http://apcocoa.uni-passau.de/wiki/api.php?action=feedcontributions&user=132.231.183.16&feedformat=atomApCoCoAWiki - User contributions [en]2024-03-29T05:19:14ZUser contributionsMediaWiki 1.35.0http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NC.SetOrdering&diff=10754ApCoCoA-1:NC.SetOrdering2010-07-14T21:28:58Z<p>132.231.183.16: </p>
<hr />
<div><command><br />
<title>NC.SetOrdering</title><br />
<short_description><br />
Set (admissible) ordering.<br />
</short_description><br />
<syntax><br />
NC.SetOrdering(Ordering)<br />
</syntax><br />
<description><br />
<itemize><br />
<item>@param <em>X</em>: an STRING which is an alphabet (indeterminates) of a free associative K-algebra. Every letter in <tt>X</tt> should occurrence only once. And the order of letters in <tt>X</tt> is important since it will induce an admissible ordering. </item><br />
</itemize><br />
<example><br />
NC.SetX(<quotes>abc</quotes>);<br />
NC.RingEnv();<br />
Coefficient ring : Q (float type in C++)<br />
Alphabet : abc<br />
<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>NC.Add</see><br />
<see>NC.GB</see><br />
<see>NC.IsGB</see><br />
<see>NC.LC</see><br />
<see>NC.LT</see><br />
<see>NC.LTIdeal</see><br />
<see>NC.Multiply</see><br />
<see>NC.NR</see><br />
<see>NC.SetFp</see><br />
<see>NC.SetRelations</see><br />
<see>NC.SetRules</see><br />
<see>NC.SetX</see><br />
<see>NC.Subtract</see><br />
<see>Gbmr.MRSubtract</see><br />
<see>Gbmr.MRMultiply</see><br />
<see>Gbmr.MRBP</see><br />
<see>Gbmr.MRIntersection</see><br />
<see>Gbmr.MRKernelOfHomomorphism</see><br />
<see>Gbmr.MRMinimalPolynomials</see><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>groebner</type><br />
</types><br />
<key>NC.SetOrdering</key><br />
<key>SetOrdering</key><br />
<wiki-category>Package_gbmr</wiki-category><br />
</command></div>132.231.183.16http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NC.SetOrdering&diff=10751ApCoCoA-1:NC.SetOrdering2010-07-14T21:25:18Z<p>132.231.183.16: New page: <command> <title>NC.SetOrdering</title> <short_description> Set (admissible) ordering. </short_description> <syntax> NC.SetOrdering(Ordering) </syntax> <description> <itemize> <item>@param...</p>
<hr />
<div><command><br />
<title>NC.SetOrdering</title><br />
<short_description><br />
Set (admissible) ordering.<br />
</short_description><br />
<syntax><br />
NC.SetOrdering(Ordering)<br />
</syntax><br />
<description><br />
<itemize><br />
<item>@param <em>X</em>: an STRING which is an alphabet (indeterminates) of a free associative K-algebra. Every letter in <tt>X</tt> should occurrence only once. And the order of letters in <tt>X</tt> is important since it will induce an admissible ordering. </item><br />
</itemize><br />
<example><br />
NC.SetX(<quotes>abc</quotes>);<br />
NC.RingEnv();<br />
Coefficient ring : Q (float type in C++)<br />
Alphabet : abc<br />
<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>NC.Add</see><br />
<see>NC.GB</see><br />
<see>NC.IsGB</see><br />
<see>NC.LC</see><br />
<see>NC.LT</see><br />
<see>NC.LTIdeal</see><br />
<see>NC.Multiply</see><br />
<see>NC.NR</see><br />
<see>NC.SetFp</see><br />
<see>NC.SetX</see><br />
<see>NC.Subtract</see><br />
<see>Gbmr.MRSubtract</see><br />
<see>Gbmr.MRMultiply</see><br />
<see>Gbmr.MRBP</see><br />
<see>Gbmr.MRIntersection</see><br />
<see>Gbmr.MRKernelOfHomomorphism</see><br />
<see>Gbmr.MRMinimalPolynomials</see><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>groebner</type><br />
</types><br />
<key>NC.SetOrdering</key><br />
<key>SetOrdering</key><br />
<wiki-category>Package_gbmr</wiki-category><br />
</command></div>132.231.183.16http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NC.NR&diff=10747ApCoCoA-1:NC.NR2010-07-14T20:57:59Z<p>132.231.183.16: </p>
<hr />
<div><command><br />
<title>NC.NR</title><br />
<short_description><br />
Normal remainder polynomial with respect to a list of polynomials over a free associative K-algebra.<br />
</short_description><br />
<syntax><br />
NC.NR(F:LIST, Polynomials:LIST):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 />
<itemize><br />
<item>Before calling the function, please set ring environment coefficient field (<tt>K</tt>), alphabet (<tt>X</tt>) and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(Ordering) respectively. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</item><br />
<item>@param <em>F</em>: a polynomial in <tt>K&lt;X&gt;</tt>. Each polynomial in <tt>K&lt;X&gt;</tt> is represented as a LIST of LISTs, which are pairs of form <tt>[c, w]</tt> where <tt>c</tt> is in <tt>K</tt> and <tt>w</tt> is a word in <tt>X*</tt>. Unit in <tt>X*</tt> is empty word represented as an empty STRING <quotes></quotes>. 0 polynomial is represented as an empty LIST <tt>[]</tt>. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as <tt>F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]</tt>.</item><br />
<item>@param <em>Polynomials</em>: a LIST of polynomials in <tt>K&lt;X&gt;</tt>.</item><br />
<item>@return: a STRING which represents normal remainder of <tt>F</tt> with respect to <tt>Polynomials</tt>.</item><br />
</itemize><br />
<example><br />
NC.RingEnv();<br />
Coefficient ring : Q (float type in C++)<br />
