# ApCoCoA-1:NC.NR

## NC.NR

Normal remainder polynomial with respect to a list of polynomials over a free associative `K`-algebra.

### Syntax

NC.NR(F:LIST, Polynomials: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.

Before calling the function, please set ring environment coefficient field `K`, alphabet `X` and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(Ordering) respectively. Default coefficient field is `Q`. Default ordering is length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

@param

*F*: a polynomial in`K<X>`. Each polynomial in`K<X>`is represented as a LIST of LISTs, which are pairs of form [c, w] where c is in`K`and w is a word in`X*`. Unit in`X*`is empty word represented as an empty STRING "".`0`polynomial is represented as an empty LIST []. For example, polynomial`F:=xy-y+1`in`K<x,y>`is represented as`F:=[[1,"xy"], [-1, "y"], [1,""]]`.@param

*Polynomials*: a LIST of polynomials in`K<X>`.@return: a LIST which represents normal remainder of

`F`with respect to`Polynomials`.

#### Example

NC.SetX(<quotes>abc</quotes>); NC.RingEnv(); Coefficient ring : Q Alphabet : abc Ordering : LLEX ------------------------------- F:=[[1,<quotes>ab</quotes>],[1,<quotes>aca</quotes>],[1,<quotes>bb</quotes>],[1,<quotes>bab</quotes>],[1,<quotes></quotes>]]; F1 := [[1,<quotes>a</quotes>],[1,<quotes>c</quotes>]]; F2 := [[1,<quotes>b</quotes>],[1,<quotes>ba</quotes>]]; Polynomials:=[F1,F2]; NC.NR(F,Polynomials); [[1, <quotes>bcb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes></quotes>]] ------------------------------- NC.SetOrdering(<quotes>ELIM</quotes>); NC.NR(F,Polynomials); [[1, <quotes>bcb</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes></quotes>]] -------------------------------

### See also