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ApCoCoA-1:NCo.MB - Revision history
2024-03-29T11:11:13Z
Revision history for this page on the wiki
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http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NCo.MB&diff=16133&oldid=prev
AndraschkoBot: replaced <quotes> tag by real quotes
2020-10-29T13:41:07Z
<p>replaced <quotes> tag by real quotes</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:41, 29 October 2020</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l13" >Line 13:</td>
<td colspan="2" class="diff-lineno">Line 13:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><par/></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><par/></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Please set ring environment <em>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>ApCoCoA-1:NCo.SetFp|NCo.SetFp</ref>, <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>, respectively, before using this function. The default coefficient field is <tt>Q</tt>, and the default ordering is the length-lexicographic ordering (<del class="diffchange diffchange-inline"><quotes></del>LLEX<del class="diffchange diffchange-inline"></quotes></del>). For more information, please check the relevant functions.</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Please set ring environment <em>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>ApCoCoA-1:NCo.SetFp|NCo.SetFp</ref>, <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>, respectively, before using this function. The default coefficient field is <tt>Q</tt>, and the default ordering is the length-lexicographic ordering (<ins class="diffchange diffchange-inline">"</ins>LLEX<ins class="diffchange diffchange-inline">"</ins>). For more information, please check the relevant functions.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><itemize></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><itemize></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K&lt;X&gt;</tt> forming a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,<del class="diffchange diffchange-inline"><quotes></del>xy<del class="diffchange diffchange-inline"></quotes></del>], [-1, <del class="diffchange diffchange-inline"><quotes></del>y<del class="diffchange diffchange-inline"></quotes></del>], [1,<del class="diffchange diffchange-inline"><quotes></quotes></del>]]. <em>Warning:</em> users should take responsibility to make sure that Gb is indeed a Groebner basis with respect to a length compatible word ordering! In the case that Gb is a partical Groebner basis, the function enumerates a pseudo basis.</item></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K&lt;X&gt;</tt> forming a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,<ins class="diffchange diffchange-inline">"</ins>xy<ins class="diffchange diffchange-inline">"</ins>], [-1, <ins class="diffchange diffchange-inline">"</ins>y<ins class="diffchange diffchange-inline">"</ins>], [1,<ins class="diffchange diffchange-inline">""</ins>]]. <em>Warning:</em> users should take responsibility to make sure that Gb is indeed a Groebner basis with respect to a length compatible word ordering! In the case that Gb is a partical Groebner basis, the function enumerates a pseudo basis.</item></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><item>@return: a LIST of words forming a Macaulay's basis of the <tt>K</tt>-algebra <tt>K&lt;X&gt;/&lt;Gb&gt;</tt>.</item></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><item>@return: a LIST of words forming a Macaulay's basis of the <tt>K</tt>-algebra <tt>K&lt;X&gt;/&lt;Gb&gt;</tt>.</item></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l24" >Line 24:</td>
<td colspan="2" class="diff-lineno">Line 24:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></itemize></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></itemize></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><example></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><example></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>NCo.SetX(<del class="diffchange diffchange-inline"><quotes></del>xyzt<del class="diffchange diffchange-inline"></quotes></del>);</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>NCo.SetX(<ins class="diffchange diffchange-inline">"</ins>xyzt<ins class="diffchange diffchange-inline">"</ins>);</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>NCo.SetOrdering(<del class="diffchange diffchange-inline"><quotes></del>LLEX<del class="diffchange diffchange-inline"></quotes></del>);</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>NCo.SetOrdering(<ins class="diffchange diffchange-inline">"</ins>LLEX<ins class="diffchange diffchange-inline">"</ins>);</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Gb:= [[[1, <del class="diffchange diffchange-inline"><quotes></del>yt<del class="diffchange diffchange-inline"></quotes></del>], [-1, <del class="diffchange diffchange-inline"><quotes></del>ty<del class="diffchange diffchange-inline"></quotes></del>]], [[1, <del class="diffchange diffchange-inline"><quotes></del>xt<del class="diffchange diffchange-inline"></quotes></del>], [-1, <del class="diffchange diffchange-inline"><quotes></del>tx<del class="diffchange diffchange-inline"></quotes></del>]], [[1, <del class="diffchange diffchange-inline"><quotes></del>xy<del class="diffchange diffchange-inline"></quotes></del>], [-1, <del class="diffchange diffchange-inline"><quotes></del>ty<del class="diffchange diffchange-inline"></quotes></del>]], [[1, <del class="diffchange diffchange-inline"><quotes></del>xx<del class="diffchange diffchange-inline"></quotes></del>], [-1, <del class="diffchange diffchange-inline"><quotes></del>yx<del class="diffchange diffchange-inline"></quotes></del>]], </div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Gb:= [[[1, <ins class="diffchange diffchange-inline">"</ins>yt<ins class="diffchange