Alphabet : abc<br />
Ordering : LLEX<br />
<br />
-------------------------------<br />
F:=[[1,<quotes>ab</quotes>],[1,<quotes>aca</quotes>],[1,<quotes>bb</quotes>],[1,<quotes>bab</quotes>],[1,<quotes></quotes>]];<br />
F1 := [[1,<quotes>a</quotes>],[1,<quotes>c</quotes>]]; <br />
F2 := [[1,<quotes>b</quotes>],[1,<quotes>ba</quotes>]];<br />
Polynomials:=[F1,F2];<br />
NC.NR(F,Polynomials);<br />
[[1, <quotes>bcb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes></quotes>]]<br />
-------------------------------<br />
NC.SetOrdering(<quotes>ELIM</quotes>);<br />
NC.NR(F,Polynomials);<br />
[[1, <quotes>bcb</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes></quotes>]]<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>NC.Add</see><br />
<see>NC.GB</see><br />
<see>NC.IsGB</see><br />
<see>NC.LC</see><br />
<see>NC.LT</see><br />
<see>NC.LTIdeal</see><br />
<see>NC.Multiply</see><br />
<see>NC.Subtract</see><br />
<see>Gbmr.MRSubtract</see><br />
<see>Gbmr.MRMultiply</see><br />
<see>Gbmr.MRBP</see><br />
<see>Gbmr.MRIntersection</see><br />
<see>Gbmr.MRKernelOfHomomorphism</see><br />
<see>Gbmr.MRMinimalPolynomials</see><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>groebner</type><br />
</types><br />
<key>NC.NR</key><br />
<key>NR</key><br />
<wiki-category>Package_gbmr</wiki-category><br />
</command></div>132.231.183.16http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NC.LC&diff=10744ApCoCoA-1:NC.LC2010-07-14T20:56:23Z<p>132.231.183.16: </p>
<hr />
<div><command><br />
<title>NC.LC</title><br />
<short_description><br />
Leading coefficient of polynomial over a free associative K-algebra.<br />
</short_description><br />
<syntax><br />
NC.LT(F:LIST):K<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 />
<itemize><br />
<item>Before calling the function, please set ring environment coefficient field (<tt>K</tt>), alphabet (<tt>X</tt>) and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(Ordering) respectively. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</item><br />
<item>@param <em>F</em>: a polynomial in <tt>K&lt;X&gt;</tt>. Each polynomial in <tt>K&lt;X&gt;</tt> is represented as a LIST of LISTs, which are pairs of form <tt>[c, w]</tt> where <tt>c</tt> is in <tt>K</tt> and <tt>w</tt> is a word in <tt>X*</tt>. Unit in <tt>X*</tt> is empty word represented as an empty STRING <quotes></quotes>. 0 polynomial is represented as an empty LIST <tt>[]</tt>. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as <tt>F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]</tt>.</item><br />
<item>@return: an element of K which is leading term of <tt>F</tt> with respect to current ordering. If <tt>F=0</tt>, then return <tt>0</tt>. </item><br />
</itemize><br />
<example><br />
NC.SetX(<quotes>abc</quotes>);<br />
NC.SetOrdering(<quotes>ELIM</quotes>);<br />
NC.RingEnv();<br />
Coefficient ring : Q (float type in C++)<br />
Alphabet : abc<br />
Ordering : ELIM<br />
<br />
-------------------------------<br />
F:=[[1,<quotes>ab</quotes>],[2,<quotes>aa</quotes>],[3,<quotes>bb</quotes>],[4,<quotes>bab</quotes>]]; <br />
NC.LC(F); -- ELIM ordering<br />
2<br />
-------------------------------<br />
NC.SetOrdering(<quotes>LLEX</quotes>); <br />
NC.LC(F); -- LLEX ordering<br />
4<br />
-------------------------------<br />
NC.LC([]);<br />
0<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>NC.Add</see><br />
<see>NC.GB</see><br />
<see>NC.IsGB</see><br />
<see>NC.LT</see><br />
<see>NC.LTIdeal</see><br />
<see>NC.Multiply</see><br />
<see>NC.NR</see><br />
<see>NC.Subtract</see><br />
<see>Gbmr.MRSubtract</see><br />
<see>Gbmr.MRMultiply</see><br />
<see>Gbmr.MRBP</see><br />
<see>Gbmr.MRIntersection</see><br />
<see>Gbmr.MRKernelOfHomomorphism</see><br />
<see>Gbmr.MRMinimalPolynomials</see><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>groebner</type><br />
</types><br />
<key>NC.LC</key><br />
<key>LC</key><br />
<wiki-category>Package_gbmr</wiki-category><br />
</command></div>132.231.183.16http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NC.IsGB&diff=10743ApCoCoA-1:NC.IsGB2010-07-14T20:55:55Z<p>132.231.183.16: </p>
<hr />
<div><command><br />
<title>NC.IsGB</title><br />
<short_description><br />
Check if a list of polynomials if Groebner basis.<br />
</short_description><br />
<syntax><br />
NC.IsGB(Polynomials:LIST):BOOL<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 />
<itemize><br />
<item>Before calling the function, please set ring environment coefficient field (<tt>K</tt>), alphabet (<tt>X</tt>) and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(Ordering) respectively. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</item><br />
<item>@param <em>Polynomials</em>: a LIST of polynomials in <tt>K&lt;X&gt;</tt>. Each polynomial in <tt>K&lt;X&gt;</tt> is represented as a LIST of LISTs, which are pairs of form <tt>[c, w]</tt> where <tt>c</tt> is in <tt>K</tt> and <tt>w</tt> is a word in <tt>X*</tt>. Unit in <tt>X*</tt> is empty word represented as an empty STRING <quotes></quotes>. 0 polynomial is represented as an empty LIST <tt>[]</tt>. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as <tt>F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]</tt>.</item><br />
