# ApCoCoA-1:NCo.NR

This article is about a function from ApCoCoA-1. |

## 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,""]] -------------------------------

### See also