# 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, 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 indeterminates) `X` and ordering through the functions NC.SetFp(Prime), NC.SetX(X) and NC.SetOrdering(Ordering), respectively, before calling the function. 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 is represented as a LIST of LISTs, which are pairs of form [C, W] where C is a coefficient and W is a word (or term). Each term is represented as a STRING. For example,`xy^2x`is represented as "xyyx", unit is represented as an empty string "". Then, polynomial`F=xy-y+1`is represented as F:=[[1,"xy"], [-1, "y"], [1,""]].`0`polynomial is represented as an empty LIST [].@param

*G*: a LIST of polynomials in`K<X>`.@return: a LIST which represents a normal remainder of F with respect to G.

#### 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>]]; G:=[F1,F2]; NC.NR(F,G); [[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,G); [[1, <quotes>bcb</quotes>], [-1, <quotes>bb</quotes>], [1, <quotes>cb</quotes>], [-1, <quotes>ccc</quotes>], [-1, <quotes></quotes>]] -------------------------------

### See also