# 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

```NC.SetFp(3);
NC.SetX(<quotes>abc</quotes>);
NC.RingEnv();
Coefficient ring : Fp = Z/(3)
Alphabet : abc
Ordering : LLEX
-------------------------------
F1 := [[2,<quotes>a</quotes>],[1,<quotes></quotes>]];
F2 := [[2,<quotes>b</quotes>],[1,<quotes>ba</quotes>]];
NC.Multiply(F1,F2); -- over F3
[[2, <quotes>aba</quotes>], [1, <quotes>ab</quotes>], [1, <quotes>ba</quotes>], [2, <quotes>b</quotes>]]
-------------------------------
NC.Multiply(F2,F1);
[[2, <quotes>baa</quotes>], [2, <quotes>ba</quotes>], [2, <quotes>b</quotes>]]
-------------------------------
NC.Multiply(F1,[]);
[ ]
-------------------------------
NC.Multiply([],F1);
[ ]
-------------------------------
NC.Multiply([],[]);
[ ]
-------------------------------
NC.UnsetFp();
NC.RingEnv();
Coefficient ring : Q
Alphabet : abc
Ordering : LLEX
-------------------------------
NC.Multiply(F1,F2); -- over Q
[[2, <quotes>aba</quotes>], [4, <quotes>ab</quotes>], [1, <quotes>ba</quotes>], [2, <quotes>b</quotes>]]
-------------------------------
NC.Multiply(F2,F1);
[[2, <quotes>baa</quotes>], [5, <quotes>ba</quotes>], [2, <quotes>b</quotes>]]
-------------------------------
```