<item>@return: a BOOL value. True if <tt>Polynomials</tt> is a GB; False, otherwise.</item><br />
</itemize><br />
<example><br />
NC.SetX(<quotes>xyt</quotes>); <br />
NC.SetOrdering(<quotes>LLEX</quotes>); <br />
F1 := [[1,<quotes>xx</quotes>], [-1,<quotes>yx</quotes>]]; <br />
F2 := [[1,<quotes>xy</quotes>], [-1,<quotes>ty</quotes>]]; <br />
F3 := [[1,<quotes>xt</quotes>], [-1, <quotes>tx</quotes>]]; <br />
F4 := [[1,<quotes>yt</quotes>], [-1, <quotes>ty</quotes>]]; <br />
Polynomials := [F1, F2,F3,F4]; <br />
NC.IsGB(Polynomials);<br />
False<br />
-------------------------------<br />
NC.SetOrdering(<quotes>ELIM</quotes>);<br />
NC.IsGB(Polynomials);<br />
False<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>NC.Add</see><br />
<see>NC.GB</see><br />
<see>NC.LC</see><br />
<see>NC.LT</see><br />
<see>NC.LTIdeal</see><br />
<see>NC.Multiply</see><br />
<see>NC.NR</see><br />
<see>NC.Subtract</see><br />
<see>Gbmr.MRSubtract</see><br />
<see>Gbmr.MRMultiply</see><br />
<see>Gbmr.MRBP</see><br />
<see>Gbmr.MRIntersection</see><br />
<see>Gbmr.MRKernelOfHomomorphism</see><br />
<see>Gbmr.MRMinimalPolynomials</see><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>groebner</type><br />
</types><br />
<key>NC.IsGB</key><br />
<key>IsGB</key><br />
<wiki-category>Package_gbmr</wiki-category><br />
</command></div>132.231.183.16http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NC.GB&diff=10742ApCoCoA-1:NC.GB2010-07-14T20:55:30Z<p>132.231.183.16: </p>
<hr />
<div><command><br />
<title>NC.GB</title><br />
<short_description><br />
Compute two-sided (partial) Groebner basis of finitely generated ideal by Buchberger's procedure.<br />
</short_description><br />
<syntax><br />
NC.GB(Polynomials:LIST[, DegreeBound:INT, LoopBound:INT, Flag: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 />
<itemize><br />
<item>Before calling the function, please set ring environment coefficient field (<tt>K</tt>), alphabet (<tt>X</tt>) and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(Ordering) respectively. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</item><br />
<item>@param <em>Polynomials</em>: a LIST of polynomials generating a two-sided ideal in <tt>K&lt;X&gt;</tt>. Each polynomial in <tt>K&lt;X&gt;</tt> is represented as a LIST of LISTs, which are pairs of form <tt>[c, w]</tt> where <tt>c</tt> is in <tt>K</tt> and <tt>w</tt> is a word in <tt>X*</tt>. Unit in <tt>X*</tt> is empty word represented as an empty STRING <quotes></quotes>. 0 polynomial is represented as an empty LIST <tt>[]</tt>. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as <tt>F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]</tt>.</item><br />
<item>@return: a LIST of polynomials, which is a Groebner basis if a finite Groebner basis exists or a partial Groebner basis.</item><br />
</itemize><br />
About the optional parameters.<br />
<itemize><br />
<item>For most of cases we don't know whether there exists a finite Groebner basis. In stead of forcing computer yelling and informing nothing valuable, the function has 3 optional parameters to control the computation. Note that at the moment all of the following 3 additional optional parameters must be used at the same time.</item><br />
<item>@param <em>DegreeBound:</em> (optional) a INT (natural number) which gives a limitation on the degree of polynomials during Buchberger's procedure. When the degree of normal remainder of some <tt>S-element</tt> reaches <tt>DegreeBound</tt>, the function stops and returns a partial Groebner basis.</item><br />
<item>@param <em>LoopBound:</em> (optional) a INT (natural number) which gives a a limitation on the main loop of Buchberger's procedure. When the main loop runs <tt>LoopBound</tt> times, the function stops and returns a partial Groebner basis.</item><br />
<item>@param <em>Flag:</em> (optional) a INT (natural number) which is a multi-switch for the output of ApCoCoAServer. If <tt>Flag=0</tt>, the server prints nothing on the screen. If <tt>Flag=1</tt>, the server prints basic information about computing procedure, such as number of S-elements has been checked and to be checked. If <tt>Flag=2</tt>, the server prints partial Groebner basis before each loop as well. Note that the initial idea of <tt>Flag</tt> is to use it for debugging and tracing the computing process.</item><br />
</itemize><br />
<example><br />
NC.SetX(<quotes>xyzt</quotes>); <br />
NC.SetOrdering(<quotes>LLEX</quotes>); <br />
F1 := [[1,<quotes>xx</quotes>], [-1,<quotes>yx</quotes>]]; <br />
F2 := [[1,<quotes>xy</quotes>], [-1,<quotes>ty</quotes>]]; <br />
F3 := [[1,<quotes>xt</quotes>], [-1, <quotes>tx</quotes>]]; <br />
F4 := [[1,<quotes>yt</quotes>], [-1, <quotes>ty</quotes>]]; <br />
Generators := [F1, F2,F3,F4]; -- computation over Q<br />
NC.GB(Generators);<br />
[[[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 />
-------------------------------<br />
NC.SetFp();<br />
NC.GB(Generators); -- computation over F2<br />
[[[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 />
-------------------------------<br />
NC.SetFp(3);<br />
NC.GB(Generators); -- computation over F3<br />
