# ApCoCoA-1:NC.LW

## NC.LW

The leading word (or term) of a non-zero polynomial in a non-commutative polynomial ring.

### Syntax

```NC.LW(F:LIST):LIST
NC.LT(F: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-zero 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 [].

• @return: a LIST, which is the leading word of F with respect to the current word ordering.

#### Example

```USE QQ[x[1..2]];
F:= [[x[1]^2], [2x[1],x[2]], [3x[2],x[1]],[4x[2]^2]]; -- x[1]^2+2x[1]x[2]+3x[2]x[1]+4x[2]^2
NC.SetOrdering(<quotes>LLEX</quotes>);
NC.LW(F);

[x[1]^2]
-------------------------------
-- Done.
-------------------------------
NC.SetOrdering(<quotes>LRLEX</quotes>);
NC.LW(F);

[x[2]^2]
-------------------------------
-- Done.
-------------------------------
NC.SetOrdering(<quotes>ELIM</quotes>);
NC.LW(F);

[x[1]^2]
-------------------------------
-- Done.
-------------------------------
NC.SetOrdering(<quotes>DEGRLEX</quotes>);
NC.LT(F);

[x[1]^2]
-------------------------------
```