Difference between revisions of "ApCoCoA-1:NC.NR"
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<title>NC.NR</title> | <title>NC.NR</title> | ||
<short_description> | <short_description> | ||
| − | Normal remainder of polynomial with respect to a | + | Normal remainder of a polynomial with respect to a LIST of polynomials in a non-commutative polynomial ring. |
</short_description> | </short_description> | ||
<syntax> | <syntax> | ||
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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 | + | 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>F</em>: a non-commutative polynomial. 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>@param <em>F</em>: a polynomial | + | <item>@param <em>G</em>: a LIST of non-zero non-commutative polynomials.</item> |
| − | <item>@param <em>G</em>: a LIST of non-zero polynomials | + | <item>@return: a LIST, which is the normal remainder of F with respect to G.</item> |
| − | <item>@return: a LIST which | ||
</itemize> | </itemize> | ||
<example> | <example> | ||
| − | + | USE QQ[x[1..2],y[1..2]]; | |
| − | NC. | + | NC.SetOrdering("LLEX"); |
| − | + | F:= [[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 | |
| − | + | G1:= [[y[1],x[2]^2], [y[2],x[2]^2]]; -- y[1]x[2]^2+y[2]x[2]^2 | |
| − | + | G2:= [[x[1],y[1]],[x[2]]]; -- x[1]y[1]+x[2] | |
| − | + | NC.NR(F,[G1,G2]); | |
| − | F:=[[1 | + | |
| − | + | [[-9y[2], x[1]^2, x[2]^3], [-x[1], y[2], x[2]^2], [5]] | |
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| − | [[1 | ||
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| − | NC.NR(F, | ||
| − | [[ | ||
------------------------------- | ------------------------------- | ||
</example> | </example> | ||
</description> | </description> | ||
<seealso> | <seealso> | ||
| − | <see> | + | <see>ApCoCoA-1:Use|Use</see> |
| − | <see>NC. | + | <see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see> |
| − | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> | |
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| − | <see> | ||
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</seealso> | </seealso> | ||
<types> | <types> | ||
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<type>non_commutative</type> | <type>non_commutative</type> | ||
</types> | </types> | ||
| − | <key> | + | <key>ncpoly.NR</key> |
<key>NC.NR</key> | <key>NC.NR</key> | ||
<key>NR</key> | <key>NR</key> | ||
| − | <wiki-category> | + | <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.NR
Normal remainder of a polynomial with respect to a LIST of polynomials in a non-commutative polynomial ring.
Syntax
NC.NR(F:LIST, 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 F: a non-commutative polynomial. 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 [].
@param G: a LIST of non-zero non-commutative polynomials.
@return: a LIST, which is the normal remainder of F with respect to G.
Example
USE QQ[x[1..2],y[1..2]];
NC.SetOrdering("LLEX");
F:= [[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
G1:= [[y[1],x[2]^2], [y[2],x[2]^2]]; -- y[1]x[2]^2+y[2]x[2]^2
G2:= [[x[1],y[1]],[x[2]]]; -- x[1]y[1]+x[2]
NC.NR(F,[G1,G2]);
[[-9y[2], x[1]^2, x[2]^3], [-x[1], y[2], x[2]^2], [5]]
-------------------------------
See also