[[[1, <quotes>yt</quotes>], [2, <quotes>ty</quotes>]], [[1, <quotes>xt</quotes>], [2, <quotes>tx</quotes>]], [[1, <quotes>xy</quotes>], [2, <quotes>ty</quotes>]], [[1, <quotes>xx</quotes>], [2, <quotes>yx</quotes>]], <br />
[[1, <quotes>tyy</quotes>], [2, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [2, <quotes>tyx</quotes>]]]<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>NC.Add</see><br />
<see>NC.IsGB</see><br />
<see>NC.LC</see><br />
<see>NC.LT</see><br />
<see>NC.LTIdeal</see><br />
<see>NC.Multiply</see><br />
<see>NC.NR</see><br />
<see>NC.Subtract</see><br />
<see>Gbmr.MRSubtract</see><br />
<see>Gbmr.MRMultiply</see><br />
<see>Gbmr.MRBP</see><br />
<see>Gbmr.MRIntersection</see><br />
<see>Gbmr.MRKernelOfHomomorphism</see><br />
<see>Gbmr.MRMinimalPolynomials</see><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>groebner</type><br />
</types><br />
<key>NC.GB</key><br />
<key>GB</key><br />
<wiki-category>Package_gbmr</wiki-category><br />
</command></div>132.231.183.16http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NC.Add&diff=10741ApCoCoA-1:NC.Add2010-07-14T20:54:24Z<p>132.231.183.16: </p>
<hr />
<div><command><br />
<title>NC.Add</title><br />
<short_description><br />
Addition of two polynomials over a free associative K-algebra.<br />
</short_description><br />
<syntax><br />
NC.Add(F1:LIST, F2:LIST):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 />
<itemize><br />
<item>Before calling the function, please set ring environment coefficient field (<tt>K</tt>), alphabet (<tt>X</tt>) and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(Ordering) respectively. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</item><br />
<item>@param <em>F1</em>: left operand of addition operator. It is a polynomial in <tt>K&lt;X&gt;</tt>. Each polynomial in <tt>K&lt;X&gt;</tt> is represented as a LIST of LISTs, which are pairs of form <tt>[c, w]</tt> where <tt>c</tt> is in <tt>K</tt> and <tt>w</tt> is a word in <tt>X*</tt>. Unit in <tt>X*</tt> is empty word represented as an empty STRING <quotes></quotes>. 0 polynomial is represented as an empty LIST <tt>[]</tt>. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as <tt>F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]</tt>.</item><br />
<item>@param <em>F2</em>: right operand of addition operator. It is a polynomial in <tt>K&lt;X&gt;</tt>.</item><br />
<item>@return: a LIST which represents a polynomial equal to <tt>F1+F2</tt>.</item><br />
</itemize><br />
<example><br />
NC.SetX(<quotes>abc</quotes>); <br />
NC.SetOrdering(<quotes>ELIM</quotes>); <br />
F1 := [[1,<quotes>a</quotes>],[1,<quotes></quotes>]]; <br />
F2 := [[1,<quotes>b</quotes>],[1,<quotes>ba</quotes>]]; <br />
NC.Add(F1,F2); -- computation over Q<br />
[[1, <quotes>ba</quotes>], [1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes></quotes>]]<br />
-------------------------------<br />
NC.RingEnv();<br />
Coefficient ring : Q (float type in C++)<br />
Alphabet : abc<br />
Ordering : ELIM<br />
<br />
-------------------------------<br />
NC.SetFp(); -- default Fp = F2<br />
NC.RingEnv();<br />
Coefficient ring : Fp = Z/(2)<br />
Alphabet : abc<br />
Ordering : ELIM<br />
<br />
-------------------------------<br />
NC.Add(F1,F2); -- computation over F2<br />
[[1, <quotes>ba</quotes>], [1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes></quotes>]]<br />
-------------------------------<br />
NC.Add(F1,F1); -- computation over F2<br />
[ ]<br />
-------------------------------<br />
NC.UnsetFp(); -- set the coefficient field back to Q<br />
NC.RingEnv();<br />
Coefficient ring : Q (float type in C++)<br />
Alphabet : abc<br />
Ordering : ELIM<br />
<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>NC.GB</see><br />
<see>NC.IsGB</see><br />
<see>NC.LC</see><br />
<see>NC.LT</see><br />
<see>NC.LTIdeal</see><br />
<see>NC.Multiply</see><br />
<see>NC.NR</see><br />
<see>NC.Subtract</see><br />
<see>Gbmr.MRSubtract</see><br />
<see>Gbmr.MRMultiply</see><br />
<see>Gbmr.MRBP</see><br />
<see>Gbmr.MRIntersection</see><br />
<see>Gbmr.MRKernelOfHomomorphism</see><br />
<see>Gbmr.MRMinimalPolynomials</see><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>groebner</type><br />
</types><br />
<key>NC.Add</key><br />
<key>Add</key><br />
<wiki-category>Package_gbmr</wiki-category><br />
</command></div>132.231.183.16http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NC.NR&diff=10736ApCoCoA-1:NC.NR2010-07-14T20:34:56Z<p>132.231.183.16: </p>
<hr />
<div><command><br />
<title>NC.NR</title><br />
<short_description><br />
Normal remainder polynomial with respect to a list of polynomials over a free associative K-algebra.<br />
</short_description><br />
<syntax><br />
NC.NR(F:LIST, Polynomials:LIST):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 />
<itemize><br />
<item>Before calling the function, please set ring environment coefficient field (<tt>K</tt>), alphabet (<tt>X</tt>) and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(Ordering) respectively. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</item><br />
