Difference between revisions of "ApCoCoA-1:NC.Interreduction"

From ApCoCoAWiki
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<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.
 
<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.
 
<par/>
 
<par/>
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>NC.SetFp</ref>, <ref>NC.SetX</ref> and <ref>NC.SetOrdering</ref>, respectively, before calling the function. The default coefficient field is <tt>Q</tt>. The default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions.
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Please set non-commutative polynomial ring (via the command <ref>Use</ref>) and word ordering (via the function <ref>NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant commands and functions.
 
<itemize>
 
<itemize>
<item>@param <em>G</em>: a LIST of polynomials in <tt>K&lt;X&gt;</tt>. 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>]]. The zero polynomial <tt>0</tt> is represented as the empty LIST [].</item>
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<item>@param <em>G</em>: a LIST of non-commutative polynomials. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial <tt>f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5</tt> is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial <tt>0</tt> is represented as the empty LIST [].</item>
<item>@return: a LIST of interreduced polynomials.</item>
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<item>@return: a LIST, which is an interreduced set of G.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>
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</description>
 
</description>
 
<seealso>
 
<seealso>
<see>NC.SetFp</see>
+
<see>Use</see>
 
<see>NC.SetOrdering</see>
 
<see>NC.SetOrdering</see>
<see>NC.SetX</see>
 
 
<see>Introduction to CoCoAServer</see>
 
<see>Introduction to CoCoAServer</see>
 
</seealso>
 
</seealso>
 
<types>
 
<types>
 
<type>apcocoaserver</type>
 
<type>apcocoaserver</type>
<type>groebner</type>
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<type>polynomial</type>
 
<type>non_commutative</type>
 
<type>non_commutative</type>
<type>ideal</type>
 
 
</types>
 
</types>
 
<key>ncpoly.Interreduction</key>
 
<key>ncpoly.Interreduction</key>

Revision as of 12:29, 26 April 2013

NC.Interreduction

Interreduction of a LIST of polynomials in a non-commutative polynomial ring.

Note that, given an admissible ordering Ordering, a set of non-zero polynomial G is called interreduced w.r.t. Ordering if no element of Supp(g) is contained in LT(G\{g}) for all g in G.

Syntax

NC.Interreduction(G:LIST):LIST

Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

Please set non-commutative polynomial ring (via the command Use) and word ordering (via the function NC.SetOrdering) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions.

  • @param G: a LIST of non-commutative polynomials. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5 is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial 0 is represented as the empty LIST [].

  • @return: a LIST, which is an interreduced set of G.

Example

NC.SetX(<quotes>abc</quotes>);
NC.SetOrdering(<quotes>ELIM</quotes>);
G:=[[[1,<quotes>ba</quotes>]], [[1,<quotes>b</quotes>],[1,<quotes></quotes>]], [[1,<quotes>c</quotes>]]];
NC.Interreduction(G);

[[[1, <quotes>a</quotes>]], [[1, <quotes>b</quotes>], [1, <quotes></quotes>]], [[1, <quotes>c</quotes>]]]
-------------------------------

See also

Use

NC.SetOrdering

Introduction to CoCoAServer