# ApCoCoA-1:NC.Mul

## NC.Mul

Multiplication of two polynomials in a non-commutative polynomial ring.

### Syntax

```NC.Mul(F1:LIST, F2: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 F1, F2: two non-commutative polynomials, which are left and right operands of multiplication respectively. 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 [].

• @return: a LIST which represents the polynomial equal to F1*F2.

#### Example

```USE ZZ/(31)[x[1..2],y[1..2]];
F1:= [[2x,x], [13y], ]; -- 2xx+13y+5
F2:= [[2y,y], [19y], ]; -- 2yy+19y+2
NC.Mul(F1,F2);

[[4x, x, y, y], [7x, x, y], [4x, x], [-5y, y, y], [10y, y], [-y^2], [-3y], ]
-------------------------------
NC.Mul(F2,F1);

[[4y, y, x, x], [7y, x, x], [4x, x], [-5y, y^2], [10y, y], [-y^2], [-3y], ]
-------------------------------
NC.Mul([],F1);

[ ]
-------------------------------
```