diffchange-inline">"</ins>], [-1, <ins class="diffchange diffchange-inline">"</ins>ty<ins class="diffchange diffchange-inline">"</ins>]], [[1, <ins class="diffchange diffchange-inline">"</ins>xt<ins class="diffchange diffchange-inline">"</ins>], [-1, <ins class="diffchange diffchange-inline">"</ins>tx<ins class="diffchange diffchange-inline">"</ins>]], [[1, <ins class="diffchange diffchange-inline">"</ins>xy<ins class="diffchange diffchange-inline">"</ins>], [-1, <ins class="diffchange diffchange-inline">"</ins>ty<ins class="diffchange diffchange-inline">"</ins>]], [[1, <ins class="diffchange diffchange-inline">"</ins>xx<ins class="diffchange diffchange-inline">"</ins>], [-1, <ins class="diffchange diffchange-inline">"</ins>yx<ins class="diffchange diffchange-inline">"</ins>]], </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[1, <del class="diffchange diffchange-inline"><quotes></del>tyy<del class="diffchange diffchange-inline"></quotes></del>], [-1, <del class="diffchange diffchange-inline"><quotes></del>tty<del class="diffchange diffchange-inline"></quotes></del>]], [[1, <del class="diffchange diffchange-inline"><quotes></del>yyx<del class="diffchange diffchange-inline"></quotes></del>], [-1, <del class="diffchange diffchange-inline"><quotes></del>tyx<del class="diffchange diffchange-inline"></quotes></del>]]];</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[1, <ins class="diffchange diffchange-inline">"</ins>tyy<ins class="diffchange diffchange-inline">"</ins>], [-1, <ins class="diffchange diffchange-inline">"</ins>tty<ins class="diffchange diffchange-inline">"</ins>]], [[1, <ins class="diffchange diffchange-inline">"</ins>yyx<ins class="diffchange diffchange-inline">"</ins>], [-1, <ins class="diffchange diffchange-inline">"</ins>tyx<ins class="diffchange diffchange-inline">"</ins>]]];</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>NCo.MB(Gb,3);</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>NCo.MB(Gb,3);</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[<del class="diffchange diffchange-inline"><quotes></quotes></del>], [<del class="diffchange diffchange-inline"><quotes></del>t<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>z<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>y<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>x<del class="diffchange diffchange-inline"></quotes></del>], [<del class="diffchange diffchange-inline"><quotes></del>tt<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>tz<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>ty<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>tx<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>zt<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>zz<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>zy<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>zx<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>yz<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>yy<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>yx<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>xz<del class="diffchange diffchange-inline"></quotes></del>], </div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[<ins class="diffchange diffchange-inline">""</ins>], [<ins class="diffchange diffchange-inline">"</ins>t<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>z<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>y<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>x<ins class="diffchange diffchange-inline">"</ins>], [<ins class="diffchange diffchange-inline">"</ins>tt<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>tz<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>ty<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>tx<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>zt<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>zz<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>zy<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>zx<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>yz<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>yy<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>yx<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>xz<ins class="diffchange diffchange-inline">"</ins>], </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[<del class="diffchange diffchange-inline"><quotes></del>ttt<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>ttz<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>tty<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>ttx<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>tzt<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>tzz<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>tzy<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>tzx<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>tyz<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>tyx<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>txz<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>ztt<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>ztz<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>zty<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>ztx<del class="diffchange diffchange-inline"></quotes></del>,</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[<ins class="diffchange diffchange-inline">"</ins>ttt<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>ttz<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>tty<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>ttx<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>tzt<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>tzz<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>tzy<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>tzx<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>tyz<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>tyx<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>txz<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>ztt<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>ztz<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>zty<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>ztx<ins