# ApCoCoA-1:NCo.NR

## NCo.NR

The normal remainder of a polynomial with respect to a LIST of polynomials in a free monoid ring.

### Syntax

```NCo.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 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 F: a polynomial in K<X>. 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 [].

• @param G: a LIST of non-zero polynomials in K<X>.

• @return: a LIST which represents the normal remainder of F with respect to G.

#### Example

```NCo.SetX("abc");
NCo.RingEnv();
Coefficient ring : Q
Alphabet : abc
Ordering : LLEX
-------------------------------
F:=[[1,"ab"],[1,"aca"],[1,"bb"],[1,"bab"],[1,""]];
F1 := [[1,"a"],[1,"c"]];
F2 := [[1,"b"],[1,"ba"]];
G:=[F1,F2];
NCo.NR(F,G);
[[-1,"bcb"], [1,"ccc"], [1,"bb"], [-1,"cb"], [1,""]]
-------------------------------
NCo.SetOrdering("ELIM");
NCo.NR(F,G);
[[-1,"bcb"], [1,"bb"], [-1,"cb"], [1,"ccc"], [1,""]]
-------------------------------
```