ApCoCoA-1:NC.Add: Difference between revisions

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{{Version|1}}
<command>
<command>
<title>NC.Add</title>
<title>NC.Add</title>
<short_description>
<short_description>
Addition of two polynomials over a free associative <tt>K</tt>-algebra.
Addition of two polynomials in a non-commutative polynomial ring.
</short_description>
</short_description>
<syntax>
<syntax>
Line 10: Line 11:
<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 coefficient field <tt>K</tt>, alphabet (or indeterminates) <tt>X</tt> and ordering through the functions <ref>NC.SetFp</ref>(Prime), <ref>NC.SetX</ref>(X) and <ref>NC.SetOrdering</ref>(Ordering), 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.
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>
<item>@param <em>F1, F2:</em> two polynomials in <tt>K&lt;X&gt;</tt>, which are left and right operands of addition respectively. Each polynomial is represented as a LIST of LISTs, which are pairs of form [C, W] where C is a coefficient and W is a word (or term). Each term is represented as a STRING. For example, <tt>xy^2x</tt> is represented as <quotes>xyyx</quotes>, unit is represented as an empty string <quotes></quotes>. Then, polynomial <tt>F=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <tt>0</tt> polynomial is represented as an empty LIST [].</item>
<item>@param <em>F1, F2:</em> two non-commutative polynomials, which are left and right operands of addition respectively. 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 which represents a polynomial equal to F1+F2.</item>
<item>@return: a LIST which represents the polynomial equal to <tt>F1+F2</tt>.</item>
</itemize>
</itemize>
<example>
<example>
NC.SetX(<quotes>abc</quotes>);
USE ZZ/(31)[x[1..2],y[1..2]];
NC.SetOrdering(<quotes>ELIM</quotes>);  
F1:= [[2x[1],x[2]], [13y[2]], [5]]; -- 2x[1]x[2]+13y[2]+5
NC.RingEnv();
F2:= [[2y[1],y[2]], [19y[2]], [2]]; -- 2y[1]y[2]+19y[2]+2
Coefficient ring : Q
NC.Add(F1,F2);
Alphabet : abc
Ordering : ELIM


-------------------------------
[[2x[1], x[2]], [2y[1], y[2]], [y[2]], [7]]
F1 := [[1,<quotes>a</quotes>],[1,<quotes></quotes>]];
F2 := [[1,<quotes>b</quotes>],[1,<quotes>ba</quotes>]];
NC.Add(F1,F2); -- over Q
[[1, <quotes>ba</quotes>], [1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes></quotes>]]
-------------------------------
NC.SetFp(); -- set default Fp = F2
NC.RingEnv();
Coefficient ring : Fp = Z/(2)
Alphabet : abc
Ordering : ELIM
 
-------------------------------
NC.Add(F1,F2); -- over F2
[[1, <quotes>ba</quotes>], [1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes></quotes>]]
-------------------------------
NC.Add(F1,F1); -- over F2
[ ]
-------------------------------
-------------------------------
</example>
</example>
</description>
</description>
<seealso>
<seealso>
<see>NC.BP</see>
<see>ApCoCoA-1:Use|Use</see>
<see>NC.Deg</see>
<see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see>
<see>NC.FindPolynomials</see>
<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
<see>NC.GB</see>
<see>NC.Intersection</see>
<see>NC.IsGB</see>
<see>NC.KernelOfHomomorphism</see>
<see>NC.LC</see>
<see>NC.LT</see>
<see>NC.LTIdeal</see>
<see>NC.MRAdd</see>
<see>NC.MRBP</see>
<see>NC.MRIntersection</see>
<see>NC.MRKernelOfHomomorphism</see>
<see>NC.MRMinimalPolynomials</see>
<see>NC.MRMultiply</see>
<see>NC.MRReducedBP</see>
<see>NC.MRSubtract</see>
<see>NC.MinimalPolynomial</see>
<see>NC.Multiply</see>
<see>NC.NR</see>
<see>NC.ReducedBP</see>
<see>NC.ReducedGB</see>
<see>NC.SetFp</see>
<see>NC.SetOrdering</see>
<see>NC.SetRelations</see>
<see>NC.SetRules</see>
<see>NC.SetX</see>
<see>NC.Subtract</see>
<see>NC.UnsetFp</see>
<see>NC.UnsetOrdering</see>
<see>NC.UnsetRelations</see>
<see>NC.UnsetRules</see>
<see>NC.UnsetX</see>
<see>Introduction to CoCoAServer</see>
</seealso>
</seealso>
<types>
<types>
<type>apcocoaserver</type>
<type>apcocoaserver</type>
<type>groebner</type>
<type>polynomial</type>
<type>non_commutative</type>
</types>
</types>
<key>gbmr.Add</key>
<key>ncpoly.Add</key>
<key>NC.Add</key>
<key>NC.Add</key>
<key>Add</key>
<key>Add</key>
<wiki-category>Package_gbmr</wiki-category>
<wiki-category>ApCoCoA-1:Package_ncpoly</wiki-category>
</command>
</command>

Latest revision as of 13:33, 29 October 2020

This article is about a function from ApCoCoA-1.

NC.Add

Addition of two polynomials in a non-commutative polynomial ring.

Syntax

NC.Add(F1:LIST, F2: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 F1, F2: two non-commutative polynomials, which are left and right operands of addition respectively. 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 represents the polynomial equal to F1+F2.

Example

USE ZZ/(31)[x[1..2],y[1..2]];
F1:= [[2x[1],x[2]], [13y[2]], [5]]; -- 2x[1]x[2]+13y[2]+5
F2:= [[2y[1],y[2]], [19y[2]], [2]]; -- 2y[1]y[2]+19y[2]+2
NC.Add(F1,F2);

[[2x[1], x[2]], [2y[1], y[2]], [y[2]], [7]]
-------------------------------

See also

Use

NC.SetOrdering

Introduction to CoCoAServer




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