class="diffchange diffchange-inline">"</ins>,</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"><quotes></del>zzt<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>zzz<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>zzy<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>zzx<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>zyz<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>zyy<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>zyx<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>zxz<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>yzt<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>yzz<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>yzy<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>yzx<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>yyz<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>yyy<del class="diffchange diffchange-inline"></quotes></del>,</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">"</ins>zzt<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>zzz<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>zzy<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>zzx<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>zyz<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>zyy<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>zyx<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>zxz<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>yzt<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>yzz<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>yzy<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>yzx<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>yyz<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>yyy<ins class="diffchange diffchange-inline">"</ins>,</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> <del class="diffchange diffchange-inline"><quotes></del>yxz<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>xzt<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>xzz<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>xzy<del class="diffchange diffchange-inline"></quotes></del>, <del class="diffchange diffchange-inline"><quotes></del>xzx<del class="diffchange diffchange-inline"></quotes></del>]]</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> <ins class="diffchange diffchange-inline">"</ins>yxz<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>xzt<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>xzz<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>xzy<ins class="diffchange diffchange-inline">"</ins>, <ins class="diffchange diffchange-inline">"</ins>xzx<ins class="diffchange diffchange-inline">"</ins>]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>-------------------------------</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>-------------------------------</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></example></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></example></div></td></tr>
</table>
AndraschkoBot
http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NCo.MB&diff=15590&oldid=prev
AndraschkoBot: insert version info
2020-10-07T10:21:23Z
<p>insert version info</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:21, 7 October 2020</td>
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<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{Version|1}}</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><command></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><command></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><title>NCo.MB</title></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><title>NCo.MB</title></div></td></tr>
</table>
AndraschkoBot
http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NCo.MB&diff=15226&oldid=prev
AndraschkoBot: fixed links to namespace ApCoCoA
2020-10-07T08:30:27Z
<p>fixed links to namespace ApCoCoA</p>
<table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 08:30, 7 October 2020</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></syntax></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></syntax></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><description></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><description></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Given a two-sided ideal <tt>I</tt> in a free monoid ring <tt>K&lt;X&gt;</tt>, we can consider the <tt>K</tt>-algebra <tt>K&lt;X&gt;/I</tt> as a <tt>K</tt>-vector space. Moreover, let <tt>G</tt> be a Groebner basis of <tt>I</tt>, and let <tt>B</tt> be the set of all words which are not a multiple of any word in the leading word set <tt>LW{G}</tt>. Then the residue class of the words in <tt>B</tt> form a <tt>K</tt>-basis, called a <em>Macaulay's basis</em>, of <tt>K&lt;X&gt;/I</tt>. For the sake of computing the values of the Hilbert function (see <ref>NCo.HF</ref>) of <tt>K&lt;X&gt;/I</tt>, in this function we require that <tt>G</tt> has to be a Groebner basis with respect to a length compatible word ordering (see <ref>NCo.SetOrdering</ref>).</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Given a two-sided ideal <tt>I</tt> in a free monoid ring <tt>K&lt;X&gt;</tt>, we can consider the <tt>K</tt>-algebra <tt>K&lt;X&gt;/I</tt> as a <tt>K</tt>-vector space. Moreover, let <tt>G</tt> be a Groebner basis of <tt>I</tt>, and let <tt>B</tt> be the set of all words which are not a multiple of any word in the leading word set <tt>LW{G}</tt>. Then the residue class of the words in <tt>B</tt> form a <tt>K</tt>-basis, called a <em>Macaulay's basis</em>, of <tt>K&lt;X&gt;/I</tt>. For the sake of computing the values of the Hilbert function (see <ref><ins class="diffchange diffchange-inline">ApCoCoA-1:NCo.HF|</ins>NCo.HF</ref>) of <tt>K&lt;X&gt;/I</tt>, in this function we require that <tt>G</tt> has to be a Groebner basis with respect to a length compatible word ordering (see <ref><ins class="diffchange diffchange-inline">ApCoCoA-1:NCo.SetOrdering|</ins>NCo.SetOrdering</ref>).