<item>@param <em>F</em>: a polynomial in <tt>K&lt;X&gt;</tt>. Each polynomial in <tt>K&lt;X&gt;</tt> is represented as a LIST of LISTs, which are pairs of form <tt>[c, w]</tt> where <tt>c</tt> is in <tt>K</tt> and <tt>w</tt> is a word in <tt>X*</tt>. Unit in <tt>X*</tt> is empty word represented as an empty STRING <quotes></quotes>. 0 polynomial is represented as an empty LIST <tt>[]</tt>. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as <tt>F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]</tt>.</item><br />
<item>@param <em>Polynomials</em>: a LIST of polynomials in <tt>K&lt;X&gt;</tt>.</item><br />
<item>@return: a STRING which represents normal remainder of <tt>F</tt> with respect to <tt>Polynomials</tt>.</item><br />
</itemize><br />
<example><br />
NC.RingEnv();<br />
Coefficient ring : Q (float type in C++)<br />
Alphabet : abc<br />
Ordering : LLEX<br />
<br />
-------------------------------<br />
F:=[[1,<quotes>ab</quotes>],[1,<quotes>aca</quotes>],[1,<quotes>bb</quotes>],[1,<quotes>bab</quotes>],[1,<quotes></quotes>]];<br />
F1 := [[1,<quotes>a</quotes>],[1,<quotes>c</quotes>]]; <br />
F2 := [[1,<quotes>b</quotes>],[1,<quotes>ba</quotes>]];<br />
Polynomials:=[F1,F2];<br />
NC.NR(F,Polynomials);<br />
[[1, <quotes>bcb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes></quotes>]]<br />
-------------------------------<br />
NC.SetOrdering(<quotes>ELIM</quotes>);<br />
NC.NR(F,Polynomials);<br />
[[1, <quotes>bcb</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes></quotes>]]<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>NC.Add</see><br />
<see>NC.GB</see><br />
<see>NC.IsGB</see><br />
<see>NC.LC</see><br />
<see>NC.LT</see><br />
<see>NC.Multiply</see><br />
<see>NC.Subtract</see><br />
<see>Gbmr.MRSubtract</see><br />
<see>Gbmr.MRMultiply</see><br />
<see>Gbmr.MRBP</see><br />
<see>Gbmr.MRIntersection</see><br />
<see>Gbmr.MRKernelOfHomomorphism</see><br />
<see>Gbmr.MRMinimalPolynomials</see><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>groebner</type><br />
</types><br />
<key>NC.NR</key><br />
<key>NR</key><br />
<wiki-category>Package_gbmr</wiki-category><br />
</command></div>132.231.183.16http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NC.IsGB&diff=10734ApCoCoA-1:NC.IsGB2010-07-14T20:33:44Z<p>132.231.183.16: </p>
<hr />
<div><command><br />
<title>NC.IsGB</title><br />
<short_description><br />
Check if a list of polynomials if Groebner basis.<br />
</short_description><br />
<syntax><br />
NC.IsGB(Polynomials:LIST):BOOL<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 />
<itemize><br />
<item>Before calling the function, please set ring environment coefficient field (<tt>K</tt>), alphabet (<tt>X</tt>) and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(Ordering) respectively. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</item><br />
<item>@param <em>Polynomials</em>: a LIST of polynomials in <tt>K&lt;X&gt;</tt>. Each polynomial in <tt>K&lt;X&gt;</tt> is represented as a LIST of LISTs, which are pairs of form <tt>[c, w]</tt> where <tt>c</tt> is in <tt>K</tt> and <tt>w</tt> is a word in <tt>X*</tt>. Unit in <tt>X*</tt> is empty word represented as an empty STRING <quotes></quotes>. 0 polynomial is represented as an empty LIST <tt>[]</tt>. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as <tt>F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]</tt>.</item><br />
<item>@return: a BOOL value. True if <tt>Polynomials</tt> is a GB; False, otherwise.</item><br />
</itemize><br />
<example><br />
NC.SetX(<quotes>xyt</quotes>); <br />
NC.SetOrdering(<quotes>LLEX</quotes>); <br />
F1 := [[1,<quotes>xx</quotes>], [-1,<quotes>yx</quotes>]]; <br />
F2 := [[1,<quotes>xy</quotes>], [-1,<quotes>ty</quotes>]]; <br />
F3 := [[1,<quotes>xt</quotes>], [-1, <quotes>tx</quotes>]]; <br />
F4 := [[1,<quotes>yt</quotes>], [-1, <quotes>ty</quotes>]]; <br />
Polynomials := [F1, F2,F3,F4]; <br />
NC.IsGB(Polynomials);<br />
False<br />
-------------------------------<br />
NC.SetOrdering(<quotes>ELIM</quotes>);<br />
NC.IsGB(Polynomials);<br />
False<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>NC.Add</see><br />
<see>NC.GB</see><br />
<see>NC.LC</see><br />
<see>NC.LT</see><br />
<see>NC.Multiply</see><br />
<see>NC.NR</see><br />
<see>NC.Subtract</see><br />
<see>Gbmr.MRSubtract</see><br />
<see>Gbmr.MRMultiply</see><br />
<see>Gbmr.MRBP</see><br />
<see>Gbmr.MRIntersection</see><br />
<see>Gbmr.MRKernelOfHomomorphism</see><br />
<see>Gbmr.MRMinimalPolynomials</see><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>groebner</type><br />
</types><br />
<key>NC.IsGB</key><br />
<key>IsGB</key><br />
<wiki-category>Package_gbmr</wiki-category><br />
</command></div>132.231.183.16http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NC.GB&diff=10733ApCoCoA-1:NC.GB2010-07-14T20:33:00Z<p>132.231.183.16: </p>
<hr />
<div><command><br />
<title>NC.GB</title><br />
<short_description><br />
Compute two-sided (partial) Groebner basis of finitely generated ideal by Buchberger's procedure.<br />
</short_description><br />
<syntax><br />
NC.GB(Polynomials:LIST[, DegreeBound:INT, LoopBound:INT, Flag: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 />
<itemize><br />
<item>Before calling the function, please set ring environment coefficient field (<tt>K</tt>), alphabet (<tt>X</tt>) and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(Ordering) respectively. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</item><br />
