# 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=2xyx^2-9yx^2x^3+5 is represented as F:=[[2x,y,x^2], [-9y,x^2,x^3], ]. 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,y,x^2], [-9y,x^2,x^3],]; -- xyx^2-9yx^2x^3+5
G1:= [[y,x^2], [y,x^2]]; -- yx^2+yx^2
G2:= [[x,y],[x]]; -- xy+x
NC.NR(F,[G1,G2]);

[[-9y, x^2, x^3], [-x, y, x^2], ]
-------------------------------
```