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><par/></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><par/></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><par/></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><par/></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Please set ring environment <em>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>NCo.SetFp</ref>, <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before using this function. The default coefficient field is <tt>Q</tt>, and the default ordering is the length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Please set ring environment <em>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref><ins class="diffchange diffchange-inline">ApCoCoA-1:NCo.SetFp|</ins>NCo.SetFp</ref>, <ref><ins class="diffchange diffchange-inline">ApCoCoA-1:NCo.SetX|</ins>NCo.SetX</ref> and <ref><ins class="diffchange diffchange-inline">ApCoCoA-1:NCo.SetOrdering|</ins>NCo.SetOrdering</ref>, respectively, before using this function. The default coefficient field is <tt>Q</tt>, and the default ordering is the length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><itemize></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><itemize></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K&lt;X&gt;</tt> forming a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <em>Warning:</em> users should take responsibility to make sure that Gb is indeed a Groebner basis with respect to a length compatible word ordering! In the case that Gb is a partical Groebner basis, the function enumerates a pseudo basis.</item></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K&lt;X&gt;</tt> forming a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <em>Warning:</em> users should take responsibility to make sure that Gb is indeed a Groebner basis with respect to a length compatible word ordering! In the case that Gb is a partical Groebner basis, the function enumerates a pseudo basis.</item></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></description></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></description></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><seealso></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><seealso></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.HF</see></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><see><ins class="diffchange diffchange-inline">ApCoCoA-1:NCo.HF|</ins>NCo.HF</see></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.IsGB</see></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><see><ins class="diffchange diffchange-inline">ApCoCoA-1:NCo.IsGB|</ins>NCo.IsGB</see></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.LW</see></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><see><ins class="diffchange diffchange-inline">ApCoCoA-1:NCo.LW|</ins>NCo.LW</see></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.SetFp</see></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><see><ins class="diffchange diffchange-inline">ApCoCoA-1:NCo.SetFp|</ins>NCo.SetFp</see></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.SetOrdering</see></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><see><ins class="diffchange diffchange-inline">ApCoCoA-1:NCo.SetOrdering|</ins>NCo.SetOrdering</see></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.SetX</see></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><see><ins class="diffchange diffchange-inline">ApCoCoA-1:NCo.SetX|</ins>NCo.SetX</see></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><see>Introduction to CoCoAServer</see></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><see><ins class="diffchange diffchange-inline">ApCoCoA-1:Introduction to CoCoAServer|</ins>Introduction to CoCoAServer</see></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></seealso></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></seealso></div></td></tr>
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AndraschkoBot
http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NCo.MB&diff=14824&oldid=prev
AndraschkoBot: Bot: Category moved
2020-10-02T16:27:32Z
<p>Bot: Category moved</p>
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<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><wiki-category>Package_gbmr</wiki-category></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><wiki-category><ins class="diffchange diffchange-inline">ApCoCoA-1:</ins>Package_gbmr</wiki-category></div></td></tr>
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AndraschkoBot
http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NCo.MB&diff=13465&oldid=prev
Stadler at 19:59, 14 May 2013
2013-05-14T19:59:46Z
<p></p>
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</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l3" >Line 3:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><short_description></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><short_description></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Enumerate a Macaulay's basis of a finitely generated <tt>K</tt>-algebra.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Enumerate a Macaulay's basis of a finitely generated <tt>K</tt>-algebra.</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"><par/></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">Given a two-sided ideal <tt>I</tt> in a free monoid ring <tt>K&lt;X&gt;</tt>, we can consider the <tt>K</tt>-algebra <tt>K&lt;X&gt;/I</tt> as a <tt>K</tt>-vector space. Moreover, let <tt>G</tt> be a Groebner basis of <tt>I</tt>, and let <tt>B</tt> be the set of all words which are not a multiple of any word in the leading word set <tt>LW{G}</tt>. Then the residue class of the words in <tt>B</tt> form a <tt>K</tt>-basis, called a <em>Macaulay's basis</em>, of <tt>K&lt;X&gt;/I</tt>. For the sake of computing the values of the Hilbert function (see <ref>NCo.HF</ref>) of <tt>K&lt;X&gt;/I</tt>, in this function we require that <tt>G</tt> has to be a Groebner basis with respect to a length compatible word ordering (see <ref>NCo.SetOrdering</ref>).