<item>@param <em>Polynomials</em>: a LIST of polynomials generating a two-sided ideal in <tt>K&lt;X&gt;</tt>. Each polynomial in <tt>K&lt;X&gt;</tt> is represented as a LIST of LISTs, which are pairs of form <tt>[c, w]</tt> where <tt>c</tt> is in <tt>K</tt> and <tt>w</tt> is a word in <tt>X*</tt>. Unit in <tt>X*</tt> is empty word represented as an empty STRING <quotes></quotes>. 0 polynomial is represented as an empty LIST <tt>[]</tt>. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as <tt>F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]</tt>.</item><br />
<item>@return: a LIST of polynomials, which is a Groebner basis if a finite Groebner basis exists or a partial Groebner basis.</item><br />
</itemize><br />
About the optional parameters.<br />
<itemize><br />
<item>For most of cases we don't know whether there exists a finite Groebner basis. In stead of forcing computer yelling and informing nothing valuable, the function has 3 optional parameters to control the computation. Note that at the moment all of the following 3 additional optional parameters must be used at the same time.</item><br />
<item>@param <em>DegreeBound:</em> (optional) a INT (natural number) which gives a limitation on the degree of polynomials. When the degree of normal remainder of some S-element reaches <tt>DegreeBound</tt>, the function stops and returns a partial Groebner basis.</item><br />
<item>@param <em>LoopBound:</em> (optional) a INT (natural number) which gives a a limitation on the main loop of Buchberger's procedure. When the main loop runs <tt>LoopBound</tt> times, the function stops and returns a partial Groebner basis.</item><br />
<item>@param <em>Flag:</em> (optional) a INT (natural number) which is a multi-switch for the output of ApCoCoAServer. If <tt>Flag=0</tt>, the server prints nothing on the screen. If <tt>Flag=1</tt>, the server prints basic information about computing procedure, such as number of S-elements has been checked and to be checked. If <tt>Flag=2</tt>, the server prints partial Groebner basis before each loop as well. Note that the initial idea of <tt>Flag</tt> is to use it for debugging and tracing the computing process.</item><br />
</itemize><br />
<example><br />
NC.SetX(<quotes>xyzt</quotes>); <br />
NC.SetOrdering(<quotes>LLEX</quotes>); <br />
F1 := [[1,<quotes>xx</quotes>], [-1,<quotes>yx</quotes>]]; <br />
F2 := [[1,<quotes>xy</quotes>], [-1,<quotes>ty</quotes>]]; <br />
F3 := [[1,<quotes>xt</quotes>], [-1, <quotes>tx</quotes>]]; <br />
F4 := [[1,<quotes>yt</quotes>], [-1, <quotes>ty</quotes>]]; <br />
Generators := [F1, F2,F3,F4]; -- computation over Q<br />
NC.GB(Generators);<br />
[[[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 />
-------------------------------<br />
NC.SetFp();<br />
NC.GB(Generators); -- computation over F2<br />
[[[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 />
-------------------------------<br />
NC.SetFp(3);<br />
NC.GB(Generators); -- computation over F3<br />
[[[1, <quotes>yt</quotes>], [2, <quotes>ty</quotes>]], [[1, <quotes>xt</quotes>], [2, <quotes>tx</quotes>]], [[1, <quotes>xy</quotes>], [2, <quotes>ty</quotes>]], [[1, <quotes>xx</quotes>], [2, <quotes>yx</quotes>]], <br />
[[1, <quotes>tyy</quotes>], [2, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [2, <quotes>tyx</quotes>]]]<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>NC.Add</see><br />
<see>NC.IsGB</see><br />
<see>NC.LC</see><br />
<see>NC.LT</see><br />
<see>NC.Multiply</see><br />
<see>NC.NR</see><br />
<see>NC.Subtract</see><br />
<see>Gbmr.MRSubtract</see><br />
<see>Gbmr.MRMultiply</see><br />
<see>Gbmr.MRBP</see><br />
<see>Gbmr.MRIntersection</see><br />
<see>Gbmr.MRKernelOfHomomorphism</see><br />
<see>Gbmr.MRMinimalPolynomials</see><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>groebner</type><br />
</types><br />
<key>NC.GB</key><br />
<key>GB</key><br />
<wiki-category>Package_gbmr</wiki-category><br />
</command></div>132.231.183.16http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NC.Add&diff=10732ApCoCoA-1:NC.Add2010-07-14T20:32:08Z<p>132.231.183.16: </p>
<hr />
<div><command><br />
<title>NC.Add</title><br />
<short_description><br />
Addition of two polynomials over a free associative K-algebra.<br />
</short_description><br />
<syntax><br />
NC.Add(F1:LIST, F2:LIST):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 />
<itemize><br />
<item>Before calling the function, please set ring environment coefficient field (<tt>K</tt>), alphabet (<tt>X</tt>) and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(Ordering) respectively. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</item><br />
<item>@param <em>F1</em>: left operand of addition operator. It is a polynomial in <tt>K&lt;X&gt;</tt>. Each polynomial in <tt>K&lt;X&gt;</tt> is represented as a LIST of LISTs, which are pairs of form <tt>[c, w]</tt> where <tt>c</tt> is in <tt>K</tt> and <tt>w</tt> is a word in <tt>X*</tt>. Unit in <tt>X*</tt> is empty word represented as an empty STRING <quotes></quotes>. 0 polynomial is represented as an empty LIST <tt>[]</tt>. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as <tt>F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]</tt>.</item><br />
