copyright (C) Manfred Minimair, 2015
    module DRes:

        # Compose a function that computes the Dixon resultant.
        # extractor ... function, with Dixon matrix input, to extract the Dixon
        # resultant, or list of functions that will be composed
        # measures ... output parameter, a table. If measures is a name then a
        # table will be assigned to it. If measures is undefined then no timing
        # will be produced.
        # Return the timed Dixon resultant function.
        ComposeDixon( extractor := Extract:-MaxMinorElim,
                      { measures_::{name, table, undefined} := undefined } )

        # Gaussian elimination based minor extraction.
        MaxMinor( pols::list(polynom), vars::list(symbol),
                  measures::{name, table, undefined} := undefined )

        # Maximal-rank submatrix extraction and MaxMinor function.
        RankMaxMinor( pols::list(polynom), vars::list(symbol),
                      measures::{name, table, undefined} := undefined )

        # Maximal-rank submatrix extraction and Maple determinant function.
        RankMapleDet( pols::list(polynom), vars::list(symbol),
                      measures::{name, table, undefined} := undefined )

        # Maximal-rank submatrix extraction and Maple determinant function with
        # PreDF.
        RankFactorsMapleDet( pols::list(polynom), vars::list(symbol),
                             measures::{name, table, undefined} := undefined )

        # Gaussian elimination based minor extraction with ColNumersFactors
        # detection.
        MaxMinorColNumers( pols::list(polynom), vars::list(symbol),
                           measures::{name, table, undefined} := undefined )

        # Gaussian elimination based minor extraction with ColDenomsFactors
        # detection.
        MaxMinorColDenoms( pols::list(polynom), vars::list(symbol),
                           measures::{name, table, undefined} := undefined )

        # Gaussian elimination based minor extraction with ColDenomsFactors
        # detection.
        MaxMinorCol( pols::list(polynom), vars::list(symbol),
                     measures::{name, table, undefined} := undefined )

        # Gaussian elimination based minor extraction with
        # PivotRowNumersFactors detection.
        MaxMinorPivotRowNumers( pols::list(polynom), vars::list(symbol),
                                measures::{name, table, undefined} := undefined
                              )

        # Gaussian elimination based minor extraction with
        # PivotRowDenomsFactors detection.
        MaxMinorPivotRowDenoms( pols::list(polynom), vars::list(symbol),
                                measures::{name, table, undefined} := undefined
                              )

        # Gaussian elimination based minor extraction with
        # PivotRowNumersFactors and PivotRowDenomsFactors detection.
        MaxMinorPivotRow( pols::list(polynom), vars::list(symbol),
                          measures::{name, table, undefined} := undefined )

        # Gaussian elimination based minor extraction with ColNumersFactors,
        # ColDenomsFactors, PivotRowNumersFactors and PivotRowDenomsFactors
        # detection.
        MaxMinorColPivotRow( pols::list(polynom), vars::list(symbol),
                             measures::{name, table, undefined} := undefined )

        # Gaussian elimination based minor extraction with ColNumersFactors,
        # ColDenomsFactors, PivotRowNumersFactors and PivotRowDenomsFactors
        # detection.
        MaxMinorColRow( pols::list(polynom), vars::list(symbol),
                        measures::{name, table, undefined} := undefined )

        # Gaussian elimination based minor extraction with
        # Pivot/RestRowNumersFactors and Pivo/ResttRowDenomsFactors detection
        # on RankSubMatrix
        RankRow( pols::list(polynom), vars::list(symbol),
                 measures::{name, table, undefined} := undefined )

        # Gaussian elimination based minor extraction with ColNumersFactors,
        # ColDenomsFactors, Pivot/RestRowNumersFactors and
        # Pivot/RestRowDenomsFactors detection on RankSubMatrix.
        RankColRow( pols::list(polynom), vars::list(symbol),
                    measures::{name, table, undefined} := undefined )

        # Maximal-rank submatrix extraction and MaxMinor function with
        # PivotRowFactors detection.
        RankMaxMinorPivotRow( pols::list(polynom), vars::list(symbol),
                              measures::{name, table, undefined} := undefined )

        # Maximal-rank submatrix extraction and MaxMinor function with
        # Col/PivotRowFactors detection.
        RankMaxMinorColPivotRow( pols::list(polynom), vars::list(symbol),
                                 measures::{name, table, undefined} :=
                                   undefined
                               )

        # Return the first argument passed to the function.
        GetFirst( )

        # Detect failure case.
        # msg ... error message
        # Return rest arguments or, if msg is a non-empty string with the error
        # message msg.
        DetectFail( msg::string )

    # Generate polynomials.
    # Author: Manfred Minimair
    # E-mail: manfred@minimair.org
    # Released under GPL Version 3
    # Copyright (C) Manfred Minimair, 2015
    module GenPoly:

        # Random parametric total-degree integer polynomial.
        # deg ... total degree or list of total degrees
        # vars ... list of variables
        # paramDeg ... total degree of the coefficient polynomials in the
        # parameters
        # params ... list of parameters
        # sparse ... if true generate sparse polynomials and sparse parametric
        # coefficients; otherwise dense.Return a random dense integer
        # polynomial of total degree deg, or a list of such polynomials if deg
        # is a list with corresponding degrees. The degrees of the parameters
        # are of total degree paramDeg. If parmDeg=0 or params=[] then create
        # random ineger coefficients.
        RandParamTotalDeg( deg::{integer, list(integer)}, vars::list,
                           paramDeg::integer, params::list,
                           { sparse::truefalse := true } )

        # Random dense parametric multi-degree integer polynomial.
        # degs ... multi-degree list or list of multi-degree lists
        # vars ... list of variables
        # paramDeg ... total degree of the coefficient polynomials in the
        # parameters
        # params ... list of parameters
        # Return a random dense integer polynomial of multi-degree degs, or a
        # list of such polynomials if degs is a list with corresponding
        # multi-degrees. The degrees of the parameters are of total degree
        # paramDeg. If parmDeg=0 or params=[] then create random ineger
        # coefficients.
        RandParamMultiDeg( degs, vars, paramDeg, params ) :: polynom

    # Collection of sample polynomial systems.
    # To iterate over repo use: for sys in op(op(repo)) do ... Get(lhs(sys))
    # ... end do
    # or iterator over the list Available()
    # Author: Manfred Minimair
    # E-mail: manfred@minimair.org
    # Released under GPL Version 3
    # Copyright (C) Manfred Minimair, 2015
    module Samples:

        # Retrieve a polynomial system for use in Dixon resultant computation
        # Return name, system, degrees, variables, param degree, parameters
        Get( sysName::string )

        repo::table = ...

        # List names of available sample systems.
        # paramters are sequence of arbitrary strings; list of names of systems
        # to be considered; if undefined, then consider all in repo
        # include ... list of names of systems or single name to be considered;
        # if undefined, then consider all in repo
        # exclude ... list of names of systems to be excluded be default
        # Return a list of system names available.
        Available( { exclude::list(string) := [],
                     include::{string, undefined, list(string)} := undefined }
                 ) :: list

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