</del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></short_description></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></short_description></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><syntax></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><syntax></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></syntax></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></syntax></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><description></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><description></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Given a two-sided ideal <tt>I</tt> in a free monoid ring <tt>K&lt;X&gt;</tt>, we can consider the <tt>K</tt>-algebra <tt>K&lt;X&gt;/I</tt> as a <tt>K</tt>-vector space. Moreover, let <tt>G</tt> be a Groebner basis of <tt>I</tt>, and let <tt>B</tt> be the set of all words which are not a multiple of any word in the leading word set <tt>LW{G}</tt>. Then the residue class of the words in <tt>B</tt> form a <tt>K</tt>-basis, called a <em>Macaulay's basis</em>, of <tt>K&lt;X&gt;/I</tt>. For the sake of computing the values of the Hilbert function (see <ref>NCo.HF</ref>) of <tt>K&lt;X&gt;/I</tt>, in this function we require that <tt>G</tt> has to be a Groebner basis with respect to a length compatible word ordering (see <ref>NCo.SetOrdering</ref>).</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><par/></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><par/></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><par/></div></td></tr>
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Stadler
http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NCo.MB&diff=13232&oldid=prev
Xiu at 18:11, 30 April 2013
2013-04-30T18:11:52Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:11, 30 April 2013</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4" >Line 4:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Enumerate a Macaulay's basis of a finitely generated <tt>K</tt>-algebra.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Enumerate a Macaulay's basis of a finitely generated <tt>K</tt>-algebra.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><par/></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><par/></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Given a two-sided ideal <tt>I</tt> in a free monoid ring <tt>K&lt;X&gt;</tt>, we can consider the <tt>K</tt>-algebra <tt>K&lt;X&gt;/I</tt> as a <tt>K</tt>-vector space. Moreover, let <tt>G</tt> be a Groebner basis of <tt>I</tt>, and let <tt>B</tt> be the set of all words which are not a multiple of any word in the leading word set <tt>LW{G}</tt>. Then the residue class of the words in <tt>B</tt> form a <tt>K</tt>-basis, called <em>Macaulay's basis</em>, of <tt>K&lt;X&gt;/I</tt>. For the sake of computing the values of the Hilbert function (see <ref>NCo.HF</ref>) of <tt>K&lt;X&gt;/I</tt>, in this function we require that <tt>G</tt> has to be a Groebner basis with respect to a length compatible word ordering (see <ref>NCo.SetOrdering</ref>).</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Given a two-sided ideal <tt>I</tt> in a free monoid ring <tt>K&lt;X&gt;</tt>, we can consider the <tt>K</tt>-algebra <tt>K&lt;X&gt;/I</tt> as a <tt>K</tt>-vector space. Moreover, let <tt>G</tt> be a Groebner basis of <tt>I</tt>, and let <tt>B</tt> be the set of all words which are not a multiple of any word in the leading word set <tt>LW{G}</tt>. Then the residue class of the words in <tt>B</tt> form a <tt>K</tt>-basis, called <ins class="diffchange diffchange-inline">a </ins><em>Macaulay's basis</em>, of <tt>K&lt;X&gt;/I</tt>. For the sake of computing the values of the Hilbert function (see <ref>NCo.HF</ref>) of <tt>K&lt;X&gt;/I</tt>, in this function we require that <tt>G</tt> has to be a Groebner basis with respect to a length compatible word ordering (see <ref>NCo.SetOrdering</ref>).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></short_description></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></short_description></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><syntax></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><syntax></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l16" >Line 16:</td>
<td colspan="2" class="diff-lineno">Line 16:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K&lt;X&gt;</tt> forming a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <em>Warning:</em> users should take responsibility to make sure that Gb is indeed a Groebner basis with respect to a length compatible word ordering! In the case that Gb is a partical Groebner basis, the function enumerates a pseudo basis.</item></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K&lt;X&gt;</tt> forming a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <em>Warning:</em> users should take responsibility to make sure that Gb is indeed a Groebner basis with respect to a length compatible word ordering! In the case that Gb is a partical Groebner basis, the function enumerates a pseudo basis.</item></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><item>@return: a LIST of words forming a Macaulay's basis of the K-algebra <tt>K&lt;X&gt;/&lt;Gb&gt;</tt>.</item></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><item>@return: a LIST of words forming a Macaulay's basis of the <ins class="diffchange diffchange-inline"><tt></ins>K<ins class="diffchange diffchange-inline"></tt></ins>-algebra <tt>K&lt;X&gt;/&lt;Gb&gt;</tt>.</item></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></itemize></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></itemize></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Optional parameter: </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Optional parameter: </div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l38" >Line 38:</td>