<item>@param <em>F2</em>: right operand of addition operator. It is a polynomial in <tt>K&lt;X&gt;</tt>.</item><br />
<item>@return: a LIST which represents a polynomial equal to <tt>F1+F2</tt>.</item><br />
</itemize><br />
<example><br />
NC.SetX(<quotes>abc</quotes>); <br />
NC.SetOrdering(<quotes>ELIM</quotes>); <br />
F1 := [[1,<quotes>a</quotes>],[1,<quotes></quotes>]]; <br />
F2 := [[1,<quotes>b</quotes>],[1,<quotes>ba</quotes>]]; <br />
NC.Add(F1,F2); -- computation over Q<br />
[[1, <quotes>ba</quotes>], [1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes></quotes>]]<br />
-------------------------------<br />
NC.RingEnv();<br />
Coefficient ring : Q (float type in C++)<br />
Alphabet : abc<br />
Ordering : ELIM<br />
<br />
-------------------------------<br />
NC.SetFp(); -- default Fp = F2<br />
NC.RingEnv();<br />
Coefficient ring : Fp = Z/(2)<br />
Alphabet : abc<br />
Ordering : ELIM<br />
<br />
-------------------------------<br />
NC.Add(F1,F2); -- computation over F2<br />
[[1, <quotes>ba</quotes>], [1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes></quotes>]]<br />
-------------------------------<br />
NC.Add(F1,F1); -- computation over F2<br />
[ ]<br />
-------------------------------<br />
NC.UnsetFp(); -- set the coefficient field back to Q<br />
NC.RingEnv();<br />
Coefficient ring : Q (float type in C++)<br />
Alphabet : abc<br />
Ordering : ELIM<br />
<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>NC.GB</see><br />
<see>NC.IsGB</see><br />
<see>NC.LC</see><br />
<see>NC.LT</see><br />
<see>NC.Multiply</see><br />
<see>NC.NR</see><br />
<see>NC.Subtract</see><br />
<see>Gbmr.MRSubtract</see><br />
<see>Gbmr.MRMultiply</see><br />
<see>Gbmr.MRBP</see><br />
<see>Gbmr.MRIntersection</see><br />
<see>Gbmr.MRKernelOfHomomorphism</see><br />
<see>Gbmr.MRMinimalPolynomials</see><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>groebner</type><br />
</types><br />
<key>NC.Add</key><br />
<key>Add</key><br />
<wiki-category>Package_gbmr</wiki-category><br />
</command></div>132.231.183.16http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NC.LC&diff=10729ApCoCoA-1:NC.LC2010-07-14T20:23:41Z<p>132.231.183.16: </p>
<hr />
<div><command><br />
<title>NC.LC</title><br />
<short_description><br />
Leading coefficient of polynomial over a free associative K-algebra.<br />
</short_description><br />
<syntax><br />
NC.LT(F:LIST):K<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 />
<itemize><br />
<item>Before calling the function, please set ring environment coefficient field (<tt>K</tt>), alphabet (<tt>X</tt>) and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(Ordering) respectively. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</item><br />
<item>@param <em>F</em>: a polynomial in <tt>K&lt;X&gt;</tt>. Each polynomial in <tt>K&lt;X&gt;</tt> is represented as a LIST of LISTs, which are pairs of form <tt>[c, w]</tt> where <tt>c</tt> is in <tt>K</tt> and <tt>w</tt> is a word in <tt>X*</tt>. Unit in <tt>X*</tt> is empty word represented as an empty STRING <quotes></quotes>. 0 polynomial is represented as an empty LIST <tt>[]</tt>. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as <tt>F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]</tt>.</item><br />
<item>@return: an element of K which is leading term of <tt>F</tt> with respect to current ordering. If <tt>F=0</tt>, then return <tt>0</tt>. </item><br />
</itemize><br />
<example><br />
NC.SetX(<quotes>abc</quotes>);<br />
NC.SetOrdering(<quotes>ELIM</quotes>);<br />
NC.RingEnv();<br />
Coefficient ring : Q (float type in C++)<br />
Alphabet : abc<br />
Ordering : ELIM<br />
<br />
-------------------------------<br />
F:=[[1,<quotes>ab</quotes>],[2,<quotes>aa</quotes>],[3,<quotes>bb</quotes>],[4,<quotes>bab</quotes>]]; <br />
NC.LC(F); -- ELIM ordering<br />
2<br />
-------------------------------<br />
NC.SetOrdering(<quotes>LLEX</quotes>); <br />
NC.LC(F); -- LLEX ordering<br />
4<br />
-------------------------------<br />
NC.LC([]);<br />
0<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>NC.Add</see><br />
<see>NC.GB</see><br />
<see>NC.IsGB</see><br />
<see>NC.LT</see><br />
<see>NC.Multiply</see><br />
<see>NC.NR</see><br />
<see>NC.Subtract</see><br />
<see>Gbmr.MRSubtract</see><br />
<see>Gbmr.MRMultiply</see><br />
<see>Gbmr.MRBP</see><br />
<see>Gbmr.MRIntersection</see><br />
<see>Gbmr.MRKernelOfHomomorphism</see><br />
<see>Gbmr.MRMinimalPolynomials</see><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>groebner</type><br />
</types><br />
<key>NC.LC</key><br />
<key>LC</key><br />
<wiki-category>Package_gbmr</wiki-category><br />
</command></div>132.231.183.16http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NC.LC&diff=10728ApCoCoA-1:NC.LC2010-07-14T20:23:05Z<p>132.231.183.16: </p>
<hr />
<div><command><br />
<title>NC.LC</title><br />
<short_description><br />
Leading coefficient of polynomial over a free associative K-algebra.<br />
</short_description><br />
<syntax><br />
NC.LT(F:LIST):K<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 />
<itemize><br />
<item>Before calling the function, please set ring environment coefficient field (<tt>K</tt>), alphabet (<tt>X</tt>) and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(Ordering) respectively. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</item><br />
