# ApCoCoA-1:NC.NR

## 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