# ApCoCoA-1:NC.Interreduction

## NC.Interreduction

Interreduction of a LIST of polynomials in a non-commutative polynomial ring.

### Syntax

```NC.Interreduction(G:LIST):LIST
```

### Description

Note that, given a word ordering, a set of non-zero polynomial G is called interreduced if, for all g in G, no element of Supp(g) is a multiple of any element in LW{G\{g}}.

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 non-commutative polynomial ring (via the command Use) and word ordering (via the function NC.SetOrdering) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions.

• @param G: a LIST of non-commutative polynomials. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial f=2xyx^2-9yx^2x^3+5 is represented as F:=[[2x,y,x^2], [-9y,x^2,x^3], ]. The zero polynomial 0 is represented as the empty LIST [].

• @return: a LIST, which is an interreduced set of G.

#### Example

```USE QQ[x[1..2],y[1..2]];
NC.SetOrdering(<quotes>LLEX</quotes>);
F1:= [[x,y,x^2], [-9y,x^2,x^3],]; -- xyx^2-9yx^2x^3+5
F2:= [[y,x^2], [y,x^2]]; -- yx^2+yx^2
F3:= [[x,y],[x]]; -- xy+x
NC.Interreduction([F1,F2,F3]);

[[[y, x^2, x^3], [1/9x, y, x^2], [-5/9]], [[y, x^2], [y, x^2]], [[x, y], [x]]]
-------------------------------
```