<td colspan="2" class="diff-lineno">Line 38:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.HF</see></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.HF</see></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.IsGB</see></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.IsGB</see></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><see>NCo.LW</see></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.SetFp</see></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.SetFp</see></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.SetOrdering</see></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.SetOrdering</see></div></td></tr>
</table>
Xiu
http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NCo.MB&diff=13193&oldid=prev
Xiu at 12:34, 30 April 2013
2013-04-30T12:34:37Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 12:34, 30 April 2013</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l2" >Line 2:</td>
<td colspan="2" class="diff-lineno">Line 2:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><title>NCo.MB</title></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><title>NCo.MB</title></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><short_description></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><short_description></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Enumerate Macaulay basis of a <tt>K</tt>-algebra.</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Enumerate <ins class="diffchange diffchange-inline">a </ins>Macaulay<ins class="diffchange diffchange-inline">'s </ins>basis of a <ins class="diffchange diffchange-inline">finitely generated </ins><tt>K</tt>-algebra<ins class="diffchange diffchange-inline">.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"><par/></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Given a two-sided ideal <tt>I</tt> in a free monoid ring <tt>K&lt;X&gt;</tt>, we can consider the <tt>K</tt>-algebra <tt>K&lt;X&gt;/I</tt> as a <tt>K</tt>-vector space. Moreover, let <tt>G</tt> be a Groebner basis of <tt>I</tt>, and let <tt>B</tt> be the set of all words which are not a multiple of any word in the leading word set <tt>LW{G}</tt>. Then the residue class of the words in <tt>B</tt> form a <tt>K</tt>-basis, called <em>Macaulay's basis</em>, of <tt>K&lt;X&gt;/I</tt>. For the sake of computing the values of the Hilbert function (see <ref>NCo.HF</ref>) of <tt>K&lt;X&gt;/I</tt>, in this function we require that <tt>G</tt> has to be a Groebner basis with respect to a length compatible word ordering (see <ref>NCo.SetOrdering</ref>)</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></short_description></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></short_description></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><syntax></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><syntax></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>NCo.MB(Gb:LIST<del class="diffchange diffchange-inline">):LIST</del></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>NCo.MB(Gb:LIST<ins class="diffchange diffchange-inline">[</ins>, <ins class="diffchange diffchange-inline">DB</ins>:INT<ins class="diffchange diffchange-inline">]</ins>):LIST</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline">NCo.MB(Gb:LIST</del>, <del class="diffchange diffchange-inline">DegreeBound</del>:INT):LIST</div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></syntax></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></syntax></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><description></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><description></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><par/></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><par/></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Please set ring environment <em>coefficient field</em> <tt>K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>NCo.SetFp</ref>, <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before <del class="diffchange diffchange-inline">calling the </del>function. <del class="diffchange diffchange-inline">Default </del>coefficient field is <tt>Q</tt><del class="diffchange diffchange-inline">. Default </del>ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Please set ring environment <em>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>NCo.SetFp</ref>, <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before <ins class="diffchange diffchange-inline">using this </ins>function. <ins class="diffchange diffchange-inline">The default </ins>coefficient field is <tt>Q</tt><ins class="diffchange diffchange-inline">, and the default </ins>ordering is <ins class="diffchange diffchange-inline">the </ins>length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><itemize></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><itemize></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K&lt;X&gt;</tt> <del class="diffchange diffchange-inline">which is </del>a Groebner basis <del class="diffchange diffchange-inline">(w.r.t. </del>a <del class="diffchange diffchange-inline"><tt></del>length compatible<del class="diffchange diffchange-inline"></tt> admissible </del>ordering<del class="diffchange diffchange-inline">, say <tt>Ordering</tt>) of the two-sided ideal generated by Gb</del>. Each polynomial is represented as a LIST of monomials, which are <del class="diffchange diffchange-inline">pairs </del>of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt><del class="diffchange diffchange-inline">F</del>=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <em>Warning:</em> users should take responsibility to make sure that Gb is indeed a Groebner basis <del class="diffchange diffchange-inline">w.r.t. <tt>Ordering</tt></del>! In the case that Gb is a partical Groebner basis, the function enumerates a pseudo basis.