# ApCoCoA-1:NCo.Multiply

 This article is about a function from ApCoCoA-1.

## NCo.Multiply

Multiplication of two polynomials in a free monoid ring.

### Syntax

```NCo.Multiply(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 ring environment coefficient field K, alphabet (or set of indeterminates) X and ordering via the functions NCo.SetFp, NCo.SetX and NCo.SetOrdering, respectively, before using this function. The default coefficient field is Q, and the default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

• @param F1, F2: two polynomials in K<X>, which are left and right operands of multiplication respectively. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <X> and C is the coefficient of W. For example, the polynomial f=xy-y+1 is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. The zero polynomial 0 is represented as the empty LIST [].

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

#### Example

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