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

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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 K-algebra.
+
Addition of two polynomials in a non-commutative polynomial ring.
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
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<description>
 
<description>
 
<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/>
 +
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>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>
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<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>
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<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 [c, w] where c is in <tt>K</tt> and w is a word in <tt>X*</tt>.  Unit in <tt>X*</tt> is empty word represented as an empty STRING <quotes></quotes>. <tt>0</tt> polynomial is represented as an empty LIST []. For example, polynomial <tt>F:=xy-y+1</tt> in <tt>K&lt;x,y&gt;</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]].</item>
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<item>@return: a LIST which represents the polynomial equal to <tt>F1+F2</tt>.</item>
−
<item>@param <em>F2</em>: right operand of addition operator. It is a polynomial in <tt>K&lt;X&gt;</tt>.</item>
 
−
<item>@return: a LIST which represents polynomial <tt>F1+F2</tt>.</item>
 
 
</itemize>
 
</itemize>
 
<example>
 
<example>
−
NC.SetX(<quotes>abc</quotes>);
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USE ZZ/(31)[x[1..2],y[1..2]];
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NC.SetOrdering(<quotes>ELIM</quotes>);
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F1:= [[2x[1],x[2]], [13y[2]], [5]]; -- 2x[1]x[2]+13y[2]+5
−
F1 := [[1,<quotes>a</quotes>],[1,<quotes></quotes>]];
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F2:= [[2y[1],y[2]], [19y[2]], [2]]; -- 2y[1]y[2]+19y[2]+2
−
F2 := [[1,<quotes>b</quotes>],[1,<quotes>ba</quotes>]];
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NC.Add(F1,F2);
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NC.Add(F1,F2); -- computation over Q
 
−
[[1, <quotes>ba</quotes>], [1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes></quotes>]]
 
−
-------------------------------
 
−
NC.RingEnv();
 
−
Coefficient ring : Q (float type in C++)
 
−
Alphabet : abc
 
−
Ordering : ELIM
 
−
 
 
−
-------------------------------
 
−
NC.SetFp(); -- default Fp = F2
 
−
NC.RingEnv();
 
−
Coefficient ring : Fp = Z/(2)
 
−
Alphabet : abc
 
−
Ordering : ELIM
 
−
 
 
−
-------------------------------
 
−
NC.Add(F1,F2); -- computation over F2
 
−
[[1, <quotes>ba</quotes>], [1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes></quotes>]]
 
−
-------------------------------
 
−
NC.Add(F1,F1); -- computation over F2
 
−
[ ]
 
−
-------------------------------
 
−
NC.UnsetFp(); -- set the coefficient field back to Q
 
−
NC.RingEnv();
 
−
Coefficient ring : Q (float type in C++)
 
−
Alphabet : abc
 
−
Ordering : ELIM
 
  
 +
[[2x[1], x[2]], [2y[1], y[2]], [y[2]], [7]]
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
 
</description>
 
</description>
 
<seealso>
 
<seealso>
−
<see>NC.GB</see>
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<see>ApCoCoA-1:Use|Use</see>
−
<see>NC.IsGB</see>
+
<see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see>
−
<see>NC.LC</see>
+
<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
−
<see>NC.LT</see>
 
−
<see>NC.LTIdeal</see>
 
−
<see>NC.Multiply</see>
 
−
<see>NC.NR</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>Gbmr.MRSubtract</see>
 
−
<see>Gbmr.MRMultiply</see>
 
−
<see>Gbmr.MRBP</see>
 
−
<see>Gbmr.MRIntersection</see>
 
−
<see>Gbmr.MRKernelOfHomomorphism</see>
 
−
<see>Gbmr.MRMinimalPolynomials</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>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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