</item></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K&lt;X&gt;</tt> <ins class="diffchange diffchange-inline">forming </ins>a Groebner basis <ins class="diffchange diffchange-inline">with respect to </ins>a length compatible <ins class="diffchange diffchange-inline">word </ins>ordering. Each polynomial is represented as a LIST of monomials, which are <ins class="diffchange diffchange-inline">LISTs </ins>of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt><ins class="diffchange diffchange-inline">f</ins>=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <em>Warning:</em> users should take responsibility to make sure that Gb is indeed a Groebner basis <ins class="diffchange diffchange-inline">with respect to a length compatible word ordering</ins>! In the case that Gb is a partical Groebner basis, the function enumerates a pseudo basis.</item<ins class="diffchange diffchange-inline">></ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><item>@param <em><del class="diffchange diffchange-inline">DegreeBound</del>:</em> <del class="diffchange diffchange-inline">(optional) </del>a positive <del class="diffchange diffchange-inline">integer </del>which is a degree bound of <del class="diffchange diffchange-inline">Hilbert funtion</del>. <del class="diffchange diffchange-inline"><em></del>Note that<del class="diffchange diffchange-inline"></em> </del>we set <del class="diffchange diffchange-inline"><tt>DegreeBound</del>=32<del class="diffchange diffchange-inline"></tt> </del>by default. Thus <del class="diffchange diffchange-inline">to compute the whole Macaulay basis</del>, <del class="diffchange diffchange-inline">it is necessary to set <tt>DegreeBound</tt> to a larger enough number.</item></del></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del class="diffchange diffchange-inline"><item>@return: a LIST of terms, which forms </del>the <del class="diffchange diffchange-inline">Macaulay basis of the K-algebra </del><tt>K&lt;X&gt;/<del class="diffchange diffchange-inline">(</del>Gb<del class="diffchange diffchange-inline">)</tt> w.r.t. <tt>Ordering</del></tt>.</item></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"><item>@return: a LIST of words forming a Macaulay's basis of the K-algebra <tt>K&lt;X&gt;/&lt;Gb&gt;</tt>.</item></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"></itemize></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">Optional parameter: </ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"><itemize</ins>></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><item>@param <em><ins class="diffchange diffchange-inline">DB</ins>:</em> a positive <ins class="diffchange diffchange-inline">INT </ins>which is a degree bound of <ins class="diffchange diffchange-inline">the lengths of words</ins>. Note that we set <ins class="diffchange diffchange-inline">DB</ins>=32 by default. Thus, <ins class="diffchange diffchange-inline">in </ins>the <ins class="diffchange diffchange-inline">case that </ins><tt>K&lt;X&gt;/<ins class="diffchange diffchange-inline">&lt;</ins>Gb<ins class="diffchange diffchange-inline">&gt;</ins></tt> <ins class="diffchange diffchange-inline">has a finite Macaulay's basis, it is necessary to set DB to a large enough INT in order to compute the whole Macaulay's basis</ins>.</item></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></itemize></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></itemize></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><example></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><example></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l31" >Line 31:</td>
<td colspan="2" class="diff-lineno">Line 36:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></description></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></description></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><seealso></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><seealso></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><see>NCo.HF</see></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><see>NCo.IsGB</see></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.SetFp</see></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.SetFp</see></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.SetOrdering</see></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><see>NCo.SetOrdering</see></div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l38" >Line 38:</td>
<td colspan="2" class="diff-lineno">Line 45:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><types></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><types></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><type>apcocoaserver</type></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><type>apcocoaserver</type></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><type>ideal</type></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><type>groebner</type></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><type>groebner</type></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"><type>ideal</type></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><type>non_commutative</type></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><type>non_commutative</type></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></types></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></types></div></td></tr>
</table>
Xiu
http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NCo.MB&diff=13105&oldid=prev
Xiu: New page: <command> <title>NCo.MB</title> <short_description> Enumerate Macaulay basis of a <tt>K</tt>-algebra. </short_description> <syntax> NCo.MB(Gb:LIST):LIST NCo.MB(Gb:LIST, DegreeBound:INT):LI...
