# Difference between revisions of "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>]]
-------------------------------
```