Difference between revisions of "ApCoCoA-1:NC.NR"
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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 non-commutative polynomial ring (via the command <ref>Use</ref>) and word ordering (via the function <ref>NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant commands and 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 (<quotes>LLEX</quotes>). 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 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> | ||
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</description> | </description> | ||
<seealso> | <seealso> | ||
| − | <see>Use</see> | + | <see>ApCoCoA-1:Use|Use</see> |
| − | <see>NC.SetOrdering</see> | + | <see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see> |
| − | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |
</seealso> | </seealso> | ||
<types> | <types> | ||
Revision as of 08:24, 7 October 2020
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(<quotes>LLEX</quotes>); 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