2013-04-25T17:14:19Z
<p>New page: <command> <title>NCo.MB</title> <short_description> Enumerate Macaulay basis of a <tt>K</tt>-algebra. </short_description> <syntax> NCo.MB(Gb:LIST):LIST NCo.MB(Gb:LIST, DegreeBound:INT):LI...</p>
<p><b>New page</b></p><div><command><br />
<title>NCo.MB</title><br />
<short_description><br />
Enumerate Macaulay basis of a <tt>K</tt>-algebra.<br />
</short_description><br />
<syntax><br />
NCo.MB(Gb:LIST):LIST<br />
NCo.MB(Gb:LIST, DegreeBound:INT):LIST<br />
</syntax><br />
<description><br />
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.<br />
<par/><br />
Please set ring environment <em>coefficient field</em> <tt>K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>NCo.SetFp</ref>, <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before calling the function. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.<br />
<itemize><br />
<item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K&lt;X&gt;</tt> which is a Groebner basis (w.r.t. a <tt>length compatible</tt> admissible ordering, say <tt>Ordering</tt>) of the two-sided ideal generated by Gb. Each polynomial is represented as a LIST of monomials, which are pairs of the form [C, W] where W is a word in <tt>&lt;X&gt;</tt> and C is the coefficient of W. For example, the polynomial <tt>F=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <em>Warning:</em> users should take responsibility to make sure that Gb is indeed a Groebner basis w.r.t. <tt>Ordering</tt>! In the case that Gb is a partical Groebner basis, the function enumerates a pseudo basis.</item><br />
<item>@param <em>DegreeBound:</em> (optional) a positive integer which is a degree bound of Hilbert funtion. <em>Note that</em> we set <tt>DegreeBound=32</tt> by default. Thus to compute the whole Macaulay basis, it is necessary to set <tt>DegreeBound</tt> to a larger enough number.</item><br />
<item>@return: a LIST of terms, which forms the Macaulay basis of the K-algebra <tt>K&lt;X&gt;/(Gb)</tt> w.r.t. <tt>Ordering</tt>.</item><br />
</itemize><br />
<example><br />
NCo.SetX(<quotes>xyzt</quotes>);<br />
NCo.SetOrdering(<quotes>LLEX</quotes>);<br />
Gb:= [[[1, <quotes>yt</quotes>], [-1, <quotes>ty</quotes>]], [[1, <quotes>xt</quotes>], [-1, <quotes>tx</quotes>]], [[1, <quotes>xy</quotes>], [-1, <quotes>ty</quotes>]], [[1, <quotes>xx</quotes>], [-1, <quotes>yx</quotes>]], <br />
[[1, <quotes>tyy</quotes>], [-1, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [-1, <quotes>tyx</quotes>]]];<br />
NCo.MB(Gb,3);<br />
[[<quotes></quotes>], [<quotes>t</quotes>, <quotes>z</quotes>, <quotes>y</quotes>, <quotes>x</quotes>], [<quotes>tt</quotes>, <quotes>tz</quotes>, <quotes>ty</quotes>, <quotes>tx</quotes>, <quotes>zt</quotes>, <quotes>zz</quotes>, <quotes>zy</quotes>, <quotes>zx</quotes>, <quotes>yz</quotes>, <quotes>yy</quotes>, <quotes>yx</quotes>, <quotes>xz</quotes>], <br />
[<quotes>ttt</quotes>, <quotes>ttz</quotes>, <quotes>tty</quotes>, <quotes>ttx</quotes>, <quotes>tzt</quotes>, <quotes>tzz</quotes>, <quotes>tzy</quotes>, <quotes>tzx</quotes>, <quotes>tyz</quotes>, <quotes>tyx</quotes>, <quotes>txz</quotes>, <quotes>ztt</quotes>, <quotes>ztz</quotes>, <quotes>zty</quotes>, <quotes>ztx</quotes>,<br />
<quotes>zzt</quotes>, <quotes>zzz</quotes>, <quotes>zzy</quotes>, <quotes>zzx</quotes>, <quotes>zyz</quotes>, <quotes>zyy</quotes>, <quotes>zyx</quotes>, <quotes>zxz</quotes>, <quotes>yzt</quotes>, <quotes>yzz</quotes>, <quotes>yzy</quotes>, <quotes>yzx</quotes>, <quotes>yyz</quotes>, <quotes>yyy</quotes>,<br />
<quotes>yxz</quotes>, <quotes>xzt</quotes>, <quotes>xzz</quotes>, <quotes>xzy</quotes>, <quotes>xzx</quotes>]]<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>NCo.SetFp</see><br />
<see>NCo.SetOrdering</see><br />
<see>NCo.SetX</see><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>groebner</type><br />
<type>ideal</type><br />
<type>non_commutative</type><br />
</types><br />
<key>gbmr.MB</key><br />
<key>NCo.MB</key><br />
<key>MB</key><br />
<wiki-category>Package_gbmr</wiki-category><br />
</command></div>
Xiu