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

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{{Version|1}}
 
<command>
 
<command>
 
<title>NC.Interreduction</title>
 
<title>NC.Interreduction</title>
 
<short_description>
 
<short_description>
−
Interreduce a list (set) of polynomials in a free monoid ring. Note that, given an admissible ordering <tt>Ordering</tt>, a set of non-zero polynomial <tt>G</tt> is called <em>interreduced</em> w.r.t. <tt>Ordering</tt> if no element of <tt>Supp(g)</tt> is contained in <tt>LT(G\{g})</tt> for all <tt>g</tt> in <tt>G</tt>.
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Interreduction of a LIST of polynomials in a non-commutative polynomial ring.  
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
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</syntax>
 
</syntax>
 
<description>
 
<description>
 +
Note that, given a word ordering, a set of non-zero polynomial <tt>G</tt> is called <em>interreduced</em> if, for all <tt>g</tt> in <tt>G</tt>, no element of <tt>Supp(g)</tt> is a multiple of any element in <tt>LW{G\{g}}</tt>.
 +
<par/>
 
<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>ApCoCoA-1:Use|Use</ref>) and word ordering (via the function <ref>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions.
 
<itemize>
 
<itemize>
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<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>
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<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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NC.SetX(<quotes>abc</quotes>);
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USE QQ[x[1..2],y[1..2]];
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NC.SetOrdering(<quotes>ELIM</quotes>);
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NC.SetOrdering("LLEX");
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G:=[[[1,<quotes>ba</quotes>]], [[1,<quotes>b</quotes>],[1,<quotes></quotes>]], [[1,<quotes>c</quotes>]]];
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F1:= [[x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3],[5]]; -- x[1]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5
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NC.Interreduction(G);
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F2:= [[y[1],x[2]^2], [y[2],x[2]^2]]; -- y[1]x[2]^2+y[2]x[2]^2
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F3:= [[x[1],y[1]],[x[2]]]; -- x[1]y[1]+x[2]
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NC.Interreduction([F1,F2,F3]);
  
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[[[1, <quotes>a</quotes>]], [[1, <quotes>b</quotes>], [1, <quotes></quotes>]], [[1, <quotes>c</quotes>]]]
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[[[y[2], x[1]^2, x[2]^3], [1/9x[1], y[2], x[2]^2], [-5/9]], [[y[1], x[2]^2], [y[2], x[2]^2]], [[x[1], y[1]], [x[2]]]]
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
 
</description>
 
</description>
 
<seealso>
 
<seealso>
−
<see>NC.Add</see>
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<see>ApCoCoA-1:Use|Use</see>
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<see>NC.Deg</see>
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<see>ApCoCoA-1:NC.LW|NC.LW</see>
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<see>NC.FindPolynomials</see>
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<see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see>
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<see>NC.GB</see>
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<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
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<see>NC.HF</see>
 
−
<see>NC.Interreduction</see>
 
−
<see>NC.Intersection</see>
 
−
<see>NC.IsFinite</see>
 
−
<see>NC.IsGB</see>
 
−
<see>NC.IsHomog</see>
 
−
<see>NC.KernelOfHomomorphism</see>
 
−
<see>NC.LC</see>
 
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<see>NC.LT</see>
 
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<see>NC.LTIdeal</see>
 
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<see>NC.MB</see>
 
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<see>NC.MinimalPolynomial</see>
 
−
<see>NC.Multiply</see>
 
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<see>NC.NR</see>
 
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<see>NC.ReducedGB</see>
 
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<see>NC.SetFp</see>
 
−
<see>NC.SetOrdering</see>
 
−
<see>NC.SetRelations</see>
 
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<see>NC.SetRules</see>
 
−
<see>NC.SetX</see>
 
−
<see>NC.Subtract</see>
 
−
<see>NC.TruncatedGB</see>
 
−
<see>NC.UnsetFp</see>
 
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<see>NC.UnsetOrdering</see>
 
−
<see>NC.UnsetRelations</see>
 
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<see>NC.UnsetRules</see>
 
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<see>NC.UnsetX</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>
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<type>ideal</type>
 
 
</types>
 
</types>
−
<key>gbmr.Interreduction</key>
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<key>ncpoly.Interreduction</key>
 
<key>NC.Interreduction</key>
 
<key>NC.Interreduction</key>
 
<key>Interreduction</key>
 
<key>Interreduction</key>
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<wiki-category>Package_gbmr</wiki-category>
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<wiki-category>ApCoCoA-1:Package_ncpoly</wiki-category>
 
</command>
 
</command>

Latest revision as of 13:34, 29 October 2020

This article is about a function from ApCoCoA-1.

NC.Interreduction

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

Syntax

NC.Interreduction(G:LIST):LIST

Description

Note that, given a word ordering, a set of non-zero polynomial G is called interreduced if, for all g in G, no element of Supp(g) is a multiple of any element in LW{G\{g}}.

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

USE QQ[x[1..2],y[1..2]];
NC.SetOrdering("LLEX");
F1:= [[x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3],[5]]; -- x[1]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5
F2:= [[y[1],x[2]^2], [y[2],x[2]^2]]; -- y[1]x[2]^2+y[2]x[2]^2
F3:= [[x[1],y[1]],[x[2]]]; -- x[1]y[1]+x[2]
NC.Interreduction([F1,F2,F3]);

[[[y[2], x[1]^2, x[2]^3], [1/9x[1], y[2], x[2]^2], [-5/9]], [[y[1], x[2]^2], [y[2], x[2]^2]], [[x[1], y[1]], [x[2]]]]
-------------------------------

See also

Use

NC.LW

NC.SetOrdering

Introduction to CoCoAServer




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