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

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(New page: <command> <title>NC.Multiply</title> <short_description> Multiplication of two polynomials in a non-commutative polynomial ring. </short_description> <syntax> NC.Mul(F1:LIST, F2:LIST):LIST...)
 
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{{Version|1}}
 
<command>
 
<command>
<title>NC.Multiply</title>
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<title>NC.Mul</title>
 
<short_description>
 
<short_description>
 
Multiplication of two polynomials in a non-commutative polynomial ring.
 
Multiplication of two polynomials in a non-commutative polynomial ring.
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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>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 multiplication respectively. 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>F1, F2:</em> two non-commutative polynomials, which are left and right operands of multiplication 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 the polynomial equal to <tt>F1*F2</tt>.</item>
 
<item>@return: a LIST which represents the polynomial equal to <tt>F1*F2</tt>.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>
NC.SetFp(3);
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USE ZZ/(31)[x[1..2],y[1..2]];
NC.SetX(<quotes>abc</quotes>);
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F1:= [[2x[1],x[2]], [13y[2]], [5]]; -- 2x[1]x[2]+13y[2]+5
NC.RingEnv();
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F2:= [[2y[1],y[2]], [19y[2]], [2]]; -- 2y[1]y[2]+19y[2]+2
Coefficient ring : Fp = Z/(3)
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NC.Mul(F1,F2);
Alphabet : abc
+
 
Ordering : LLEX
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[[4x[1], x[2], y[1], y[2]], [7x[1], x[2], y[2]], [4x[1], x[2]], [-5y[2], y[1], y[2]], [10y[1], y[2]], [-y[2]^2], [-3y[2]], [10]]
 
-------------------------------
 
-------------------------------
F1 := [[2,<quotes>a</quotes>],[1,<quotes></quotes>]];
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NC.Mul(F2,F1);
F2 := [[2,<quotes>b</quotes>],[1,<quotes>ba</quotes>]];
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NC.Multiply(F1,F2); -- over F3
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[[4y[1], y[2], x[1], x[2]], [7y[2], x[1], x[2]], [4x[1], x[2]], [-5y[1], y[2]^2], [10y[1], y[2]], [-y[2]^2], [-3y[2]], [10]]
[[2, <quotes>aba</quotes>], [1, <quotes>ab</quotes>], [1, <quotes>ba</quotes>], [2, <quotes>b</quotes>]]
 
 
-------------------------------
 
-------------------------------
NC.Multiply(F2,F1);
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NC.Mul([],F1);
[[2, <quotes>baa</quotes>], [2, <quotes>ba</quotes>], [2, <quotes>b</quotes>]]
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-------------------------------
 
NC.Multiply(F1,[]);
 
[ ]
 
-------------------------------
 
NC.Multiply([],F1);
 
[ ]
 
-------------------------------
 
NC.Multiply([],[]);
 
 
[ ]
 
[ ]
-------------------------------
 
NC.UnsetFp();
 
NC.RingEnv();
 
Coefficient ring : Q
 
Alphabet : abc
 
Ordering : LLEX
 
-------------------------------
 
NC.Multiply(F1,F2); -- over Q
 
[[2, <quotes>aba</quotes>], [4, <quotes>ab</quotes>], [1, <quotes>ba</quotes>], [2, <quotes>b</quotes>]]
 
-------------------------------
 
NC.Multiply(F2,F1);
 
[[2, <quotes>baa</quotes>], [5, <quotes>ba</quotes>], [2, <quotes>b</quotes>]]
 
 
-------------------------------
 
-------------------------------
 
</example>
 
</example>
 
</description>
 
</description>
 
<seealso>
 
<seealso>
<see>Introduction to CoCoAServer</see>
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<see>ApCoCoA-1:Use|Use</see>
 +
<see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see>
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<see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
 
</seealso>
 
</seealso>
 
<types>
 
<types>
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<key>NC.Mul</key>
 
<key>NC.Mul</key>
 
<key>Mul</key>
 
<key>Mul</key>
<wiki-category>Package_ncpoly</wiki-category>
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<wiki-category>ApCoCoA-1:Package_ncpoly</wiki-category>
 
</command>
 
</command>

Latest revision as of 13:35, 29 October 2020

This article is about a function from ApCoCoA-1.

NC.Mul

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

Syntax

NC.Mul(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 multiplication 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.Mul(F1,F2);

[[4x[1], x[2], y[1], y[2]], [7x[1], x[2], y[2]], [4x[1], x[2]], [-5y[2], y[1], y[2]], [10y[1], y[2]], [-y[2]^2], [-3y[2]], [10]]
-------------------------------
NC.Mul(F2,F1);

[[4y[1], y[2], x[1], x[2]], [7y[2], x[1], x[2]], [4x[1], x[2]], [-5y[1], y[2]^2], [10y[1], y[2]], [-y[2]^2], [-3y[2]], [10]]
-------------------------------
NC.Mul([],F1);

[ ]
-------------------------------

See also

Use

NC.SetOrdering

Introduction to CoCoAServer