<item>@param <em>F</em>: a polynomial in <tt>K&lt;X&gt;</tt>. Each polynomial in <tt>K&lt;X&gt;</tt> is represented as a LIST of LISTs, which are pairs of form <tt>[c, w]</tt> where <tt>c</tt> is in <tt>K</tt> and <tt>w</tt> is a word in <tt>X*</tt>. Unit in <tt>X*</tt> is empty word represented as an empty STRING <quotes></quotes>. 0 polynomial is represented as an empty LIST <tt>[]</tt>. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as <tt>F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]</tt>.</item><br />
<item>@return: an element of K which is leading term of <tt>F</tt> with respect to current ordering. If <tt>F=0</tt>, then return <tt>0</tt>. </item><br />
</itemize><br />
<example><br />
NC.SetAlphabet(<quotes>abc</quotes>);<br />
NC.SetOrdering(<quotes>ELIM</quotes>);<br />
NC.RingEnv();<br />
ERROR: Unknown operator $apcocoa/gbmr.SetAlphabet<br />
CONTEXT: $apcocoa/gbmr.SetAlphabet(<quotes>abc</quotes>)<br />
-------------------------------<br />
Coefficient ring : Q (float type in C++)<br />
<br />
-------------------------------<br />
NC.SetX(<quotes>abc</quotes>);<br />
NC.SetOrdering(<quotes>ELIM</quotes>);<br />
NC.RingEnv();<br />
Coefficient ring : Q (float type in C++)<br />
Alphabet : abc<br />
Ordering : ELIM<br />
<br />
-------------------------------<br />
F:=[[1,<quotes>ab</quotes>],[2,<quotes>aa</quotes>],[3,<quotes>bb</quotes>],[4,<quotes>bab</quotes>]]; <br />
NC.LC(F); -- ELIM ordering<br />
2<br />
-------------------------------<br />
NC.SetOrdering(<quotes>LLEX</quotes>); <br />
NC.LC(F); -- LLEX ordering<br />
4<br />
-------------------------------<br />
NC.LC([]);<br />
0<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>NC.Add</see><br />
<see>NC.GB</see><br />
<see>NC.IsGB</see><br />
<see>NC.LT</see><br />
<see>NC.Multiply</see><br />
<see>NC.NR</see><br />
<see>NC.Subtract</see><br />
<see>Gbmr.MRSubtract</see><br />
<see>Gbmr.MRMultiply</see><br />
<see>Gbmr.MRBP</see><br />
<see>Gbmr.MRIntersection</see><br />
<see>Gbmr.MRKernelOfHomomorphism</see><br />
<see>Gbmr.MRMinimalPolynomials</see><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>groebner</type><br />
</types><br />
<key>NC.LC</key><br />
<key>LC</key><br />
<wiki-category>Package_gbmr</wiki-category><br />
</command></div>132.231.183.16http://apcocoa.uni-passau.de/wiki/index.php?title=ApCoCoA-1:NC.LC&diff=10727ApCoCoA-1:NC.LC2010-07-14T20:18:16Z<p>132.231.183.16: New page: <command> <title>NC.LC</title> <short_description> Leading coefficient of polynomial over a free associative K-algebra. </short_description> <syntax> NC.LT(F:LIST):K </syntax> <description...</p>
<hr />
<div><command><br />
<title>NC.LC</title><br />
<short_description><br />
Leading coefficient of polynomial over a free associative K-algebra.<br />
</short_description><br />
<syntax><br />
NC.LT(F:LIST):K<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 />
<itemize><br />
<item>Before calling the function, please set ring environment coefficient field (<tt>K</tt>), alphabet (<tt>X</tt>) and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(Ordering) respectively. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.</item><br />
<item>@param <em>F</em>: a polynomial in <tt>K&lt;X&gt;</tt>. Each polynomial in <tt>K&lt;X&gt;</tt> is represented as a LIST of LISTs, which are pairs of form <tt>[c, w]</tt> where <tt>c</tt> is in <tt>K</tt> and <tt>w</tt> is a word in <tt>X*</tt>. Unit in <tt>X*</tt> is empty word represented as an empty STRING <quotes></quotes>. 0 polynomial is represented as an empty LIST <tt>[]</tt>. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as <tt>F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]</tt>.</item><br />
<item>@return: an element of K which is leading term of <tt>F</tt> with respect to current ordering. If <tt>F=0</tt>, then return <tt>0</tt>. </item><br />
</itemize><br />
<example><br />
NC.UnsetFp();<br />
NC.RingEnv();<br />
Coefficient ring : Q (float type in C++)<br />
Alphabet : abc<br />
Ordering : ELIM<br />
<br />
-------------------------------<br />
F:=[[1,<quotes>ab</quotes>],[1,<quotes>aa</quotes>],[1,<quotes>bb</quotes>],[1,<quotes>bab</quotes>]];<br />
NC.LT(F); -- ELIM ordering<br />
aa<br />
-------------------------------<br />
NC.SetOrdering(<quotes>LLEX</quotes>);<br />
NC.LT(F); -- LLEX ordering<br />
bab<br />
-------------------------------<br />
</example><br />
</description><br />
<seealso><br />
<see>NC.Add</see><br />
<see>NC.GB</see><br />
<see>NC.IsGB</see><br />
<see>NC.LT</see><br />
<see>NC.Multiply</see><br />
<see>NC.NR</see><br />
<see>NC.Subtract</see><br />
<see>Gbmr.MRSubtract</see><br />
<see>Gbmr.MRMultiply</see><br />
<see>Gbmr.MRBP</see><br />
<see>Gbmr.MRIntersection</see><br />
<see>Gbmr.MRKernelOfHomomorphism</see><br />
<see>Gbmr.MRMinimalPolynomials</see><br />
<see>Introduction to CoCoAServer</see><br />
</seealso><br />
<types><br />
<type>apcocoaserver</type><br />
<type>groebner</type><br />
</types><br />
<key>NC.LC</key><br />
<key>LC</key><br />
<wiki-category>Package_gbmr</wiki-category><br />
</command></div>132.231.183.16